# Application of Gap Metric to LADRC Design in Multilinear Model of SDR

Authors: Hongfu Wang; Qinghua Zeng; An Wang; Zongyu Zhang
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

The solid ducted rocket ramjet (SDR) system faces many disturbances in the process of operation, and the linear active disturbance rejection controller (LADRC) has been widely used in engineering to solve such problems. However, the SDR also has strong nonlinearity, which brings a great challenge to the application of the LADRC in the gas flow regulation of the SDR. And the problem of fast adjustment of the “compensation factor” is one of the main difficulties in LADRC. In this paper, under the LADRC frame, the gas generator system’s closed-loop stability of the SDR was analyzed and the range of compensation factors had been calculated, and then, the “gap factor” was introduced and the “cross-iteration” method was used to quickly map out the “compensation factor” in the multilinear model controller based on the variation of the zero-point position of the system and the gap metric between adjacent set points. This greatly simplifies the parameter tuning process of the LADRC when it was applied to strongly nonlinear systems. Finally, through the comparison of simulation with adaptive PI controller and model-assisted LADRC (M-LADRC), the results have shown that the control method designed in this paper can obtain satisfactory performance and has a good engineering application prospect.


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## Body

## 1. Introduction

The solid ducted rocket ramjet (SDR) does not need to carry additional oxidizer, and its specific impulse is 3-5 times that of the traditional solid rocket engine, which is an ideal powerplant for supersonic vehicles. While the gas flow regulation technology can realize the adjustable thrust of SDR, this is important for achieving its wide envelope and large maneuverability [1].With the concerted efforts of many scholars, the gas flow regulation theory of SDR has made certain research progress. Since the gas flow at high temperature and pressure is difficult to measure directly, the common practice is to regulate it indirectly by adjusting the pressure inside the gas generator (GG). Niu et al., for the strong nonlinearity and variable parameter characteristics of the pneumatic gas flow regulation system, proposed a dual-loop control scheme with high gain feedback to suppress nonlinearity, and the experimental results showed a good pressure response [2], but the ultimate purpose of regulating the pressure is to regulate the gas flow, and the pressure-to-flow response process has nonminimum phase (NMP) characteristics, so the pressure regulation loop alone could not completely determine the flow regulation process. In Ref. [3], a new feedback variable was constructed by fusing information from the working pressure of GG and the combustor entrance, which increased the stability margin of the system to some extent. In Ref. [4], an adaptive controller was designed for overcoming the effect of valve friction on the flow regulation process. Reference [5] designed a closed-loop model reference adaptive controller, which could better adapt to the nonlinearity of the pressure regulation system compared to the PI controller and exhibited smaller overshoot during longer and wider range of pressure regulation, but the biggest shortcoming of the paper is its experiments with a cold gas system simulating the GG, which has a large difference between them. So, recent studies have shown the difficulty of gas flow regulation in SDR, on the one hand, due to the strong nonlinearity and time-varying nature of parameters [6] such as throat area, characteristic velocity of gas, and gas constants in SDR. On the other hand, because there is a NMP characteristic in the response process from GG pressure to gas flow, we cannot focus only on the regulation effect of pressure but also consider the limitation of the undershoot of gas flow.Active disturbance rejection control (ADRC) technology treats system uncertainties as disturbances and is well adapted to them, so it is still a hot topic of research today. Reference [7] has shown that any linear finite-dimensional controller could be implemented via the LADRC structure, and it is a general-purposed control structure. And the transfer function of LADRC was derived in Ref. [8]. An attitude control method for nonlinear missile system was proposed in Ref. [9], which combined backstepping technique with LADRC. In addition, the LADRC was integrated with U-control theory as an aeroengine speed controller [10]. However, for a system such as SDR, which has strong nonlinearity in addition to uncertainty (the common practice is to make “small perturbation” assumptions and build small perturbation linearization models near the set points), the design and tuning of the control law become complicated and tedious