CN105116914B - A kind of stratospheric airship analytic modell analytical model predicted path tracking and controlling method - Google Patents
A kind of stratospheric airship analytic modell analytical model predicted path tracking and controlling method Download PDFInfo
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Abstract
A kind of stratospheric airship analytic modell analytical model predicted path tracking and controlling method, steps are as follows：It is given it is expected pursuit gain：It is given it is expected space arbitrary parameter path and desired dirigible speed；Stratospheric airship models：Dynamic Modeling is carried out to certain model stratospheric airship, obtains six degree of freedom nonlinear model；Guidance Law calculates：At each moment according to the calculating of the angular speed desired value at the reference point locations progress current time in the position of current stratospheric airship, posture and expectation reference path；Control law calculates：The Guidance Law and the quantity of state by being obtained by sensor measurements such as combined inertial nevigations that are calculated according to previous step and output can be surveyed, control law is calculated using analytic modell analytical model predictive control algorithm, obtain the controlled quentity controlled variable of rudder face and airscrew thrust, the input quantity being calculated is directly acted on into dirigible propeller, rudder and elevator, you can complete the path following control of dirigible.
Description
Technical Field
The invention provides a stratospheric airship analysis model prediction path tracking control method, provides a new control method for tracking a spatial parameterized path without online rolling optimization and in a closed analysis solution form for an under-actuated stratospheric airship, and belongs to the technical field of automatic control.
Background
The stratospheric airship is an aerostat which is parked in the air by virtue of air buoyancy and continuously works all day long in the stratosphere far away from the earth surface, has the advantages of moderate flight height, long task execution time, strong viability, large carrying payload and the like, and has wide military and civil prospects in the fields of communication, monitoring, traffic management and the like. The key technology for developing stratospheric airship relates to many fields such as materials, structures, energy sources, navigation and control. Among these key technologies, path tracking control of stratospheric airships is a very important part. The stratospheric airship path tracking control means that the stratospheric airship can track and finally stabilize on a preset reference path under the action of a controller. The stratospheric airship path tracking control system has the characteristics of under-actuated characteristic, strong nonlinearity, susceptibility to model parameter change, external interference influence and the like, and is a typical nonlinear system.
Aiming at the characteristics of the stratospheric airship path tracking control system, the invention provides a stratospheric airship analysis model prediction path tracking control method and provides a nonlinear airship model-based path tracking control method. The method integrates a guidance algorithm based on coordinate transformation and an analytic model predictive control theory. According to the method and the controller designed by theory, the problem of space path tracking control of the stratospheric airship can be well solved, the under-actuated stratospheric airship is gradually stabilized on a set reference path, and an effective design means is provided for engineering realization of path tracking control of the stratospheric airship.
Disclosure of Invention
(1) The purpose is as follows: the invention aims to provide a stratospheric airship analytical model prediction path tracking control method, and a control engineer can realize space path tracking control of a stratospheric airship according to theoretical steps of the method and by combining actual system parameters in actual design.
(2) The technical scheme is as follows: the invention relates to a prediction path tracking control method of an analytic model of an stratospheric airship, which mainly comprises the following steps: firstly, performing guidance navigation calculation by a given expected tracking path to generate an expected angular velocity; and performing dynamic modeling on a certain stratospheric airship to obtain a six-degree-of-freedom nonlinear model. And then taking the forward speed, the lateral speed, the yaw rate, the pitch angle rate and the roll angle rate as output variables, solving the input quantity by applying an analytical model predictive control algorithm, and acting the input quantity on a system to realize path tracking control. In practical application, the state quantities of the airship such as position, attitude, speed and the like are measured by sensors such as a combined inertial navigation system, and the control quantity calculated by the method is transmitted to actuating devices such as a steering engine and a propulsion propeller to realize the path tracking function of the airship on the stratosphere.
