Adaptive saturated tracking control for solid launch vehicles in ascending based on differential inclusion stabilization

Fei Liu,Song-yan Wang,Tao Chao,Ming Yang

Control and Simulation Center,Harbin Institute of Technology,Harbin,150090,China

Keywords:Solid launch vehicles Trajectory tracking Adaptive controller Differential inclusion

ABSTRACT The large-range uncertainties of specific impulse,mass flow per second,aerodynamic coefficients and atmospheric density during rapid turning in solid launch vehicles(SLVs)ascending leads to the deviation of the actual trajectory from the reference one.One of the traditional trajectory tracking methods is to observe the uncertainties by Extended State Observer (ESO) and then modify the control commands.However,ESO cannot accurately estimate the uncertainties when the uncertainty ranges are large,which reduces the guidance accuracy.This paper introduces differential inclusion (DI) and designs a controller to solve the large-range parameter uncertainties problem.When above uncertainties have large ranges,it can be combined with the ascent dynamic equation and described as a DI system in the mathematical form of a set.If the DI system is stabilized,all the subsets are stabilized.Different from the traditional controllers,the parameters of the designed controller are calculated by the uncertain boundaries.Therefore,the controller can solve the problem of large-range parameter uncertainties of in ascending.Firstly,the ascent deviation system is obtained by linearization along the reference trajectory.The trajectory tracking system with engine parameters and aerodynamic uncertainties is described as an ascent DI system with respect to state deviation,which is called DI system.A DI adaptive saturation tracking controller (DIAST) is proposed to stabilize the DI system.Secondly,an improved barrier Lyapunov function (named time-varying tangent-log barrier Lyapunov function) is proposed to constrain the state deviations.Compared with traditional barrier Lyapunov function,it can dynamically adjust the boundary of deviation convergence,which improve the convergence rate and accuracy of altitude,velocity and LTIA deviation.In addition,the correction amplitudes of angle of attack(AOA)and angle of sideslip(AOS)need to be limited in order to guarantee that the overload constraint is not violated during actual flight.In this paper,a fixed time adaptive saturation compensation auxiliary system is designed to shorten the saturation time and accelerate the convergence rate,which eliminates the adverse effects caused by the saturation.Finally,it is proved that the state deviations are ultimately uniformly bounded under the action of DIAST controller.Simulation results show that the DI ascent tracking system is stabilized within the given uncertainty boundary values.The feasible bounds of uncertainty is broadened compared with Integrated Guidance and Control algorithm.Compared with Robust Gain-Scheduling Control method,the robustness to the engine parameters are greatly improved and the control variable is smoother.

1.Introduction

Guidance for SLVs has been an active area of research for decades [1].The dynamics parameters of solid rocket are highly complex nonlinear and time-varying.During the actual dynamic flight,the different working environment of the engine,such as temperature,charging amount and machining error,leads to the difference between the specific impulse and mass flow per second of the engine and the ground test [2].On the other hand,the limitations of theoretical calculations and the differences between wind tunnel experiments and actual flight conditions make it difficult for the obtained aerodynamic characteristics to accurately reflect the actual characteristics[3].These factors make a challenge to guide the SLV to the reference trajectory.For the SLVs,the thrust is not controllable and exhaust shutdown is adopted.Therefore,tracking control can only be carried out by adjusting the AOA and AOS.During the actual flight,the uncertainty boundaries are generally obtained through online identification and prediction methods.It is necessary to have good tracking performance within the given uncertainty boundaries.In addition,the ascending phase is a process of fast speed and rapid trajectory turning.The controller must control the actual trajectory not to deviate too much from the reference trajectory.

Many high-performance control methods have been applied to different types of aircraft to improve their trajectory tracking performance.Based on the error system of reference trajectory[4-6],adopt linear quadratic adjustment(LQR)theory to realize trajectory tracking.However,LQR is based on time-invariant model and cannot guarantee the global stability of trajectory error system.Sliding mode variable structure control has also been applied to the trajectory tracking [7],which is a common nonlinear control method.Although it has good robustness and anti-interference ability,the large and high frequency chattering control signal may not be conducive to the command tracking of the actuator.Considering the good comprehensive capability,Utkin et al.[8]proposed a Linear parameter change (LPV) robust gain scheduling control method.Adaptive controller gain can be achieved by measuring or estimating time-varying model parameters [9].However,according to the existing literature,this method has not been applied to the trajectory tracking of SLVs.

Trajectory tracking method based on estimation-compensation is very practical in engineering practice [10].The extended state observer(ESO)can observe the disturbance caused by uncertainties in the system,and the compensation of the disturbance can improve the control effect.ESO can be divided into linear extended state observer (LESO) and nonlinear extended state observer(NESO) according to different structures [11].NESO has a complex structure and the nonlinear functions in the observer often rely on empirical design.So it is difficult for observer design and application[12].LESO,on the other hand,is easy to design for observer due to its simple structure and few design parameters [13].If ESO has good convergence for error observation,the accuracy of the estimation-compensation method is higher.However,when the SLV ascent model contains a large range of parameter uncertainties,it is often difficult for ESO to accurately estimate the uncertainties in each channel,resulting in a decrease in the robustness of the algorithm.Another idea is to treat the uncertainty and dynamics system as a whole,and extend it to a DI system.DI system has the following two characteristics: (1) According to the principle of DI system,if the DI system is stabilized,any subset of its inclusion can be stabilized.(2) The parameters of the DI system stabilizing controller are related to the uncertainty boundaries,which ensures the applicability of the controller parameters in the case of maximum uncertainties and suitable for any cases in the range of uncertainties.