due to the large number of set points. Regarding the parameter tuning problem of LADRC, there are tuning methods based on model information, for example, Ref. [11] used some known disturbances for feedforward compensation, which could reduce the observation burden of linear extended state observer. The third-order LADRC tuning formulas for oscillatory systems were derived from the internal model controllers (IMC) [12]; essentially, it also made use of model information. An actuator model was integrated into the LADRC to improve its performance [13]. And a new tuning method for second-order LADRC based on relay feedback was proposed in Ref. [14]; it also belongs to a model-assisted approach. Fu and Tan proposed a tuning method for reduced-order active disturbance rejection controller (RADRC) using model information as well as the generalized active disturbance rejection control (GADRC) tuning method [15]. However, if the model information is completely unknown then it needs to be identified, such as the forgetting factor recursive least-squares (FFRLS) method used in Ref. [16] and the unscented Kalman filter (UKF) method used in Ref. [17], but it is not suitable for systems with high experimental cost and strong nonlinearity. There are also parameter optimization methods based on intelligent algorithms, for example, the fuzzy control theory was combined with LADRC to adjust the control parameters of LADRC online [18, 19]. Similar LADRC tuning methods are the differential evolution (DE) algorithm designed in Ref. [20], the genetic algorithm (GA) designed in Ref. [21], the particle swarm optimization algorithm (PSO) designed in Ref. [22], the pigeon inspired optimization (PIO) designed in Ref. [23], the simultaneous heat transfer search (SHTS) algorithm designed in Ref. [24], the adaptive algorithm designed in Ref. [25], and the one optimization methods under the constraint of robustness metric described in Ref. [26]. In contrast, Ref. [27, 28] selected suitable parameters according to the frequency domain response under different parameters of LADRC. Since Gao proposed the LADRC [29, 30], the “compensation factor” has become one of the most critical parameters. And Chen also has pointed out that online estimation of the compensation factor is one of the further research directions of LADRC [31].The gap metric is used to describe the “distance” between two linear systems, and its smaller value indicates that the dynamic characteristics of the two models are closer [32]. Therefore, the method of dividing complex nonlinear systems into multilinear model based on the gap metric theory [33, 34] has been proposed and gradually developed by scholars. However, the partitioning of set points and the design of control laws remain independent of each other. In fact, the variation of the gap metric between linear models largely reflects the variation of the nonlinear system characteristics. Therefore, some design basis of the control law must be included in the gap metric. This paper combined the advantages of the gap metric theory and the LADRC, not only divided the nonlinear model into multilinear model based on the gap metric theory but also introduced the variation of the gap metric among multilinear model into the design and tuning of the LADRC for the first time. The main contributions of this paper are as follows:
(a)
The nonlinearity of the GG was analyzed, and the nonlinear model of GG was divided into multilinear model based on the gap metric theory(b)
The stability of GG was analyzed based on Hurwitz theorem in the LADRC control frame, and the range of compensation factors that could stabilize the LADRC was obtained(c)
The compensation factor of the “base point” was calculated based on the root locus method and further mapping the other compensation factors directly based on the gap metric between adjacent set points and the variation of the zero point, which greatly simplifies the design process of the control law and further broadens the application scope of LADRC(d)
The multivariate metrics quantitative analysis method was used to compare the control effects of the three control laws (GM-LADRC, adaptive PI controller, and M-LADRC) and we have analyzed the advantages and disadvantages of the GM-LADRC which was proposed in this paperThe rest of this paper is organized as follows. The mathematical model of SDR and its nonlinear characteristics are described in Section2, how to divide the nonlinear model into multilinear models based on the gap metric is described in Section 3, and how to calculate the “compensation factor” in LADRC based on the gap metric and the stability analysis of the system are described in Section 4, while the principle of LADRC for gas flow multimodel control is described in Section 5, followed by Section 6, where the three control laws are compared from several aspects, and finally, the conclusion and outlook are given in Section 7.


