Model reference double boundary projection adaptive control method for input-constrained

Abstract

The present invention discloses a model reference double - boundary projection adaptive control method for input - constrained situations, which includes the following steps: obtaining the input characteristics of an engine and designing a reference model; designing a standard adaptive update law; upgrading the standard adaptive update law to a projection adaptive update law; and establishing a control parameter projection domain by using the input limit. The present invention realizes the adaptive update of the controller parameters in the case of input constraints through a double - boundary projection operator to solve the problem of parameter drift of traditional model reference adaptive controllers in the case of input constraints. In this method, the approach of establishing a parameter projection domain by using the input limit can be applied to the design of model reference projection adaptive controllers in various input - constrained situations and has general applicability to different power mechanical systems.

Description

Technical Field

[0001] The present invention relates to a model reference double - boundary projection adaptive control method for input - constrained, which belongs to the field of aero - engine control. Background Art

[0002] Aero - engines are a class of complex strongly - nonlinear control objects, and it is difficult to design an engine control system with linear control methods. Traditional control methods for aero - engines are designed based on linear models, such as proportional - integral control methods based on transfer - function models and multivariable control methods based on state - space models. However, in actual operation, the system parameters of the object not only change with its own state but also are related to the external environment (such as altitude, Mach number, etc.), so its system parameters are uncertain, resulting in poor control effects of methods based on linear models.

[0003] Currently, model reference adaptive control is a relatively mature method to solve system - parameter uncertainty. It updates control parameters through an adaptive law so that the response of the closed - loop system meets the expected performance requirements, and it can be designed only with relevant input characteristics without detailed object information, eliminating the process of system identification or modeling of the object. However, when the input is restricted, the adaptive mechanism in model reference adaptive control still continuously updates the parameters to make the output closer to the reference model. At this time, the updated parameters do not affect the response and may even cause parameter drift, and the result may lead to system instability. The model reference double - boundary projection adaptive control method for input - constrained, while retaining the good adaptive ability of model reference adaptive control, realizes the upper and lower - bound limitation of control parameters when the input is restricted through a projection operator, thus ensuring the stability of the system. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to overcome the problem that the control parameters drift in traditional model reference adaptive control when the input is restricted, resulting in poor control effects or even system instability. On the basis of standard model reference adaptive control, the adaptive law is upgraded to a projection adaptive law, and the projection domain of the projection operator adopts a double - boundary scheme, so that the limitation of the projection operator on the control parameters is upgraded from only limiting the maximum value to upper and lower - bound limitations, meeting the requirements of engine input limitations. The limitation of the control parameters ensures that the control quantity calculated by the controller does not exceed the input limitation, thus realizing model reference adaptive control of aero - engines under input - constrained conditions.

[0005] To achieve the above object, a model reference double - boundary projection adaptive control method for input - constrained includes the following steps:

[0006] Step 1: Obtain the input characteristics of the engine and design a reference model;

[0007] Step 2: Design a standard adaptive update law;

[0008] Step 3: Upgrade the standard adaptive update law to a projection adaptive update law;

[0009] Step 4: Establish a control parameter projection domain using the input limit quantity.

[0010] Furthermore, the specific steps in Step 1 are as follows:

[0011] Step 1-1: The aero-engine can be expressed in the form of Equation (1)

[0012]

[0013] where x p (t) is the state of the controlled object, u(t) is the control input, a p and b p are unknown parameters with uncertainties. b p characterizes the input characteristics of the system. By performing a step on the input and analyzing the response curve, the sign information of b p can be obtained.

[0014] Step 1-2: Construct a reference signal according to the preset performance requirements, which is generated by the reference model (2)

[0015]

[0016] where x m (t) is the state of the reference model, r(t) is a bounded and piecewise continuous reference command, a m < 0, b m are known constants selected by the designer according to the performance requirements.

[0017] Furthermore, the specific steps in Step 2 are as follows:

[0018] Step 2-1: Define the state tracking error as

[0019] e(t) = x p (t) - x m (t) (3)

[0020] The control objective is to achieve global uniform asymptotic tracking of the object state to the reference model state, that is

[0021] Step 2-2: Adopt a control law of state feedback plus command feedforward, and its form is as follows

[0022] u(t) = K 1 (t)x p(t) + K 2 (t)r(t) (4)

[0023] Where K 1 (t) is the adjustable feedback gain, and K 2 (t) is the adjustable feedforward gain. Assume there exists a constant satisfying When and the object output is exactly the same as the reference signal.