The invention discloses a method for controlling prediction path tracking of an analytic model of an airship on an stratosphere, which comprises the following specific steps of:
step one given the desired tracking value: giving an arbitrary parameterized path in a desired space and a desired airship speed;
modeling a stratospheric airship: performing dynamic modeling on a certain type of stratospheric airship to obtain a six-degree-of-freedom nonlinear model;
step three, calculation of guidance law: calculating the angular velocity expected value at the current moment according to the position and the posture of the current stratospheric airship and the position of a reference point on an expected reference path at each moment;
step four, calculating a control law: calculating a control law by using an analytic model predictive control algorithm according to the guidance law obtained by calculation in the last step to obtain the control quantity of the thrust of the control surface and the propeller;
wherein given the desired tracking value as described in step one, itThe tracking values include a desired path and a desired velocity value. X given an arbitrary parameterized spatial expectation pathp＝xp(l),yp＝yp(l),zp＝zp(l)，l∈[0,max]And l is a path parameter.
Given a desired velocity vd＝[ud,wd]T＝[C,0]T(C＞0)，ud,wdThe decomposition of the desired speed along the hull coordinate system.
Wherein, in the modeling of the stratospheric airship in the step two, the calculation method is as follows:
define position vector ζ ═ x, y, z]TThe coordinates of the mass center of the airship in an inertial system are obtained; velocity vector v ═ u, v, w]TIs the component of the airship's speed on the aircraft system; angular velocity vector ω ═ p, q, r]TIs the component of the angular velocity of the airship on the airplane system; euler angle γ ═ θ, ψ, φ]TThe components of (a) represent pitch, yaw and roll angles, respectively.
The kinematic equation of position is:
wherein R isgAnd (gamma) is a conversion matrix from the body coordinate system to the inertial system.
The attitude kinematics equation is:
the kinetic equation is:
whereinIs a matrix of constants, and the matrix of constants,is a non-linear function matrix with respect to ω and v. Mu.sFAnd muδThe propeller thrust and rudder deflection angle.
Wherein, in the guidance law calculation in the third step, the calculation method is as follows:
as shown in FIG. 1 of the drawings, an inertial coordinate system { I }, a velocity coordinate system { W } and a path coordinate system { F } are first defined. Definition of pFIs the projection of the distance of the airship centroid relative to a reference point on the path coordinate system. The kinematic equation for the position error is:
in the above formulaThe speed of movement of the reference point is tracked for the desired path. OmegaF/IIs the projection of the path coordinate system on the path coordinate system relative to the angular velocity of the inertial system. An auxiliary coordinate system D is defined, which is a pose used to describe the approach of the airship to the target path. Definition ofA transformation matrix from a speed coordinate system to an auxiliary coordinate system;error in true value. The kinematic equation of the attitude error is as follows:
in the above formulaIn order to be an attitude error,is composed ofThe elements of the first row and the third column of the matrix,an element of a first row and a second column of the matrix; a transformation matrix from a path coordinate system to an auxiliary coordinate system; { omega [ [ omega ] ]F/I}FThe projection of the path coordinate system on the path coordinate system relative to the angular velocity of the inertia system; { omega [ [ omega ] ]D/F}DIs the projection of the auxiliary coordinate system on the auxiliary coordinate system relative to the angular velocity of the path coordinate system; q and r are pitch and yaw rates in a velocity coordinate system.