## 2. The Mathematical Model and Nonlinear Characteristics of GG

### 2.1. Mathematical Model of GG

The schematic diagram of SDR is shown in Figure1. The propellant is burned in the GG to produce primary gas, then the primary gas enters the ram combustor through the throat, and after the secondary combustion occurs with the air flowing into the intake, the gas will be ejected through the nozzleless booster to produce thrust. The ratio of air to (primary) gas is different (air-fuel ratio), the degree of combustion will be different, and the thrust generated also would be different, so in order to make the ramjet generate different thrust, it is necessary to regulate the mass flow of primary gas. Since the mass flow rate of high temperature and pressure gas is not easily measured directly, so it is generally controlled indirectly by controlling the pressure in the GG. We can change the area of the GG throat by means of the interstage valve, which in turn will cause a change in the working pressure of the GG and a further change in the combustion rate of the propellant; finally, the mass flow rate of gas (abbreviated as gas flow) would be changed.Figure 1
Schematic structure of the SDR.The basic principle of the GG mathematical model is the law of “mass conservation,” which means that the mass of gas generated by the combustion of propellant is equal to the sum of the mass of the gas inside the GG and the mass of the discharge from the throat. In this paper, we follow the mathematical model of GG established in Ref. [35], and the dynamic equilibrium differential equation within GG can be expressed in the form of
(1)dPgdt=Rg·TgV·ρb·Ab·a·Pgn−Pg·AtCr,wherePg is the gas pressure in GG, Rg is the gas constant, Tg is the gas temperature, and V represents the free volume, which means the volume between the propellant end face and the throat. ρb represents the propellant density, Ab represents burning area of the propellant, a represents the propellant combustion rate coefficient, n represents the pressure index, At represents the throat area, Cr represents the characteristic velocity of the gas, and θ represents the swing angle of the interstage valve.If the transfer function fromAt to θ is denoted by Kθ, then the transfer functions from Pg and mg to θ could be expressed in the form of equations (2) and (3). (See part A in Supplementary Materials for score analysis.)
(2)ΔPgsΔθs=−K1ts+T1t,(3)ΔṁgsΔθs=K2t·s−K3ts+T1t,where(4)T1t=−Rg0t·Tg0tV0t·ρb·Ab·a·1e−6n·n·Pg0tn−1−At0tCr0t,K1t=Kθt·Rg0t·Tg0tV0t·Pg0tCr0t,K2t=Kθt·Pg0tCr0t,K3t=K1t·At0t−Pg0t·T1t·KθtCr0t.


### 2.2. Mathematical Model of Interstage Valve

To make the study more convenient, the mathematical model of the interstage valve (also known as the valve motor) can be identified and equated to a second-order system by a frequency response test. As shown in Figure2, the resonant frequency (ωr) of the valve motor is about 65.94 rad/s and the high resonance peak Lωr is about 1.722 dB, which can finally be derived as its equivalent second-order transfer function according to equation (5), and the result is shown in equation (6).
(5)ωr=ωn∙1−2∙ξ2,Mr=Aωr=12∙ξ∙1−ξ2,20∙log10Aωr=Lωr.Figure 2
Frequency response of the valve motor.The solution isξ=0.46 and ωn=87.34.
(6)Gvs=ωn2s2+2∙ξ∙ωn∙s+ωn2≈7628s2+80.35∙s+7628,whereξ represents the damping ratio and ωn represents the undamped natural frequency. ωr represents the resonant frequency, and Mr represents the resonant peak.


### 2.3. Nonlinearity of GG

During the pressure response within the GG, the values ofPg0, V0, At, Cr, Rg, and Tg vary with the operating conditions of the GG in equation (2), which eventually leads to the nonlinear changes of K1 and T1. Taking an SDR ground test as an example, the values of K1 and T1 can be calculated based on the set points on the pressure response curve and the system parameters at the set points. As shown in Figure 3, it could be seen that K1 and T1 exhibit a strong nonlinear with the change of pressure.Figure 3
Variation of system’s parameters in a ground test. (a) Variation ofK1 and T1 in a ground test. (b) Variation of Pg and the angle of valve in a ground test.
(a)(b)