[0024] Step 2 - 3: From the object equation, control law, and reference model, the error dynamic equation can be obtained

[0025]

[0026] Define the parameter error After rearrangement, we can get

[0027]

[0028] Step 2 - 4: Select the Lyapunov function from the error dynamic equation

[0029]

[0030] Where α 1 and α 2 are both constants and α 1 , α 2 > 0. Therefore, V(t) is positive definite.

[0031] Step 2 - 5: Using the standard adaptive update law (8), it can be ensured that when the error is non - zero

[0032]

[0033] According to the second method of Lyapunov, the error system is asymptotically stable.

[0034] Furthermore, the specific steps in Step 3 are as follows:

[0035] Step 3 - 1: Define the double - boundary projection operator Proj: R n ×R n →R n as

[0036]

[0037] where S = {θ ∈ R n |g min ≤ g(θ) ≤ 0} is a convex set with a smooth boundary, which can also be called the projection domain, and g: R n→R is taken as a continuously differentiable convex function g min and θ max are constants, and both jointly determine the size of the projection domain S, where S 0 is the interior of S, and δ min (S) = {θ ∈ R n | g(θ) = g min} is the inner boundary of S, and δ max (S) = {θ ∈ R n | g(θ) = 0} is the outer boundary of S, y ∈ R n , and θ(0) is within S.

[0038] For the double - boundary projection operator, the inequality (9) holds

[0039] (θ - θ * ) T (Proj(θ, y) - y) ≤ 0, θ * ∈ R n (9)

[0040] Step 3 - 2: Apply the projection operator to the adaptive update law to ensure that the projection - adaptive update law can guarantee the asymptotic stability of the system.

[0041] The projection - adaptive update law is

[0042]

[0043] Substitute it into the derivative of the Lyapunov function and apply the inequality (9), we can obtain

[0044]

[0045] This ensures that the projection - adaptive update law can guarantee the asymptotic stability of the system.

[0046] Furthermore, the specific steps in Step 4 are as follows:

[0047] Step 4 - 1: Obtain the limit range of the control quantity at the current moment according to the limit protection module

[0048] u min (t) ≤ u(t) ≤ u max (t)

[0049] Step 4 - 2: From the parameter - adaptive update law, the quantitative relationship between control parameters can be obtained as

[0050]

[0051] Considering that the projection adaptive update law is adopted during actual update and the integral information is not completely preserved, it is more accurate to introduce the control parameters at the previous moment. The discrete quantitative relationship between control parameters can be approximated as

[0052]

[0053] where k represents the k-th moment and dstep is the step size of each step.

[0054] Step 4-3: When the control quantity reaches the limit value, the control parameter also reaches the limit value, that is

[0055] u min / max (k) = K 1min / max (k)x p (k) + K 2min / max (k)r(k) (12)

[0056] From equations (11) and (12), the limit values of the control parameters can be obtained

[0057]

[0058] Step 4-4: Design the projection domain using the limit values of the control parameters. For the update laws of K 1 and K 2 , the design parameters of its projection domain should be

[0059]

[0060] Beneficial effects: A model reference double-boundary projection adaptive control method for input constraints provided by the present invention, compared with the prior art by adopting the above technical solutions, has the following technical effects:

[0061] (1) The controller structure adopted by the present invention is a standard model reference adaptive control architecture, with a simple structure, few design parameters, and easy parameter adjustment. Also, due to its adaptive characteristics, it can ensure that the control effect is continuously improved during the actual control process;

[0062] (2) The projection operator adopted by the present invention uses a double-boundary form of the projection domain. Compared with the traditional projection operator, it can realize the upper and lower bound constraints on the control parameters. By combining the input limit value with the proportional characteristics of the adaptive law, the limit value of the control parameter is calculated, thereby realizing the calculation of the projection domain. For the case where the input constraint changes with time, the update domain of the projection operator adopted by the present invention can also be updated in real time.

[0063] (3) For the calculation of the control parameters, the present invention cleverly uses the control parameter value at the previous moment to avoid calculating the integral, thereby reducing the calculation error of the limit value of the control parameter. Description of the Drawings

[0064] Figure 1 It is the structure diagram of the model reference double - boundary projection adaptive control system for input - limited models;

[0065] Figure 2 It is the output response diagram after the object input step of 2% in the example of the present invention;

[0066] Figure 3 It is the simulation comparison diagram between the MRAC proposed by the present invention and the traditional MRAC;

[0067] Figure 4 It is the comparison diagram of the control parameter changes between the MRAC proposed by the present invention and the traditional MRAC;

[0068] Figure 5 It is the fuel flow change diagram at the ground point between the MRAC proposed by the present invention and the traditional MRAC. Detailed implementation manners

[0069] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings.