DI is the dynamic system established on the basis of a certain understanding of the system process,but incomplete determined.It is an important approach to describe the discontinuous or uncertain dynamic system,and also the generalization of differential equations [14].Therefore,a DI system can be established to describe the dynamics of the ascending with uncertainties.If the ascent DI system can be stabilized,the original ascent differential equation deviation system with uncertainties can be stabilized[15].DI is proposed to solve two types of problems: the solution of discontinuous differential equations [16-18]and the stabilization of control systems.In addition to general control problems,the control of multi-model systems and uncertain parameter systems[19,20]can also be considered as differential inclusion.The stabilization of nonlinear DI systems is a hot research topic recently[21-25].The control Lyapunov function(CLF) method can be used to solve a class of single-input NDI systems [21]and multi-input systems [22].In Ref.[23],convex hull Lyapunov function is used to construct the nonlinear feedback law for a class of NDI systems.In Ref.[24]sliding mode control method is used to solve the stabilization problem of a class of NDI systems with disturbances.A finite-time stabilizing controller is designed for a closed-loop system by an adaptive terminal sliding mode control law with unknown upper bound of perturbation is studied in Ref.[25].Ref.[26]focuses on the problem of robust stabilization of linear differential inclusions subject to actuator saturation.In Ref.[27],a frequencydomain method was proposed to analyze the asymptotic stability of DI systems with discrete and distributed delay.In Ref.[28],the stability of polyhedral differential inclusion(PDI)systems is studied by sliding mode control.The global stabilization is addressed based on fractional-order differential inclusions with the aid of Lyapunov functions and the comparison principle [29].The FTS problem of impulsive differential inclusion is studied in Ref.[30].The direct Lyapunov method for DI is used to solve the stabilization problem in the Keldysh model [31].In Refs.[32,33],several sufficient conditions ensuring local exponential stabilization based on differential inclusion theory and fractional Lyapunov stability theory.

In view of the above analysis,the idea of DI is introduced in this paper.The trajectory tracking problem of in ascending is transformed into the DI system stabilization problem,so as to achieve the purpose of being applicable to a large range of uncertainties.

The research ideas of this paper are as follows:First,a dynamic model of trajectory tracking in the ascending is established,which takes into account the uncertainties of engine specific impulse,aerodynamic coefficient and atmospheric density.Since the trajectory tracking system is a non-affine nonlinear system,it brings disadvantages to the controller design and real-time solution.Therefore,the control variables are extracted by linearizing the tracking system along the trajectory affine.After obtaining the affine deviation system,the affine system and the uncertainties(engine parameters,aerodynamic coefficients and atmospheric density uncertainties)are expressed as a set of mathematical forms to establish the DI system in ascending.Several concepts of Lie variables are proposed and the properties of Lie variables are given to prepare for the design of the controller.Then,the form and design method of stabilizing controller are proposed for the DI system.To solve the problem that the actual trajectory cannot deviate too much from the reference trajectory,a new time-varying tangent-log barrier Lyapunov function (TVTBLF) is proposed.The time-varying parameters in TVTBLF can dynamically adjust the convergence domain and convergence rate of state deviations,which improves the dynamic and steady performance of convergence.In addition,TVTBLF can be applied to a large range of initial values of the deviation,which broadens the application range of the traditional barrier Lyapunov function.Considering the overload constraint in actual flight,it is necessary to limit the amplitude of AOA and AOS corrections.A fixed time adaptive saturation compensation auxiliary system is designed and the adaptive laws of the auxiliary system parameters are given.When the corrections of AOA and AOS saturate,the compensation system produces auxiliary variables to compensate the loss of control performance caused by saturation and shorten the saturation time.When the corrections of AOA and AOS exit the saturated regions,the adaptive laws make the auxiliary variables converge to zero rapidly in a fixed time,avoiding the influence of the auxiliary variables on the original system variables.Finally,the calculation method of Lie variables and controller parameters are proposed and the adaptive fixed time DI stabilizing controller is obtained.

The major contributions of this study can be summarized as follows:

(1) The trajectory deviation system with engine specific impulse,aerodynamic coefficient and atmospheric density uncertainties is described as a DI system in the form of set,and then the trajectory tracking problem is transformed into stabilization problem of DI system.

(2) A stabilization controller is proposed for the ascent DI system.According to the uncertain boundaries,the calculation method of controller parameters is given.It is proved that under the action of the controller,the deviation is uniformly bounded when the uncertainties are within the given boundary values.The tracking near the reference trajectory is realized when the uncertainties of engine specific impulse,aerodynamic coefficient and atmospheric density are in large ranges.

(3) The error accumulation caused by the open-loop flight of the 1st stage leads to the deviation of the initial state of the 2nd stage.So a TVTBLF is proposed which can be applied to the case of unconstrained initial deviation.In addition,the timevarying characteristic of the TVTBLF parameters improves the steady-state and transient performance of deviation convergence by setting the time-varying boundary parameters of exponential decay in advance.The preset performance of the parameters can not only constrain the state deviations to ensure that the actual trajectory does not deviate too much from the reference trajectory,but also dynamically adjust the convergence domain and convergence rate of the deviations.

(4) In order to avoid violating the overload constraint during actual flight,the correction commands of AOA and AOS are saturated.A fixed-time adaptive saturation compensating auxiliary system is designed to reduce the adverse influence caused by control variable saturation.When saturation occurs,the compensator generates auxiliary variables to compensate for the control loss caused by saturation.When the control variables exit the saturated region,the compensator parameters are adjusted adaptively to accelerate the convergence rate of the auxiliary variables,which makes the auxiliary variables converge to zero quickly without affecting the convergence characteristics of the controller.