## 2.1. Mathematical Model of GG

The schematic diagram of SDR is shown in Figure1. The propellant is burned in the GG to produce primary gas, then the primary gas enters the ram combustor through the throat, and after the secondary combustion occurs with the air flowing into the intake, the gas will be ejected through the nozzleless booster to produce thrust. The ratio of air to (primary) gas is different (air-fuel ratio), the degree of combustion will be different, and the thrust generated also would be different, so in order to make the ramjet generate different thrust, it is necessary to regulate the mass flow of primary gas. Since the mass flow rate of high temperature and pressure gas is not easily measured directly, so it is generally controlled indirectly by controlling the pressure in the GG. We can change the area of the GG throat by means of the interstage valve, which in turn will cause a change in the working pressure of the GG and a further change in the combustion rate of the propellant; finally, the mass flow rate of gas (abbreviated as gas flow) would be changed.Figure 1
Schematic structure of the SDR.The basic principle of the GG mathematical model is the law of “mass conservation,” which means that the mass of gas generated by the combustion of propellant is equal to the sum of the mass of the gas inside the GG and the mass of the discharge from the throat. In this paper, we follow the mathematical model of GG established in Ref. [35], and the dynamic equilibrium differential equation within GG can be expressed in the form of
(1)dPgdt=Rg·TgV·ρb·Ab·a·Pgn−Pg·AtCr,wherePg is the gas pressure in GG, Rg is the gas constant, Tg is the gas temperature, and V represents the free volume, which means the volume between the propellant end face and the throat. ρb represents the propellant density, Ab represents burning area of the propellant, a represents the propellant combustion rate coefficient, n represents the pressure index, At represents the throat area, Cr represents the characteristic velocity of the gas, and θ represents the swing angle of the interstage valve.If the transfer function fromAt to θ is denoted by Kθ, then the transfer functions from Pg and mg to θ could be expressed in the form of equations (2) and (3). (See part A in Supplementary Materials for score analysis.)
(2)ΔPgsΔθs=−K1ts+T1t,(3)ΔṁgsΔθs=K2t·s−K3ts+T1t,where(4)T1t=−Rg0t·Tg0tV0t·ρb·Ab·a·1e−6n·n·Pg0tn−1−At0tCr0t,K1t=Kθt·Rg0t·Tg0tV0t·Pg0tCr0t,K2t=Kθt·Pg0tCr0t,K3t=K1t·At0t−Pg0t·T1t·KθtCr0t.


## 2.2. Mathematical Model of Interstage Valve

To make the study more convenient, the mathematical model of the interstage valve (also known as the valve motor) can be identified and equated to a second-order system by a frequency response test. As shown in Figure2, the resonant frequency (ωr) of the valve motor is about 65.94 rad/s and the high resonance peak Lωr is about 1.722 dB, which can finally be derived as its equivalent second-order transfer function according to equation (5), and the result is shown in equation (6).
(5)ωr=ωn∙1−2∙ξ2,Mr=Aωr=12∙ξ∙1−ξ2,20∙log10Aωr=Lωr.Figure 2
Frequency response of the valve motor.The solution isξ=0.46 and ωn=87.34.
(6)Gvs=ωn2s2+2∙ξ∙ωn∙s+ωn2≈7628s2+80.35∙s+7628,whereξ represents the damping ratio and ωn represents the undamped natural frequency. ωr represents the resonant frequency, and Mr represents the resonant peak.


## 2.3. Nonlinearity of GG

During the pressure response within the GG, the values ofPg0, V0, At, Cr, Rg, and Tg vary with the operating conditions of the GG in equation (2), which eventually leads to the nonlinear changes of K1 and T1. Taking an SDR ground test as an example, the values of K1 and T1 can be calculated based on the set points on the pressure response curve and the system parameters at the set points. As shown in Figure 3, it could be seen that K1 and T1 exhibit a strong nonlinear with the change of pressure.Figure 3
Variation of system’s parameters in a ground test. (a) Variation ofK1 and T1 in a ground test. (b) Variation of Pg and the angle of valve in a ground test.
(a)(b)