[0070] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of them. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts belong to the scope of protection of the present invention.

[0071] In order to overcome the problem that the control parameters drift in traditional model reference adaptive control under input limitation, resulting in poor control effect or even system instability, the present invention proposes a model reference double - boundary projection adaptive control method for input - limited models. On the basis of the standard model reference adaptive control, the standard adaptive law is upgraded to a projection adaptive law, and the projection domain of the projection operator adopts a double - boundary scheme, so that the limitation of the projection operator on the control parameters is upgraded from only limiting the maximum value to upper and lower - bound limitations, meeting the requirements of engine input limitations. The limitation of the control parameters ensures that the control quantity calculated by the controller will not exceed the input limitation, thus realizing the model reference adaptive control of aero - engines under input limitation.

[0072] Figure 1 The structure diagram of the model reference double - boundary projection adaptive control system for input - limited models applied by the method of the present invention is shown. The detailed implementation manners of the present invention take the design of a model reference double - boundary projection adaptive control method for a certain type of turbofan engine under input limitation as an example. The design of this method includes the following steps:

[0073] Step 1: Obtain the input characteristics of the engine and design a reference model;

[0074] Step 2: Design a standard adaptive update law;

[0075] Step 3: Upgrade the standard adaptive update law to a projection adaptive update law;

[0076] Step 4: Establish a projection domain for control parameters using the input limit.

[0077] Furthermore, the specific steps in Step 1 are as follows:

[0078] Step 1-1: The aeroengine can be expressed in the form of Equation (1)

[0079]

[0080] where x p (t) is the state of the controlled object, u(t) is the control input, a p and b p are unknown parameters with uncertainties, and b p characterizes the input characteristics of the system. By applying a step input and analyzing the response curve, the sign of b p can be obtained as +.

[0081] Step 1-2: Construct a reference signal according to the preset performance requirements, which is generated by the reference model (2)

[0082]

[0083] where x m (t) is the state of the reference model, r(t) is a bounded and piecewise continuous reference command, a m < 0, b m are known constants. According to the performance requirement that the adjustment time is no more than 3 s, let a m = -1, b m = 1.

[0084] Furthermore, the specific steps in Step 2 are as follows:

[0085] Step 2-1: Define the state tracking error as

[0086] e(t) = x p (t) - x m (t) (3)

[0087] The control objective is to achieve global uniform asymptotic tracking of the object state to the reference model state, that is

[0088] Step 2-2: Adopt a control law of state feedback plus command feedforward, which is in the following form

[0089] u(t) = K1 (t)x p (t)+K 2 (t)r(t) (4)

[0090] Wherein, K 1 (t) is the adjustable feedback gain, and K 2 (t) is the adjustable feedforward gain. Assume that there exists a constant satisfying When and the object output is exactly the same as the reference signal. Step 2-3: From the object equation, the control law and the reference model, the error dynamic equation can be obtained

[0091]

[0092] Define the parameter error After arrangement, it can be obtained

[0093]

[0094] Step 2-4: Select the Lyapunov function from the error dynamic equation

[0095]

[0096] Wherein, α 1 and α 2 are both constants and α 1 , α 2 >0, so V(t) is positive definite.

[0097] Step 2-5: Adopt the standard adaptive update law (8), then it can be ensured that when the error is not zero

[0098]

[0099] Since the characteristics of the object change with its own state

[0100] According to the second Lyapunov method, it can be ensured that the error system is asymptotically stable.

[0101] Furthermore, the specific steps in step 3 are as follows:

[0102] Step 3-1: Define the double-boundary projection operator Proj: R n ×R n →R n as

[0103]

[0104] where S = {θ ∈ R n |gmin {θ ∈ ℝ | ≤ g(θ) ≤ 0} is a convex set with a smooth boundary, also known as the projection domain, and g: ℝ n → ℝ is a continuously differentiable convex function, taken as g min and θ max are constants, and both jointly determine the size of the projection domain S. S 0 is the interior of S, and ∂ min (S) = {θ ∈ ℝ n | g(θ) = g min} is the inner boundary of S, and ∂ max (S) = {θ ∈ ℝ n | g(θ) = 0} is the outer boundary of S. y ∈ ℝ n , and θ(0) is within S.

[0105] For the double - boundary projection operator, the inequality (9) holds

[0106] (θ - θ * ) T (Proj(θ, y) - y) ≤ 0, θ * ∈ ℝ n (9)

[0107] Step 3 - 2: Apply the projection operator to the adaptive update law to ensure that the projection - adaptive update law can guarantee the asymptotic stability of the system.