Definition ofIs a generalized error vector. We want to design the conductivity such that xpfAsymptotically converging to zero. The desired angular velocity calculation formula is as follows:
wherein,is a normal number. The moving speed of the reference point on the desired path may be calculated by:
wherein k islIs a normal number. Due to qcAnd rcThe angular velocity is a desired angular velocity in the velocity coordinate system, and therefore, the angular velocity needs to be converted into a desired angular velocity in the body coordinate system. Definition of ωdThe expected angular speed of the airship under the coordinate system of the body. The conversion formula is as follows:
wherein, in the step four, the calculation method for solving the control law is as follows:
the six-degree-of-freedom kinetic model obtained in step one can be written in the form of the following matrix expression:
wherein x ═ u, v, w, p, q, r]T is a state quantity, [ mu ] T [ F ]x,Fz,δR,δEL,δER]T is the system input, i.e. the thrust of the airship propeller and the yaw angle of the rudder and elevator, and y is [ u, w, p, q, r ═ u, w, p, q, r]TThe output is controllable for the system.
f(x)＝[f1(x),f2(x),f3(x),f4(x),f5(x),f6(x)]TAnd h (x) ═ h1(x),h2(x),h3(x),h4(x),h5(x)]TIs a non-linear function vector with respect to the state quantity x, B ═ B1,b2,b3,b4,b5,b6]TIs a constant matrix of 6 rows and 5 columns.
Let us define the correlation of the system as ρ, the control order as κ, and T as the predicted time domain length. The system input quantity obtained according to the analytical model predictive control algorithm is as follows:
whereinAndis the derivative of lie, yd(t) is a reference output for the current time instant.Denotes yd(t) taking the i derivative over time. Where M isρIs given by:
where K is a matrixThe first 5 rows, and the two matrices are calculated by:
wherein
(3) The advantages and effects are as follows:
compared with the prior art, the invention discloses a prediction path tracking control method of an analytic model of an stratospheric airship, which has the advantages that:
1) the analytical model prediction control algorithm and the basic model prediction algorithm have the same basic principle, both adopt a rolling optimization strategy, and act the first component of the optimized solution at each sampling moment on the system, so that the rolling implementation can take uncertainty caused by model mismatch, time variation, interference and the like into consideration, make up in time, and establish new optimization on the actual basis all the time to keep the control practically optimal;
2) the analytical model predictive control algorithm is superior to the basic model predictive control algorithm in that the analytical solution form predictive control algorithm is provided, so that online optimization is not needed when the nonlinear control problem is solved by using the algorithm, and the calculated amount is saved, so that the analytical model predictive control algorithm can be used for a system with requirements on corresponding speed;
3) the method is directly based on the nonlinear model design of the stratospheric airship, and is simple and easy to design aiming at different airship model controllers.
4) The method has the advantages of simple algorithm structure, simple controller design and easy engineering realization.
The control engineer can give any expected path according to the actual airship in the application process, and directly transmits the control quantity calculated by the method to the executing mechanism to realize the function of path tracking control.
Drawings
FIG. 1 is a diagram of a guidance law computational geometry of the present invention;
FIG. 2 is a block diagram of the control method according to the present invention;
FIG. 3 is a schematic view of an stratospheric airship of the present invention;
the symbols in the figures are as follows:
p desired path reference points;
the current centroid position of the Q airship;
{ I } inertial frame;
an inertial coordinate system X-axis;
an inertial coordinate system Y-axis;
an inertial coordinate system Z axis;
{ F } path coordinate system;
the path coordinate system is tangential along the trajectory;
the path coordinate system is normal along the track;
the path coordinate system is normal along the track;
{ W } speed coordinate system;
the X axis of the speed coordinate system;
a speed coordinate system Y-axis;
a speed coordinate system Z axis;
v airship velocity;
xFposition of airship in path coordinate systemCoordinates;