## 3. Determining Multilinear Models with Gap Metric

The gap metric is used to characterize the distance between two linear systems, and its value ranges from 0 to 1. The smaller the value is, the closer the dynamic characteristics of the two systems are. According to the description in Ref. [32], the design of the gap threshold was subjective. In this paper, the average change rate of the open-loop gain (K1/T1) and the time constant (1/T1) between adjacent set points could be kept within 20%, so the gap threshold was set as 0.05. And the set points with the gap metric less than this threshold were clustered, which was used as a basis to obtain the linear model bank of the system. Figure 4(a) indicates the gap between adjacent set points when the free volume is constant but the pressure varies. And Figure 4(b) shows the gap between adjacent set points when the pressure is constant but the free volume varies. If P1 and P2 are two scalar transfer functions, then the gap value of the Vinnicombe method is calculated according to [36]
(7)δvP1,P2=supω∈RP1jω−P2jω1+P1jω2∙1+P2jω2.Figure 4
Gap metric between adjacent set points. (a) The gap between adjacent set points when the free volume is constant but the pressure varies. (b) The gap between adjacent set points when the pressure is constant but the free volume varies.
(a)(b)For the present system, assuming that the systematic parameters of the adjacent set points areK1N and T1N, the expression can be further written as
(8)δvP1,P2=maxBD,A2∙MM−B∙M−B+A∙C+A2∙D,A∙D≥B∙C,δvP1,P2=BD,A∙D<B∙C,where(9)A=K1N−K12,B=K1N∙T1−K1∙T1N2,C=T12+K12+T1N2+K1N2,D=T12+K12∙T1N2+K1N2,M=B2+A∙A∙D−B∙C.The final division results are shown in Figure5. Compared with the traditional “equal interval” division method, the 10 pressure points and 10 free volume points (100 set points in total) were selected for designing the control law of the flow regulation system.Figure 5
The set point classification based on gap metric. (a) The 10 selected pressure set points. (b) The 10 selected free volume set points.
(a)(b)Immediately, the change in the characteristics of the system was analyzed by calculating the system parameters under the above set points. Figure6 represents the variation law of the system parameter K3 to K2 ratio, which determines the position of the system’s zero point; the smaller the value, the closer it is to the imaginary axis, the greater the undershoot. Under the same pressure condition, the value gradually decreases with the increase of free volume, and under the same free volume condition, the value also gradually decreases with the increase of pressure. This is the inherent characteristic of the system; in order to make the undershoot of the flow response not too large, it is necessary to limit the response rate of the flow according to the changes in pressure and free volume and must not only focus on the response of the pressure alone.Figure 6
The variation law of system’s zero position.