The remainder of this paper is organized as follows.In section 2,the problem description is presented and model of the ascent DI stabilization model is established.In section 3,in view of the characteristics of the ascent DI model,a DI adaptive saturated tracking(DIAST)controller is developed and the proof of stability is presented.Next,the simulation results and discussion are given in section 4.Finally,the conclusions are summarized in section 5.

2.Ascent trajectory tracking problem formulation

In this section,the ascent trajectory tracking problem is transformed into the state deviation stabilization problem between the actual trajectory and the reference trajectory.Firstly,the ascent dynamics and parameter uncertainty model are given.Secondly,the ascent trajectory tracking system is transformed into an affine system with respect to the state deviations.Finally,combined with the parameter uncertainty boundaries and the differential equation system about state deviations,a DI system is established as the research object of controller design in the next chapter.

2.1.Ascent dynamic model

In this section,the ascent dynamic model is established for a multi-stage booster vehicle.To simplify the calculation,several assumptions are made about the model.1st,the vehicle is treated as a particle and the rotation is ignored.2nd,the effect of the oscillating nozzle is not considered.3rd,the Earth is treated as a sphere.4th,the ascent takes short flight time and ignore the centrifugal and Coriolis forces caused by the earth's rotation.The parameter uncertainties also exist during the flight.In this model,engine specific impulse,aerodynamic coefficient and atmospheric density disturbances are considered.Based on the above assumptions and parameter uncertainties,the dynamics equations of the centroid of the SLV can be derived in launching coordinate are established as follows:

In the nonlinear model Eq.(1),state variables v,hand Θ are velocity,altitude and local trajectory inclination angle (LTIA),respectively.σ is trajectory azimuth angle.α and β denote AOA and AOS respectively.Peandmare the engine thrust and the mass,respectively.R0andfMare the Earth radius and geocentric gravitational constant,respectively.Smand ρ are the characteristic area and atmosphere density,respectively.DandLare aerodynamic drag and aerodynamic lift,respectively,which are calculated as

wherecxandcyare the drag coefficient and lift coefficient respectively.

The limitations of theoretical calculation and the differences between wind tunnel experiments and actual flight conditions make it difficult to accurately reflect the actual aerodynamic characteristics.This paper considers the uncertainties of aerodynamic coefficient and atmospheric density

On the other hand,during the actual dynamic flight,the engine parameters of specific impulse and mass flow per second is different from the ground test result due to the different working environment of the engine,such as temperature,charge mass,machining error,etc.These two uncertainties are called engine parameter uncertainties

where,ΔIspandare engine specific impulse and mass flow per second respectively;and ΔIspand Δrepresent the uncertainties of their nominal values.

The above factors will inevitably lead to the deviation between the actual trajectory and the reference trajectory.The uncertainties considered in this paper are the stochastic uncertainties in model parameters.In addition,it is assumed that these uncertain parameters are independent of each other and all the uncertainties satisfy uniform distributions.

2.2.The establishment of ascent DI system

Under the condition of non-disturbance,a reference trajectory satisfying the given constraints can be obtained in advance.The ascent guidance problem can be regarded as a reference trajectory tracking problem.The control goal is to track the reference altitude velocity and LTIA in real time in this study.The guidance law should be calculated with a relatively fast speed to meet the requirement of online guidance.The terminal state errors should satisfy the given index.For the tracking problem of reference trajectory,it can be transformed into a deviation stabilization problem.

General nonlinear systems can be divided into affine and nonaffine systems.The control variable of the affine system is easier to be solved because the control variable is separated from the nonlinear term.However,the ascent dynamic system Eq.(1) is a non-affine system.Therefore,not only the deviation system should be constructed,but also the affine deviation system is guaranteed.Moreover,as it is a real-time tracking problem,the actual trajectory will not deviate too much from the reference trajectory.Based on the above analysis,the actual trajectory can be expanded by Taylor expansion along the reference trajectory.The establishment process of stabilization model in ascending is as follows:

In the dynamics of velocity and LTIA,the nonlinear term related touis formulated as

where,x ＝[v,Θ]Tand u ＝[α,β]T.

To prevent the overload constraint from being violated during actual flight,the corrections of AOA and AOS are subject to the limitation of magnitude,that is

where Δu ＝[Δα,Δβ]T;d is the high order term;sat(·) is the saturation function anduimaxis the absolute values of the predefined upper bound.

The nonlinear terms are given as follows:

Then the affine design model with input constraints can be formulated as

According to Eq.(9),the state deviation system in ascending can be expressed as

Obviously,the system Eq.(10) is an affine system with input constraint.For ease of the guidance algorithm design,some necessary assumption and lemmas are presented.

Assumption 1.The full states of system Eq.(9)can be obtained,and the states and its derivatives are bounded.

Assumption 2.The magnitudes of the d in Eq.(9) is unknown but bounded by‖d‖≤Δ,where Δ is the unknown boundary.

Assumption 3.Parameter perturbations ΔIsp,Δ,Δcx,Δcyand Δ ρ are unknown,but the boundaries are known

Then the ascent guidance system with uncertainties can be constructed as follows:

3.Ascent guidance scheme design based on DIAST controller

In this section,an adaptive saturation tracking controller is designed for the DI system established in the previous section.Aiming at the large-range uncertainty problem,the concept of Lie boundary is proposed.The framework of DI stabilization controller is constructed by taking Lie boundary as the controller parameters.In order to improve the transient and steady-state performance of the controller,an improved Barrier Lyapunov Function is proposed to dynamically adjust the convergence process of state deviations.A fixed time adaptive saturation compensator is designed to eliminate the adverse effect of potential saturation on system performance.

A design of ascent guidance scheme is proposed to deal with the reference trajectory tracking problem.From the dynamics of altitude equation＝v sin Θ,the altitude deviation can be adjusted by modifying the nominal LTIA.The emphasis of this paper is how to design tracking algorithm to realize an excellent tracking performance of modified LTIA and reference velocity commands.The ascent tracking control structure is shown as Fig.1.