[0108] The projection - adaptive update law is

[0109]

[0110] Substitute it into the derivative of the Lyapunov function and apply the inequality (9), we can obtain

[0111]

[0112] which guarantees that the projection - adaptive update law can ensure the asymptotic stability of the system.

[0113] Furthermore, the specific steps in Step 4 are as follows:

[0114] Step 4 - 1: Obtain the range of the control quantity at the current moment according to the limit protection module

[0115] u min (t) ≤ u(t) ≤ u max (t)

[0116] Step 4 - 2: From the parameter - adaptive update law, the quantitative relationship between the control parameters can be obtained as

[0117]

[0118] Considering that the projection adaptive update law is adopted during actual update and the integral information is not completely preserved, it is more accurate to introduce the control parameters at the previous moment. The quantitative relationship in discrete form between control parameters can be approximated as

[0119]

[0120] where k represents the k-th moment and dstep is the step size of each step.

[0121] Step 4-3: When the control quantity reaches the limit value, the control parameter also reaches the limit value, that is

[0122] u min / max (k) = K 1min / max (k)x p (k) + K 2min / max (k)r(k) (12)

[0123] From equations (11) and (12), the limit values of the control parameters can be obtained

[0124]

[0125] Step 4-4: Design the projection domain using the limit values of the control parameters. For the update laws of K 1 and K 2 , the design parameters of its projection domain should be

[0126]

[0127] Taking the steady-state and transient control of a certain type of turbofan engine as an example, based on the component-level model of this turbofan engine and combined with its corresponding control plan, the high-pressure rotational speed n H is selected as the state variable, and the main combustion chamber fuel flow rate W f is used as the control variable to conduct steady-state and transient control. At the steady-state operating point of H = 0 km, MA = 0, n H = 67.5%, a 2% perturbation is made to the main combustion chamber fuel flow rate, and the obtained response curve is as shown in Figure 2 . It can be seen from Figure 2 that when the input quantity suddenly steps up, the output starts to increase rapidly upward and gradually slows down with the increase of time, indicating that the input increment is proportional to the state change speed. Therefore, the sign of b p in equation (1) can be obtained as positive.

[0128] Moreover, from the prior knowledge of the engine, it is known that the characteristics of the system change with the change of the system state. Therefore, the adaptive gains α 1 and α 2 should also be designed as variable parameters with the state. The specific design values obtained after parameter adjustment change with the state as shown in Table 1:

[0129] Table 1 Interpolation Table of Adaptive Gain Varied with States

[0130]

[0131] The limit value of the control quantity is calculated by the limit protection module. Here, no further explanation will be given on how the limit value is calculated. The limit range of the control parameters can be obtained through Equation (13), and then the convex function of the projection operator and its update domain can be updated through Equation (14).

[0132] Perform performance simulation on the designed controller. At the ground point H = 0 km and MA = 0, conduct comprehensive simulation of the transient state and steady state and compare it with the standard model reference adaptive control (MRAC). The simulation comparison results are as Figure 3 shown. From Figure 3 it can be seen that for the steady state stage (35 s - 70 s), the MRAC proposed in the present invention has little difference in control effect from the standard MRAC. Both can complete the tracking of the command within 3 s, and both maintain a small overshoot and have no steady-state error. However, for the transient state stage (20 s - 30 s and 75 s - 90 s), due to the large-scale state change quickly, the input is restricted during the process. The traditional MRAC does not have the ability to handle input restriction, so a huge overshoot occurs. From Figure 4 analyzing the parameters of the traditional MRAC, it can be found that its control parameters increase (or decrease) sharply during the input restriction stage, resulting in deviation from the steady-state parameter value. The controller needs a longer time to adjust the deviated control parameters back. The manifestation of this process in the fuel flow is as Figure 5 shown, that is, the fuel flow reaches the fuel flow limit value for a longer period of time. Since the MRAC proposed in the present invention obtains the limit value of the control parameters through the input limit value and then restricts the control parameters through the double-boundary projection operator, the control quantity is not exceeded, and anti-integral saturation during the restricted period is also achieved. Analyzing the Figure 3 results can show that the MRAC proposed in the present invention can ensure that the transient regulation time is less than 5 s and the steady-state regulation time is less than 3 s. Both can ensure a small overshoot and the steady-state error is 0, proving that the controller has good steady-state and transient control qualities.

[0133] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above embodiments. What is described in the above embodiments and the specification only illustrates the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention will have various changes and improvements, and these changes and improvements all fall within the scope of the present invention claimed. The scope of protection claimed by the present invention is defined by the appended claims and their equivalents.