## 4. A “Compensation Factor” Mapping Method Based on Gap Metric and Stability Analysis

LADRC has the advantages of simple structure and easy implementation, thus making ADRC break through the bottleneck of parameter rectification and engineering. The basic idea of the LADRC algorithm is to linearize the expanded state observer (ESO) and associate its parameters with the observer bandwidth. For a first-order system, the ESO could be written as equation (10). And the expression of the control law could be written as equation (11). (See part B in Supplementary Materials for score analysis.)
(10)ż1=β1·y−z1+z2+b0·u,ż2=β2·y−z1,(11)u=ωc·R−z1−z2b0,whereβ1=2·ω0 and β2=ω02. z1 and z2 are the outputs of ESO, ωc is the controller bandwidth, and ω0 is the ESO bandwidth. y is the system output, which is equivalent to the pressure of GG, and R is the input of the linear state error feedback (LSEF).Since the response bandwidth of the interstage valve angle is much larger than that of the gas flow response, the valve motor loop can be temporarily treated as an ideal link in the control law design stage to reduce the computational effort, and then, it should be introduced to check the stability of the whole system when the control law design is completed. Similarly, the closed-loop transfer function of the system for the reduced-order ESO (without neglecting the observer error) could be obtained as shown in equation (12) by the method discussed in Ref. [37] (see part C in Supplementary Materials for score analysis.), and the equivalent structural block diagram of the system including LADRC and GG is shown in Figure 7. In this paper, the disturbed state of the system was also considered.
(12)Gcls=ωc∙G1s∙Gs/b01+G1s∙Gs∙Hs/b0,Figure 7
Equivalent structure diagram of the system containing LADRC and GG.where(13)Hs=ωc∙ω02+2ω0∙s+ω02∙ss+ω02,G1s=s+ω02s2+2ω0+ωc∙s,Gs=K1+∆K1s+T1+∆T1.Ifω0=P∙ωc, then equation (12) can be further written in the form of
(14)Gcls=ωc∙K1∙s+P∙ωc2a3∙s3+a2∙s2+a1∙s+a0,where(15)a3=b0,a2=b0∙T1+∆T1+2P+1∙ωc,a1=b0∙2P+1∙ωc∙T1+∆T1+K1+∆K1∙P2∙ωc2+2P∙ωc2,a0=K1+∆K1∙P2∙ωc3.It is well known that Lyapunov has proved the remarkable conclusion that if the linear approximation of a system is strictly stable, then the nonlinear system will be stable in some domain at the equilibrium point where the linear approximation is applied. Therefore, according to the Hurwitz theorem, the sufficient condition for the system to be stable at each equilibrium point could be found in(16)b0>0,T1+∆T1>0,K1+∆K1>0,ωc>18∙T1+∆T1.For this system, the value of the system parameterT1 at the selected set points will not exceed 50. Although ωc taken large enough could satisfy the stability condition, but too large ωc would affect the dynamic characteristics of the system and reduce the convergence speed of the system. Therefore, comprehensive consideration was made so that P=2 and ωc=10rad/s.In this paper, the “base point” was defined as the starting set point of the control law design, while the control parameters of other set points should be mapped under the premise of ensuring the stability of the system, relying on the gap metric between adjacent set points and the control parameters of the “base point.” The set point which could make the system’s “zero-pole point” closest to or farthest from the imaginary axis should be selected as the “base point,” with the aim of ensuring that the control parameters are monotonic over as large a range as possible. For example, for a system withN-dimensional set points, assuming that the control parameter at the “base point” is Ki∗,j∗,k∗,⋯, the mapping rules for the control parameters at other set points could be performed as in