Fig.1.Ascent tracking control structure diagram.

Next,several key parts of the DIAST controller are introduced in detail.

3.1.The design of DIAST controller

Based on DI system stabilization and adaptive control techniques,the detailed design procedures of the ascent adaptive robust DI control scheme of launch vehicle is presented.Moreover,the improved adaptive saturation compensator is adopted to deal with input constraints.The control structure diagram of ascent DIAST scheme is illustrated in Fig.2.

Fig.2.DIAST controller structure.

According to Fig.2,the DIASTcontroller consists of the following parts:

(1) DI stabilization model: The ascent dynamic model and parameter uncertainties are combined as a whole and transformed to a DI system,which is treated as a new object to be stabilized.The transformation process is shown in subsection 2.2.

(2) Adaptive law (30): For the loss in the affine process of the model,an adaptive law is used to estimate the ignored upper bound.The advantage of adaptive law is that it reduces the complexity of controller design without constructing the Extended State Observer.

(3) TVTBLF: A new type of time-varying tangential Barrier Lyapunov Function is proposed in subsection 3.1.1.It ensure that the actual trajectory does not deviate too much from the reference trajectory.Moreover,it improves the dynamic and steady-state performance of the tracking.

(4) Lie boundary calculation:The Lie boundary is constructed by TVTBLF and DI stabilization model.The definition of Lie boundary is given in subsection 3.1.2 and the calculation method is discussed in subsection 3.2.The main idea is to obtain the DI system boundary under the given uncertain boundary values,and introduce these boundary-related items into the control law Eq.(28).Therefore,the control law can stabilize deviations in any situation within a given uncertainty boundary,which is proved in subsection 3.1.3.

(5) Fixed-time adaptive saturation compensator: When correction commands are saturated,the compensator gain decreasing scheme is adopted to compensate the saturation loss and shorten the saturation time.After the correction commands exit the saturation region and return to normal operating conditions,the gain increasing scheme is adopted to drive the additional variable fixed time to accurately converge to 0.The tracking convergence characteristics of the original system can be restored,which effectively avoids the impact on the tracking performance of the original system.The details of adaptive scheme are shown in Eq.(31) -Eq.(34).

(6) Control law Eq.(28): The control law can stabilize the DI system and make the deviations converge uniformly.The details of the proof are shown in subsection 3.1.3.The control law consists of Lie boundary,adaptive law Eq.(30) and TVTBLF.Although the control law is conservative,it can make the deviations converge within the given uncertain boundaries.

3.1.1.Time-varying tangential Barrier Lyapunov function

To solve the state constraints problem,the traditional Logarithmic Barrier Lyapunov Function (LBLF) is expressed as Ref.[36].

whereA＞0 is the constraint boundary.The initial errorx(0) satisfies ｜x(0)｜＜A.WhenA→∞,Vln→0.

In this paper,a Tangent-Logarithmic Barrier Lyapunov Function(TLBLF) is proposed,which is expressed as

whereAT＞0 is tracking performance constants.

The derivative ofVTis

Proof.See the Appendix.

If the initial error e (0) of the system is large,the constant constraint boundary needs to be set to a larger value accordingly.It is difficult to guarantee the steady-state performance.Therefore,a time-varying tangential Barrier Lyapunov function is proposed,which is expressed as

where,the time-varying constraint boundaryATVis

Remark 2: By setting the attenuation speed parameter of the constrained boundary function,the system has better transient performance in the beginning stage of control.By setting the terminal value of the constrained boundary function,the system has better steady performance in the steady stage.

3.1.2.The Lie boundary of the ascent DI system

Define a time-varying Lyapunov function,which is a combination of several TVTBLFs

3.1.3.DI adaptive saturated tracking controller design

Before the design of controller,the following lemmas are given.

Lemma 1For any real numbera1and any real numbera2,the following inequality holds

where,tanh(·) is hyperbolic tangent function.τ ＞0 and its minimum value τ*satisfies

where,x*satisfies the equation e-2x*+1-2x*＝0.

Lemma 2F or anya1,a2∈R,the following inequality holds

The design of the control law Δu in system(13)is constructed as follows

When AOA or AOS saturation occurs,the adaptive program works to reduce ωito shorten the saturation time.When the AOA or AOS is no longer saturated,the adaptive scheme increases ωito achieve faster response speed for y tracking error e [34].The strategy of adaptive gain ω is given as follows:

where,wsandwusare the positive design parameters.

To avoid infinitely increasing of ω and keep the basic rate of convergence,the amplitude of ω is constrained as

where,the parameters ωimaxand ωiminrepresent the lower and upper bounds of ωi,respectively.

Theorem 3:For system Eq.(12),under the action of controller Eq.(27) and adaptive law Eq.(29),it can ensure the semi-global stability of the closed-loop system,and the control input and states of the closed-loop system are uniform bounded.The tracking error can be constrained within the boundary [-ATVi,ATVi].

Proof.

The closed loop Lyapunov function of system Eq.(12) is constructed as

According toLemma 1andLemma 2,there exists

According to Eq.(35)and Eq.(36),it can be further rewritten as

Both sides of Eq.(37) are multiplied by ectand integrated

Thus,the tracking erroreiis constrained within the boundary[-ATVi,ATVi].The control inputs,state variables and all closed loop signals are ultimately uniformly bounded.

Theorem 3 is proved.