yFposition of airship in path coordinate systemCoordinates;
zFposition of airship in path coordinate systemCoordinates;
pd(l) A position vector of the desired path reference point relative to the inertial coordinate system;
pIa position vector of the airship relative to an inertial coordinate system;
pFa position vector of the airship relative to the path coordinate system;
position coordinates of a Zeta airship centroid under an inertial system;
u airship forward speed;
w airship longitudinal speed;
udthe desired airship forward speed;
wdthe desired airship longitudinal speed;
ωdan expected airship angular velocity;
omega airship angular velocity;
μFpropeller thrust;
μδan airship rudder deflection angle;
a υ airship velocity vector;
gamma airship euler angles;
an ERF inertial coordinate system;
a BRF boat body coordinate system;
o airship centroid;
an X-axis of an X-ray boat body coordinate system;
a Y-axis ship body coordinate system;
a Z-axis ship body coordinate system;
Ogan inertial coordinate system origin;
xgan inertial coordinate system X-axis;
ygan inertial coordinate system Y-axis;
zgan inertial coordinate system Z axis;
p airship roll angular velocity;
q airship pitch angle velocity;
r airship yaw angular velocity;
v airship lateral velocity;
Ftpropeller thrust;
Detailed Description
The design method of each part in the invention is further explained with the attached drawings as follows:
the invention relates to a prediction path tracking control method of an analytic model of an airship on a stratosphere, which has the specific structural block diagram shown in figure 2 and comprises the following specific steps:
the method comprises the following steps: given an expected tracking value
1) As shown in FIG. 1, a relational coordinate system { I } is first established, with three coordinate axes beingAndthen given the desired spatial arbitrary parameterized path: x isp＝xp(l),yp＝yp(l),zp＝zp(l)，l∈[0,max]And l is a path parameter. And a path coordinate system F is established along the path. Defining the path coordinate system as three coordinate axesAndand the three vectors satisfy the following equations:
wherein k is1(l)、k2(l) And the curvature κ (l) and the flexibility τ (l) of the space curve satisfy the following relationship:
therefore, we can obtain the transformation matrix from the path coordinate system to the inertia systemAnd the angular velocity of the path coordinate system relative to the inertial system is expressed in the path coordinate system as
2) Given a desired velocity vd＝[ud,wd]T＝[C,0]T(C＞0)，ud,wdThe decomposition of the desired speed along the hull coordinate system.
Step two: stratospheric airship modeling
Fig. 3 is a schematic view of a stratospheric airship. The airship object adopts a traditional ellipsoid structure and is symmetrical about a longitudinal plane, the empennage adopts a cross-shaped layout with an elevator and a rudder, the nacelle is positioned below the airship capsule, and two sides of the nacelle are respectively provided with a pair of propellers.
Defining an inertial frame of coordinates ERF ═ OgxgygzgAnd a boat body coordinate system BRF ═ Oxyz }. Define position vector ζ ═ x, y, z]TThe coordinates of the mass center of the airship in an inertial system are obtained; velocity vector v ═ u, v, w]TIs the component of the airship's speed on the aircraft system; angular velocity vector ω ═ p, q, r]TIs the component of the angular velocity of the airship on the airplane system; euler angle γ ═ θ, ψ, φ]TThe components of (a) represent pitch, yaw and roll angles, respectively.
The kinematic equation of position is:
wherein R isgAnd (gamma) is a conversion matrix from the body coordinate system to the inertial system.
The attitude kinematics equation is:
the kinetic equation is:
whereinIs a matrix of constants, and the matrix of constants,is a non-linear function matrix with respect to ω and v. Mu.sFAnd muδThe propeller thrust and rudder deflection angle. Wherein muF＝[FT,x,FT,z]T，μδ＝[δR,δEL,δER]T。FT,xThe component of the thrust of the propeller on the Ox axis of the boat body coordinate system, FT,zThe component delta of the thrust of the propeller on the Oz axis of the boat body coordinate systemRIs the rudder angle, delta, of the airship rudderELRudder angle, delta, for the left elevator of an airshipERThe rudder deflection angle of the elevator on the right side of the airship. The specific values of all the items in the dynamic model equation are different with different airship structures and parameters, and are determined according to actual conditions in practical application.