(17)Ki,j,k,⋯=fUAi,j,k,⋯,UBi,j,k,⋯,UCi,j,k,⋯,⋯,i≠i∗j≠j∗k≠k∗⋯,whereUAi,j,k,⋯=gKi±1,j,k,⋯,GAPAi±1,j,k,⋯,⋯, i,j,k,⋯∈N∗ andi≠i∗.UBi,j,k,⋯=gKi,j±1,k,⋯,GAPBi,j±1,k,⋯,⋯, i,j,k,⋯∈N∗ and j≠j∗.UCi,j,k,⋯=gKi,j,k±1,⋯,GAPCi,j,k±1,⋯,⋯, i,j,k,⋯∈N∗ and k≠k∗.UAi,j,k,⋯, UBi,j,k,⋯, and UCi,j,k,⋯ denote the control parameters calculated iteratively along different dimensions, respectively.For the gas flow response of GG, in which undershoot is a stronger constraint, so the set point atPg=0.5MPa and V=0.01m3 was selected as the “base point,” whose zero point in the right half-plane is the farthest from the imaginary axis, and b01,1 is the compensation factor at the “base point.” As shown in Figure 8, after considering the valve motor model, the compensation factor of the “base point” could be calculated by the points on the system’s root locus. As shown in Figure 9, if the gain point is chosen to be 0.5, the value of b01,1 is the reciprocal of the gain, so its value is 2. Since b0∝K1∝δvP1,P2, the compensation factor could be considered positively correlated with gap metric and could be calculated as in
(18)b0i,j=maxbVi,j,bPgi,j,Figure 8
Equivalent structure of the system considering the actuator.Figure 9
The root locus of the system at the base point.wherebPgi,j=b0i−1,j+K∙GAPPgi−1,j, i∈2,10,j∈1,10.bVi,j=b0i,j−1+K∙GAPVi,j−1, i∈1,10,j∈2,10.i and j are the number of rows and columns in the set point table, respectively. From equation (18), we can see that when i=1, b0i,j is uniquely determined by bVi,j. And when j=1, b0i,j is uniquely determined by bPgi,j. Here, b0i,j is the compensation factor “b0” in equation (10). GAPPg denotes the gap between adjacent set points when the free volume is constant but the pressure varies, and GAPV denotes the gap between the adjacent set points when the pressure is constant but the free volume varies. Definition K is the gap factor, which is always constant. The adaptivity of the control law was ensured by the variation of the gap and the compensation factor.The computational process expressed in equation (18) could be more visually represented in Figure 10, where each blue “box” represents a compensation factor cell to be calculated cross-iteratively, each cell being determined jointly by GAPPg and GAPV. After the calculation, the compensation factors in the neighboring boxes were calculated in the same way; based on the variation of the zero position, the monotonic variation of b0 should be ensured as much as possible. The present method only needs to adjust the gap factor K, in the whole nonlinear domain, while the traditional method needs to adjust compensation factor based on each set point. For example, this system was divided into 100 set points, and then, the traditional way needs to tune 100 compensation factors. Eventually, the calculation results of compensation factor according to the above method are shown in Figure 11.Figure 10
Calculation method of compensation factor.Figure 11
Calculation results of compensation factor.As mentioned earlier, after introducing the valve motor model, the stability of the system needs to be reassessed. After examining the stability of the “base point” (Figure9), we also select the set point at Pg=4.48MPa and V=0.15m3 for stability analysis. As shown in Figure 12, it can be judged that the stability condition of the system at this set point is b0>1/18.7, while the stability condition at the “base point” is b0>1/5.68. Since the multilinear model of the system showed a clear regularity in the change of the zero position, it can be roughly deduced that the value of b0, which makes the system critically stable at each set point, decreases gradually with the increase of the gas pressure and free volume. Therefore, b0 mapped by the gap metric in Figure 11 all satisfy the stability requirement, where b010,10 is about 5.5 and the overshoot near this set point is about 35%, which can be reduced by equation (19).Figure 12
The root locus of the set point atPg=4.48MPa and V=0.15m3.