Remark 6In practice,the auxiliary variable of the fixed-time adaptive saturation compensator (FTASC) Eq.(30) is initially set to zero.When the saturation occurs,the compensator generates auxiliary variable χ.According to the above analysis,y can achieve uniform convergence.If the control variable leaves the saturated region,i.e.δu ＝0,Eq.(30) becomes

3.2.Parameter calculation of DIAST controller

In Section 3.1,the specific form of the ascent DIAST tracking controller is proposed.In this section,the parameters ΩM,γgand～geare calculated.

Lemma 3The maximum value of a function in the domain is taken at stationary points or the boundary of the domain.

The domain of the uncertainties in system(12) is defined as

(1).The calculation of ΩM

The expression for LfeV can be written as

wherem0is the initial mass of SLV andtis the current time.

From Eq.(46)-Eq.(50),the upper bound of LfeV is determined by ΔIsp,Δ,Δρ,Δcxand Δcy.Since ∂LfeV/∂Δpidoes not contain Δpi,there are no stationary points in D.It can be concluded that the value of LfeV has monotonicity with respect to Δpi,respectively.According toLemma 3,the upper bound of LfeV is taken in the boundary combination of ΔIsp,Δ,ΔcD,ΔcLand Δρ.Since there are 5 uncertainties,there are a total ofN＝25＝32 boundary combinations.Then ΩMcan be calculated as follows:

Bpirepresents thei-th uncertain boundary combination,which is the vertex ofD;ij＝1 andij＝2 represent the upper and lower bound of the uncertainty respectively,j＝1,2,3,4,5.

(2).The calculation of γg

The expression of LgV is calculated as follows:

Take the partial derivatives of LgV with respect to Δp

From Eq.(53),the value of LgV is independent of Δcxand Δcy.From Eq.(54)and Eq.(55),it can be concluded that the value of γihas monotonicity with respect to ΔIspand Δ,respectively.According toLemma 3and refer to the calculation of ΩM,the upper and lower bounds of γiare obtained at the boundary.

Define a set Bg,which is the boundary combination of ΔIspand Δ,

According to Eq.(22)

4.Simulation results and discussion

As a carrier of transportation,the SLV transports the load to a specified altitude at a certain velocity in a specific attitude.The background considered in the example is the 2 stages of a multistage SLV.The engine parameters of 2nd and 3rd stages are given in Table 1 (see Table 3) (see Table 4) (see Table 2).

Table 1Engine parameters of 2 stages.

Table 2Parameters setting of 5 cases.

Table 3Parameter setting of 2 kinds of BLFs.

Table 4Engine parameter and aerodynamic coefficients perturbations.

It is assumed that the 1st stage flies according to the control variable of nominal off-line programming.Whent＝120 s,the second stage of boost ends.After gliding without power for 3 s,it enters the 3rd stage of boost.Due to the existence of uncertainties,altitude,velocity and LTIA deviations are accumulated.In this paper,the trajectory tracking problems for 2nd and 3rd stages are studied under the condition of 1st stage cumulative errors and parameter uncertainties.The terminal height deviation is｜eh｜＜100m,the terminal velocity deviation is ｜ev｜＜20m/s and the terminal ballistic inclination is ｜eΘ｜＜0.5。.The simulation step is 1 ms and the control period is 10 ms.

The normal overload constraints in the longitudinal plane and lateral plane are both set to 5 g.Fig.3 shows the normal overloads in longitudinal plane and the lateral plane of the nominal trajectory.On this basis,when Δα∈[-5,+5]and Δβ∈[-5,+5],the normal overloads in the longitudinal plane and lateral plane are shown in Fig.4(a) and Fig.4(b).The overload constraints in the two planes are both less than 5 g.So ｜Δα｜＜5。and ｜Δβ｜＜5。are selected as the amplitude constraints of Δα and Δβ.

Fig.3.Normal overloads in longitudinal plane and lateral plane of nominal trajectory.

Fig.4.Normal overload with constrained Δα and Δβ: (a) Normal overload in longitudinal plane with Δα∈[-5,+5];(b) Normal overload in lateral plane with Δβ∈[-5,+5].

In this section,the effectiveness and advantages of the proposed method are verified by simulation of two sub-sections.Section 4.1 verifies the performance of the adaptive saturation compensator and TVTBLF.Section 4.2 verifies the robustness of the proposed DIAST controller with aerodynamic parameter and engine parameter perturbations.Section 4.3 verifies the adaptability of DIAST controller to large-range parameter uncertainties by comparing with two existing methods.

4.1.Controller performance analysis

4.1.1.Performance analysis of adaptive saturation compensator

In this section,to verify the excellent performance of the proposed adaptive saturation compensator,the following five scenarios are performed respectively for comparison.In the simulation of Case 1,the control variables have no input constraints.In Case 2,the input constraints are considered but the saturation compensating is not applied.In Case 3 and Case 4,the traditional compensators in Ref.[37]and Ref.[38]are used for comparative simulation respectively.Case 5 uses the designed adaptive fixed time adaptive saturation compensator.In these five scenarios,the uncertainty combination is selected as follows:

Tracking performance of the 3 cases are shown in Fig.5.

Fig.5.Tracking performance: (a) Altitude deviations in Case 1 and Case 2;(b) Altitude deviations in Case 3-Case 5;(c) Velocity deviations in Case 1 and Case 2;(d) Velocity deviations in Case 3 -Case 5;(e) LTIA deviations in Case 1 and Case 2;(f) LTIA deviations in Case 3 -Case 5;(g) Normal overload curves of Case 5.