Step three: guidance law calculation
As shown in FIG. 1 of the drawings, an inertial coordinate system { I }, a velocity coordinate system { W } and a path coordinate system { F } are first defined. Defining the projection of the distance of the airship centroid relative to a reference point on the path on a path coordinate system as { p }F}F＝[xF,yF,zF]T. From the geometric relationships in FIG. 1, p can be derivedI＝pd(l)+pF. The derivation of both sides can result in:
the kinematic equation for the position error can then be found as:
in the above formulaThe speed of movement of the reference point is tracked for the desired path. OmegaF/IIs the projection of the path coordinate system on the path coordinate system relative to the angular velocity of the inertial system. The above formula relative path coordinate system is developed to obtain:
in order to derive the attitude error equation, an auxiliary coordinate system { D } is first defined, which is used to describe the attitude of the airship approaching the target path. The auxiliary coordinate system has its origin at the center of mass of the airship and is composed of three mutually orthogonal vectorsTo indicate. The three vectors are defined as follows:
where d > 0 is a constant, a very important design parameter. From these three vectors, a transformation matrix from the auxiliary coordinate system to the path coordinate system can be calculated
Definition ofIs a transformation matrix from the velocity coordinate system to the auxiliary coordinate system, thenCan be expressed as:
definition ofThe calculation formula is as follows:
wherein, is composed ofThe first row and the first column of elements of the matrix.
The derivation is carried out on two sides of the above formula to obtain:
wherein,for attitude error, { omega }W/D}WIs a representation of the angular velocity of the velocity coordinate system relative to the secondary coordinate system in the velocity coordinate system. They are calculated by the following formulas, respectively:
substituting (27) into (26) results in the attitude error equation:
definition ofIs a generalized error vector. We want to design the conductivity such that xpfAsymptotically converging to zero. The desired angular velocity calculation formula is as follows:
wherein,is a normal number. The moving speed of the reference point on the desired path may be calculated by:
wherein k islIs a normal number. Due to qcAnd rcThe angular velocity is a desired angular velocity in the velocity coordinate system, and therefore, the angular velocity needs to be converted into a desired angular velocity in the body coordinate system. Definition of ωdThe expected angular speed of the airship under the coordinate system of the body. The conversion formula is as follows:
step four: solving the law of control
The six-degree-of-freedom kinetic model obtained in step one can be written in the form of the following matrix expression:
wherein x ═ u, v, w, p, q, r]TIs a state quantity, [ mu (t) ]x,Fz,δR,δEL,δER]TThe system input is the thrust of the airship propeller and the deflection angles of the rudder and the elevator, and y is [ u, w, p, q, r ═]TThe output is controllable for the system.
f(x)＝[f1(x),f2(x),f3(x),f4(x),f5(x),f6(x)]TAnd h (x) ═ h1(x),h2(x),h3(x),h4(x),h5(x)]TIs a non-linear function vector with respect to the state quantity x, B ═ B1,b2,b3,b4,b5,b6]TIs a constant matrix of 6 rows and 5 columns.
Firstly, defining the performance index of the rolling time domain as follows:
whereinIn order to output the predicted value,to expect to output a prediction value, T is the prediction temporal length. The model predictive control algorithm is mainly used for finding out predictive inputSo thatThe performance index takes a minimum value. Similar to other rolling horizon controls, the actual control input is the optimal control inputWhen τ is equal to 0The value of (c).
The output equation is derived as:
the relative order ρ of the system is 1 from the above equation. Assume that the control order is κ.
The system input quantity obtained according to the analytical model predictive control algorithm is as follows:
whereinAndis the derivative of lie, yd(t) is a reference output for the current time instant.Denotes yd(t) taking the i derivative over time. Where M isρIs given by:
where K is a matrixThe first 5 rows, and the two matrices are calculated by:
wherein
Where the relative order is 1. The control order is a variable parameter, the larger the control order is, the better the control precision is, but the reasonable selection can ensure the index convergence of the closed-loop system. The desired output at each time is calculated by steps 1 and 3, in which the desired values of the forward speed and the longitudinal speed are not changed to constant values, the desired value of the angular speed is calculated by the outer loop guidance law for each time, and it is assumed that the desired output is not changed in the prediction time domain. The desired output may be expressed as: y isd＝[ud,wd,pd,qd,rd]T＝[C,0,pd,qd,rd]T。
Assuming that the derivative of the desired output is zero, we can get the final system inputs as:
at each moment, the state quantity and the measurable output are measured by a sensor such as a combined inertial navigation sensor, and the system input quantity at the current moment is calculated through the measured values. And (4) directly acting the calculated input quantity on the airscrew, the rudder and the elevator of the airship, and finishing the path tracking control of the airship.