Claims

1. A model reference double - boundary projection adaptive control method for input - constrained Characterized in that: It includes the following steps: Step 1: Obtain the input characteristics of the engine and design a reference model; Step 2: Design a standard adaptive update law; Step 3: Upgrade the standard adaptive update law to a projection adaptive update law; Step 4: Establish a control parameter projection domain using the input limit; The specific steps for obtaining the input characteristics of the engine and the process of designing the reference model in Step 1 are as follows: Step 1 - 1: The aero - engine can be expressed in the form of formula (1) where x p (t) is the state of the controlled object, u(t) is the control input, a p and b p are unknown parameters with uncertainties. b p characterizes the input characteristics of the system. By performing a step on the input and analyzing the response curve, the sign information of b p can be obtained; Step 1 - 2: Construct a reference signal according to the preset performance requirements, which is generated by the reference model (2) where \(x\) m (t) is the state of the reference model, \(r(t)\) is a bounded and piecewise continuous reference command, \(a\) m \(< 0\), \(b\) m are known constants selected according to performance requirements; The process of designing the standard adaptive update law in Step 2 is as follows: Step 2 - 1: Define the state tracking error as e(t) = x p (t) - x m (t) (3) The control objective is to achieve globally uniform asymptotic tracking of the object state to the reference model state, that is Step 2 - 2: Adopt a control law of state feedback plus command feed - forward, and its form is as follows u(t) = K 1 (t)x p (t) + K 2 (t)r(t) (4) where K 1 (t) is the adjustable feedback gain, and K 2 (t) is the adjustable feedforward gain; assume that there exists a constant satisfying When and the output of the object is exactly the same as the reference signal; Step 2 - 3: Obtain the error dynamic equation from the object equation, the control law and the reference model Define parameter error It can be sorted out that Step 2 - 4: Select a Lyapunov function from the error dynamic equation where α 1 and α 2 are both constants and α 1 , α 2 > 0, so V(t) is positive definite; Step 2-5: By adopting the standard adaptive update law (8), it can be ensured that when the error is non-zero Guaranteed by the second method of Lyapunov, the error system is asymptotically stable; The upgrading process of the projection adaptive update law in Step 3 is as follows: Step 3-1: Define the double-boundary projection operator Proj: R n ×R n →R n as where \(S = \{\theta\in\mathbb{R}\ n |g min \leq g(\theta)\leq0\}\) is a convex set with a smooth boundary, also called the projection domain, and \(g:\mathbb{R}\ n \to\mathbb{R}\) is a continuously differentiable convex function, taken as g min and \(\theta max are constants, and together they determine the size of the projection domain \(S\). \(S 0 \) is the interior of \(S\), \(\partial min (S)=\{\theta\in\mathbb{R}\ n |g(\theta)=g min \}\) is the inner boundary of \(S\), and \(\partial max (S)=\{\theta\in\mathbb{R}\ n |g(\theta)=0\}\) is the outer boundary of \(S\). \(y\in\mathbb{R}\ n \), and \(\theta(0)\) is inside \(S\); For the double - boundary projection operator, the inequality (9) holds (θ - θ * ) T (Proj(θ, y) - y) ≤ 0, θ * ∈R n (9) Step 3 - 2: Apply the projection operator to the adaptive update law to ensure that the projection adaptive update law can guarantee the asymptotic stability of the system; The projection adaptive update law is Substitute it into the derivative of the Lyapunov function and use the inequality (9) to get Guarantee that the projection adaptive update law can guarantee the asymptotic stability of the system; The process of establishing the control parameter projection domain in Step 4 is as follows: Step 4 - 1: Obtain the limit range of the control quantity at the current moment according to the limit protection module u min u(t) ≤ u(t) ≤ u max (t) Step 4 - 2: Obtain the quantitative relationship between control parameters from the parameter adaptive update law as Considering that in actual update, the projection adaptive update law is adopted and the integral information is not completely preserved, so it is more accurate to introduce the control parameter at the previous moment. The approximate quantitative relationship in discrete form between control parameters is Where k represents the k - th moment and dstep is the step size of each step; Step 4 - 3: When the control quantity reaches the limit value, the control parameter also reaches the limit value, that is u min / max (k) = K 1min / max (k)x p (k) + K 2min / max (k)r(k) (12) From formula (11) and formula (12), obtain the limit value of the control parameter Step 4-4: Design the projection domain S using the control parameter limit value (13); for K 1 , K 2 's update law, the design parameters of the projection domain should be

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