As can be seen from Fig.5,altitude,velocity,and LTIA deviations are all stabilized by different controllers in Cases 1-5.The terminal velocity error without control variable constraints in Case 1 is smaller than those with control variable constraints in Case 2-Case 5.Compared Fig.5(a),Fig.5(c)and Fig.5(e)with Fig.5(b),Fig.5(d)and Fig.5(f),the overshoot in Case 1 is smallest and the convergence rate is fastest,while Case 2 has the largest overshoot and the slowest convergence rate.Part of dynamic response performance is lost when the control variable constraints are applied.In Case 3-Case 5,the saturation compensators dynamically adjust the velocity and LTIA tracking errors through feedback bounded compensation to reduce the adverse influence caused by control variable saturation.Therefore,the dynamic performance and steady-state performance are better than that in Case 2.From Fig.5(b)and Fig.5(d)and Fig.5(f),the convergence rate of altitude,velocity and LTIA in Case 5 is faster than that in Case 3 and 4,indicating that the FTASC proposed in this paper has the best compensation performance.Case 5 is taken as an example and the curves of overloads in longitudinal plane and the lateral plane are presented.The normal overloads in longitudinal plane and lateral plane do not exceed the overload limit,which meet the overload constraint requirements.

Fig.6 shows the correction commands of Case 3-Case 5.

Fig.6.AOA and AOS correction commands:(a)AOA correction commands in Case 1 and Case 2;(b)AOA correction commands in Case 3-Case 5;(c)AOS correction commands in Case 1 and Case 2;(d) AOS correction commands in Case 3 -Case 5.

From Fig.6(a) and Fig.6(c),the maximum control variable is greater than 5。and tends to be stable faster in Case 1 without control variable constraints.It indicates that the deviations converge fast.Compared with Fig.6(a) and Fig.6(c) with Fig.6(b)and Fig.6(d),the saturation time of Case 2 is significantly longer than that of Case 3-Case 5.In Fig.6(c),Case 2 even saturated twice under the condition of no saturation compensation.The saturation compensators in Case 3-Case 5 improve the desaturation rate.From Fig.6(b)and Fig.6(d)in Case 3-Case 5,the proposed FTASC has the shortest saturation time.In conclusion,the 3 saturation compensators improve the control effort,and the FTASC has the most obvious improvement effect.In addition,whent＝120 s and 123 s,Δα and Δβ have abrupt changes.It is because when the engine shuts down and enters the unpowered glide stage,both thrust and mass change abruptly.

Fig.7 shows the performance of saturation compensators of Case 3-Case 5.

Fig.7.Performance of saturation compensators:(a)δα in Case 3-Case 5;(b)δβ in Case 3-Case 5;(c)Curve of adaptive gain ω1;(d)Curve of adaptive gain ω2;(e)Curve of auxiliary variable χ1;(f) Curve of auxiliary variable χ2.

Fig.7(c) and Fig.7(d) show the adaptive gain change curves in Case 5.In order to improve the saturation compensation capability of FTASC,ω decreases when Δu is saturated and increases when Δu is unsaturated.Compared with the fixed-time anti-saturation compensator without adaptive scheme,the adaptive scheme can shorten the saturation time and improve the convergence rate of tracking errors.Take the AOA curve as an example for analysis.Whent＜82 s,δα ＞0,the auxiliary system works to compensate for the adverse effects of saturation.The decreasing of ω1makes Δα exit the saturated region faster and improves the convergence rate of tracking error.Whent＜82 s,δα ＝0 and ω1increases.FTASC can make the auxiliary variable converge to 0 faster than Case 3 and Case 4.It avoids affecting the convergence characteristics of the original closed-loop system.From Fig.7(e) and Fig.7(f) show the auxiliary variable curves.It can be concluded that when Δα and Δβ enter the saturation zones,｜χ1｜and ｜χ2｜increase under the action of compensator;when Δα and Δβ exit the saturation zones,the adaptive parameters of the compensator make χ1and χ2decay to zero rapidly.In addition,χ1and χ2strictly convergent to 0,which avoids the impact on the original dynamic system.

Fig.8 shows the Lie boundaries of F and G.

Fig.8.Lie boundaries profiles: (a) Lie boundary of F;(b) Lie boundary of G1;(c) Lie boundary of G2.

4.1.2.Performance analysis of TVTBLF

In this section,in order to demonstrate the effectiveness and superiority of the TVTBLF proposed in this paper,the following control methods are simulated and compared with traditional Logarithmic Barrier Lyapunov Function (LBLF) [36].In order to reflect fairness,the same k and fixed-time auxiliary variables are used in both cases.

Fig.9.Tracking performance with different BLFs: (a) Velocity deviations in Case 1 and Case 2;(b) LTIA deviations in Case 1 and Case 2.

Fig.10.AOA and AOS corrections with different BLFs: (a) AOS correction commands in Case 1 and Case 2;(b) AOS correction commands in Case 1 and Case 2.

4.2.Robustness analysis

4.2.1.Uncertainty combinations simulation

Simulations are conducted to verify the adaptability to complex flight environment and uncertain engine parameters.To be representative,the typical values of thrust,mass flow per second,drag and lift coefficient perturbations are selected for combined perturbation simulation.These 10 combinations are given as follow.

Fig.11 and Fig.12 show the tracking performance and correction of AOA and AOS respectively.From Figs.11(a)-Fig.11(c),it can be concluded that the controller has a good convergence performance on the altitude,velocity and LTIA deviations in the above cases.The proposed DIAST controller has good adaptability to complex flight environment and uncertain engine parameters.In the 2nd and 3rd stage,aerodynamic force decreases sharply with the increase of altitude,and the influence of aerodynamic parameters becomes more weakened.It indicates that when the engine parameters reach the boundary values,it plays a decisive role in the impact of trajectory compared to the aerodynamic parameter uncertainties.Therefore,this algorithm can suppress the great uncertainties result from engine parameters.With the increase of altitude,aerodynamic coefficient decreases gradually.In the third stage,aerodynamic force is almost zero and aerodynamic coefficient uncertainties have been eliminated.However,the engine parameter uncertainty always exists.Therefore,the terminal velocity deviation cannot be completely controlled to 0.The DIAST controller controls the velocity deviation within the range of[-2,2]m/s.The terminal velocity and LTIA of the reference trajectory are 6420 m/s and 0。,respectively.Since DIAST controller tracks Θrefand Θref(tf) ＝ 0,it can be roughly estimated that＜arctan(2/vref(tf)) ＝0.0179。.Therefore,the terminal LTIA deviation is extremely small.It can be seen from Eq.(10) that the LTIA deviation is completely controlled by Δα.Therefore,when LTIA deviation converges to near zero,the AOA also converges to near zero,which is shown in Fig.12(a).Δβ≠0 is to control the existing velocity deviation.In addition,it can be seen that the degree of convergence of the velocity deviation mainly depends on the of the engine parameter perturbation under different conditions.