## 5. LADRC’s Principle for Multilinear Model of Gas Flow Regulation System

The principle of controller is shown in Figure13. The compensation factor selector makes a set point judgment based on the feedback pressure value and interpolates a real-time b0 value from a library of compensation factor calculated by the gap metric. The LADRC has three inputs, the “processed” pressure command signal, the feedback value of the GG pressure, and the updated value of the compensation factor. The output is the control amount of the valve angle. The LADRC based on gap metric designed in this paper denoted as “GM-LADRC.”Figure 13
Principle of GM-LADRC.In this control law design process, the preprocessing of the pressure command can be described as equation (19), which has the function of softening the input signal and phase overcorrection. It corresponds to the part surrounded by the green box in Figure 13.
(19)fh=r2∙C−x1k−2∙r∙x2k,x1k+1=x1k+h∙x2k,x2k+1=x2k+h∙fh,R=x1k+1+h∙α∙x2k+1,whereC represents the pressure reference, R is the output of this module, and h is the step size. α and r are the parameters to be tuned (α=alpha), whose value varies monotonically with the free volume during the design of this control law, as shown in Figure 14.Figure 14
The regulation law ofα and r with free volume.


## 6. Simulation-Based Quantitative Analysis of Multivariate Metrics

It is also important to note that the M-LADRC does not need to tune compensation factors too; for linear systems, M-LADRC allows it to converge faster in the presence of disturbances than traditional LADRC methods; the model-assisted ESO (M-ESO) could be written in the form of equation (20) [38]. In order to analyze the performance of GM-LADRC, this paper compared it with the adaptive PI controller and M-LADRC. For the consideration of undershoot, all three controllers described in this paper preprocessed the pressure command with the assistance of equation (19). Matlab/Simulink was used as the simulation software, and the simulation conditions are given in Table 1.
(20)ż1=l1·y−z1+z2−K1·u,ż2=l2·y−z1−T1·z2+K1·T1·u,Table 1
Simulation conditions.
GG model parametersValue (unit)GM-LADRC’s parametersValueAb0.0924m2ωc10a0.0063m/sh0.01n0.53K5ρb1630kg/m3b01,12where(21)l1=2·ω0−T1,l2=ω0−T12.Since the ablation of the interstage valve, the deposition of particles in the molten state, and the change of the load torque all eventually lead to the change of the gain factor from the motor output angle to the throat area, the gain factor was made to change from 1 to 0.9 at the 100th second and from 0.9 to 1 at the 160th second in the simulation to reflect the uncertainty change of the valve as well as the load. As depicted in Figure15, the disturbance rejection ability of GM-LADRC and M-LADRC was relatively close, with a maximum fluctuation of about 0.1 MPa, but the M-LADRC caused a maximum overshoot of about 5%. The adaptive PI controller had the worst disturbance rejection ability, with a maximum fluctuation of about 0.2 MPa caused by a change in the gain factor. The adaptive PI control law had the best steady-state accuracy, followed by the GM-LADRC.Figure 15
Step response of pressure under external disturbance.If the above changes to the gain coefficients could be considered as external disturbance, then next we would make deviations to the internal parameters of the system in order to study the performance of the three controllers under the internal disturbance of the system. When the pressure command was constant at 1 MPa, we made the system maintain the standard condition for the first 100 seconds, and at the 100th second, the deviation was generated for the three parameters inside the system according to no. 1 (in Table2), which continued until the 150th second when it became the deviation corresponding to no. 2 (in Table 2), as shown in Table 2. The response results are shown in Figure 16, and it could be seen that the best performance still belonged to the GM-LADRC, and the maximum deviation of the pressure response is counted in Table 3.Table 2
Deviations corresponding to various internal disturbances.
Deviation conditionsaCrnNormal condition+0%+0%+0%1+10%+15%−10%2+10%−15%+10%Figure 16
Step response of pressure under internal disturbance.Table 3
Maximum deviation of the pressure response.
Form of disturbancePositionAdaptive PIM-LADRCGM-LADRCExternal disturbancePosition 10.1990 MPa0.1240 MPa0.0990 MPaPosition 20.1909 MPa0.0941 MPa0.0984 MPaInternal disturbancePosition 10.2692 MPa0.1835 MPa0.1472 MPaPosition 20.0881 MPa0.0420 MPa0.0434 MPaFigure17 shows that the response of the gas flow had different degrees of undershoot due to the NMP characteristic of the pressure-to-gas flow response process. The adaptive PI controller had the smallest undershoot due to its slowest response, and the undershoot of the remaining two control laws was comparable, with GM-LADRC being slightly smaller than M-LADRC at 13% and adaptive PI at about 6.5%. After about 214 seconds, the propellant was burned out.Figure 17
The pressure and gas flow response under square wave signal.The above simulation results had shown that each of the three control laws had its own advantages and disadvantages. In order to obtain the comprehensive performance of each control law in more dimensions, the four indexes of “ disturbance rejection ability,” “rapidity,” “accuracy,” and “overshoot” of the three control laws were “scored,” respectively.The scoring method [39] was carried out according to equation (22), and the simulations were tested from the lowest pressure response to 0.6,2.5 MPa in the method shown in Figures 15 and 16, respectively. For a total of 40 sets of simulations (20 groups each for internal disturbance simulation and external disturbance simulation), the final score was taken as the average score of the 40 sets of simulation results. The results are shown in Table 4, and finally, “GM-LADRC” won the highest score, indicating that it had the best overall performance. Figure 18 is a more visual depiction of it. (See Tables S1–S6 in Supplementary Materials for score analysis.)
(22)Sco=1n∙∑i=1n5−4∙xj−minxjmaxxj−minxj,Sco∈1,5,j∈1,3,Table 4
Scoring of each indicator of the three control laws.
Control lawDisturbance rejection abilityRapidityAccuracySmall overshootTotalGM-LADRC4.42.21.01.08.6M-LADRC4.64.94.45.018.9Adaptive-PI1.02.35.05.013.3Figure 18
Comparison of the advantages and disadvantages of the three control laws.wherexj are the control indexes of the three control laws under the same instruction and simulation conditions and maxxj is the maximum value of the control index of the three control laws. n is the total number of simulation experiments, and n is 80 because two observed points were set for the “disturbance rejection ability.”. For the rest of the indicators, n is 40.


## 7. Conclusion

In this paper, the GM-LADRC control law was designed for the gas flow regulation problem of SDR. Based on the multilinear model divided by the gap metric method, the gap factorK was introduced, and the compensation factor in the whole operation area was mapped by the gap metric of adjacent set points. According to this method, the parameter tuning of LADRC could be greatly simplified. Compared with the adaptive PI controller, this method still maintained the strong disturbance rejection ability of LADRC, while compared with M-LADRC this method ensured better adaptability to the system and higher control accuracy. In addition, the method described in this article is suitable for systems with strong nonlinearity but little difference in the control bandwidth requirements of the set points. Otherwise, the parameters to be mapped in the LADRC will increase, which will also increase the difficulty of parameter tuning.However, when the LADRC method as described in this paper was used for gas flow regulation in SDR, the steady-state error did not converge strictly to zero, or the error converged to zero very slowly; further research is needed for this problem. In addition, on the base of the LADRC frame described in this article, in view of the SDR model uncertainty and actuator dead zone, the use of intelligent algorithms for further online compensation is also a direction worthy of future studies.



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