Fig.11.Tracking performance of typical uncertainty values:(a)Altitude deviations;(b)Velocity deviations;(c) LTIA deviations.

Fig.12.AOA and AOS correction commands: (a) AOS correction commands;(b) AOS correction commands.

4.2.2.Monte Carlo experiment

To further verify the robustness of the controller,Monte Carlo simulation experiments are performed.The parameter values of the distributions are given as follows:

where,Ustands for uniform distribution.

The altitude,velocity and LTIA curves along 500 dispersed ascent trajectories are presented in Fig.13(a)-Fig.13(c).It can be seen that although the uncertainty values are different,all the ascent trajectories satisfy the terminal altitude,velocity and LTIA constraints accurately.Fig.14(a) and Fig.14(b) show the distribution of terminal velocity,altitude and LTIA.The terminal errors meet the accuracy index.It can be concluded that the ascent DI tacking system is stabilized under the action of DIAST controller when the uncertainties are within the given boundary values.Since each part in control law Eq.(27) is related to the uncertainty boundary values,the DIAST controller can stabilize all uncertainty combinations within the boundary.If each uncertainty combination is regarded as a sub-differential equation system of the ascent DI system,the result can further show that the stabilization process of the DI system is that of any sub-differential equation system it contains.To further illustrate the high precision of the terminal errors,the ranges of terminal state errors are shown in Table 5.To describe the terminal accuracy,the definition of error ratio is given here:

Table 5Terminal error distributions with engine parameter and aerodynamic coefficient perturbations.

Fig.13.Monte Carlo result of tracking performance: (a) Altitude deviations;(b) Velocity deviations;(c) LTIA deviations.

Fig.14.Monte Carlo result of terminal errors: (a) Terminal h-v distribution;(b) Terminal Θ-v distribution.

The first-order Taylor expansion of altitude change rate has the following approximate relationship:

Whent→tf,Θref→0.It can be approximated as cos Θref≈1 and sin Θref≈Θref.When the controller makes Δ→0，there exists

Under the action of the controller,the terminal velocity deviation is opposite to that of LTIA deviation,so as to ensure the convergence of the altitude deviation.So the distributions of vf-Θfare concentrated in the second and fourth quadrants,which are shown in Fig.14(b).

4.3.Comparison with RGSC and IGC methods

In this section,the comparison of DIAST with Integrated Guidance and Control (IGC) [39]and Robust Gain-Scheduling Control method (RGSC) 40 method are discussed,respectively.In Ref.[39]and Ref.[40],only aerodynamic force and atmospheric density deviations were considered.In order to achieve the comparison effect,the specific impulse and mass flow per second uncertainties are also taken into account.

4.3.1.Comparison with Integrated Guidance and Control method

To quantitatively describe the dynamic effects of various uncertainties on flight,an index to measure the uncertainty is given as follows,which reflects the magnitude of the uncertainties

The two items in indexpdescribe the degree of dynamic effect by various uncertainties on velocity and LTIA respectively.Case 5-Case 8 in subsection 4.2.1 are selected to compare the tracking performance of DIAST,RGSC and IGC under different magnitudes of uncertainties.The change curves ofpvs.tin Case 5-Case 8 are shown as Fig.15.

Fig.15. p vs. t curves in Case 5-Case 8.

It can be seen thatpchanges dynamically during the flight.The first stage is in the dense atmosphere.With the increase of velocity,the aerodynamic coefficient uncertainties makepincrease.When entering the second stage,the atmospheric density decreases sharply due to the increase of altitude,and the influence of aerodynamic coefficient uncertainties on flight dynamics also decreases.During the third stage,the flight dynamics are completely affected by the uncertainty of engine parameters.

A comparison of DIAST with IGC is shown in Fig.16.The disturbance caused by all uncertainties is treated as a whole,which is estimated by designing the ESO and then compensated.From Fig.16,it can be seen that the tracking performance of IGC in Case 5 and Case 6 is better than that of DIAST.The convergence rate of altitude and velocity in IGC algorithm is faster than that in DIAST.At the same time,IGC algorithm has a high terminal accuracy.It is because when the value ofpis small,ESO can accurately estimate the uncertainty and make precise compensation,which are shown in Fig.17(a) and Fig.17(b).When the magnitude of uncertainty increases further,such as 8,the parameters of ESO are not suitable for large uncertainties.From the above analysis,the of engine parameter uncertainties have great impact on ascent dynamics.If the possible fluctuation ranges of engine parameters are large,the ESO with fixed parameters cannot guarantee good observation performance for a wide uncertainty range.It can be seen that,in Fig.17(c) and Fig.17(d),the estimation errors of ESO for disturbances in velocity and LTIA channels increase gradually with time.In addition,the larger the value ofp,the greater the estimation errors are.Therefore,when the boundary ranges of uncertainties are large,it leads to the poor robustness of the algorithm in Case 8.Compared with the traditional estimation-compensation methods,although the proposed DIAST method is conservative,it can make the deviations converge when the uncertainties are in the given boundaries.

Fig.16.Tracking performance of DIAST and IGC:(a)Altitude deviations of DIAST;(b)Altitude deviations of IGC;(c)Velocity deviations of DIAST;(d)Velocity deviations of IGC;TIA deviations of DIAST;(f) LTIA deviations of IGC.

Fig.17.Tracking profiles of ESO in Case 5 and Case 8:(a)Uncertainty of the velocity channel in Case 5;(b)Uncertainty of the LTIA channel in Case 5;(c)Uncertainty of the velocity channel in Case 8;(d) Uncertainty of the LTIA channel in Case 8.

4.3.2.Comparison with robust gain-scheduling control method

In this sub-section,DIAST is compared with robust gainscheduling control (RGSC),which is developed based on the linear parameter varying system.Since altitude and LTIA are only tracked in the longitudinal plane [40],the model in paper is extended to three-dimensional trajectory.The control quantities are extended to [Δα,Δβ],and the amplitude is saturated without compensations.According to the convexity of the system,meshing is conducted on the dependent parameters in a time-varying linear matrix inequality and solved by Yalmip toolbox.

Fig.18 and Fig.19 show the tracking performance and control variable corrections of DIAST and RGSC,respectively.From(Fig.17a)-Fig.17(c),both algorithms can track altitude,velocity and LTIA deviations.The proposed DIASTalgorithm has smaller overshoot,faster convergence rate.RGSC only considers the aerodynamic perturbation without the perturbation of engine parameters.However,during the second and third stages,the uncertainties of engine parameters play a dominant role compared with those of aerodynamic coefficients and atmospheric density which is shown in Fig.14.The uncertainty of aerodynamic coefficients and atmospheric density gradually disappears with the increase of altitude.Therefore,the suppression effect of the persistent large perturbation is poor,which leads to the slower tracking accuracy to LTIA and velocity deviation.In addition,it can be observed from Fig.19(a) and Fig.19(b)that the control variables of DIAST algorithm are smoother than those of RGSC,which is more conducive to the tracking of subsequent attitude control system.Table 6 presents the value and radio of the terminal state errors when using the DIAST and RGSC.It is obvious that terminal errors of DIAST are more accurate.Meanwhile,the data in Table 6 explain the conclusions of Fig.18.

Table 6Terminal errors of comparison between DIAST and RGSC.

Fig.18.Tracking performance of DIAST and RGSC:(a)Altitude deviations;(b)Velocity deviations;(c) LTIA deviations.

Fig.19.AOA and AOS corrections of DIAST and RGSC: (a) AOS correction commands;(b) AOS correction commands.

5.Conclusions

This paper develops a stabilizing controller for an affine nonlinear DI system,which is applicable to the ascent trajectory tracking of SLVs.The simulation results show that the proposed DIAST algorithm performs well and can be implemented in ascent guidance problems.The main conclusions of this paper are summarized as follows:

(1) In view of the uncertainties of engine parameters and aerodynamic coefficients,the DI description of the ascent stabilization problem is proposed and the ascent trajectory tracking problem is converted to a DI stabilization problem.

(2) For the ascent DI system,a kind of adaptive saturated tracking controller is proposed and the closed-loop stability is proved.Within the given uncertainty boundaries,the controller can realize the velocity and LTIA deviation convergent and uniformly bounded,which ensures the actual trajectory is near the reference trajectory.The terminal velocity,LTIA and altitude errors are less than 2 m/s,0.01。and 10 m respectively,all of which meet the index requirements.

(3) Comparing with the traditional Logarithmic Barrier Lyapunov Function,the proposed time-varying Tangential-Logarithmic Barrier Lyapunov Function can dynamically limit the boundary of velocity and LTIA inclination deviation,which improves the dynamic and steady-state performance of the controller by setting the time-varying boundary parameters of exponential decay in advance.The fast and accurate tracking control of the reference ascent trajectory is realized,and the output constraint performance of the system is guaranteed.Compared with the traditional Logarithmic Barrier Lyapunov Function,the terminal velocity accuracy is improved by about 6 m/s.

(4) A fixed-time saturation compensator is designed,which shortens the saturation time of Δα and Δβ and improves the convergence rate of the tracking error.When Δα and Δβ are not saturated,the auxiliary variables of the compensator can accurately converge within a fixed time,avoiding the impact on the convergence characteristics of the original closedloop system.Compared with saturation compensator in[45,46],the saturation time of Δα and Δβ is shorten by about 2 s.

(5) Compared with Robust Gain-Scheduling Control method(RGSC) method,when the uncertainty of engine parameters reaches the limit,the control accuracy of altitude,velocity and LTIA is increased by about 70,50 and 90 times respectively when the uncertainty of engine parameters reaches the limit values.In addition,Δα and Δβ are smoother,which is beneficial to the tracking control of attitude control system.Compared with Integrated Guidance and Control (IGC)method,although the proposed DIAST method is conservative in terminal accuracy under small uncertainties,it has better robustness than IGC for large uncertainties.Under the given uncertainty quantification index,the application scope of uncertainty is broadened by 20% on the premise of the terminal accuracy index.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was supported by the National Natural Science Foundation of China(Grant Nos.61627810,61790562 and 61403096).

Appendix A

Proof sketch ofTheorem 1

Theorem 1 is proved.

Proof sketch ofTheorem 2.

According to Eq.(22) andDefinitions 1,for ∀fe∈F(e),it is obviously that Ω ＜ΩM.Thus,ΩMis the Lie boundary of F.

For ∀ge∈G,

Fig.A1. and γM i distribution diagram.

Theorem 2 is proved.