Claims (4)
1. A stratospheric airship analytic model prediction path tracking control method is characterized by comprising the following steps: the method comprises the following specific steps:
step one given the desired tracking value: giving an arbitrary parameterized path in a desired space and a desired airship speed;
step two, modeling of stratospheric airship: performing dynamic modeling on a certain type of stratospheric airship to obtain a six-degree-of-freedom nonlinear model;
step three, calculation of guidance law: calculating the angular velocity expected value at the current moment according to the position and the posture of the current stratospheric airship and the position of a reference point on an expected reference path at each moment;
step four, calculating a control law: calculating a control law by using an analytic model predictive control algorithm according to the guidance law obtained by calculation in the last step to obtain the control quantity of the thrust of the control surface and the propeller;
given the expected tracking value in step one, it is specifically:
1) firstly, a relation coordinate system { I } is established, and three coordinate axes thereof areAndthen given the desired spatial arbitrary parameterized path: x isp＝xp(l),yp＝yp(l),zp＝zp(l)，l∈[0,max]L is a path parameter; and establishing a path coordinate system { F } along the path; defining the path coordinate system as three coordinate axesAndand the three vectors satisfy the following equations:
wherein k is1(l)、k2(l) And the curvature κ (l) and the flexibility τ (l) of the space curve satisfy the following relationship:
transformation matrix from path coordinate system to inertial systemAnd the angular velocity of the path coordinate system relative to the inertial system is expressed in the path coordinate system as
2) Given a desired velocity vd＝[ud,wd]T＝[C,0]T(C＞0)，ud,wdThe decomposition of the desired speed along the hull coordinate system.
2. The stratospheric airship analytical model prediction path tracking control method according to claim 1, characterized in that: in the modeling of the stratospheric airship in the step two, the calculation method is as follows:
defining an inertial frame of coordinates ERF ═ OgxgygzgA boat body coordinate system BRF ═ Oxyz }; define position vector ζ ═ x, y, z]TThe coordinates of the mass center of the airship in an inertial system are obtained; velocity vector v ═ u, v, w]TIs the component of the airship's speed on the aircraft system; angular velocity vector ω ═ p, q, r]TIs the component of the angular velocity of the airship on the airplane system; euler angle γ ═ θ, ψ, φ]TThe components of (a) represent pitch angle, yaw angle and roll angle, respectively;
the kinematic equation of position is:
wherein R isg(gamma) is a conversion matrix from a body coordinate system to an inertial system;
the attitude kinematics equation is:
the kinetic equation is:
whereinIs a matrix of constants, and the matrix of constants,a non-linear function matrix with respect to ω and υ; mu.sFAnd muδThe propeller thrust and rudder deflection angle; wherein muF＝[FT,x,FT,z]T，μδ＝[δR,δEL,δER]T；FT,xThe component of the thrust of the propeller on the Ox axis of the boat body coordinate system, FT,zThe component delta of the thrust of the propeller on the Oz axis of the boat body coordinate systemRIs the rudder angle, delta, of the airship rudderELThe rudder deflection angle of the elevator on the left side of the airship is the rudder deflection angle of the elevator on the right side of the airship; the specific values of all the items in the dynamic model equation are different with different airship structures and parameters, and are determined according to actual conditions in practical application.
3. The stratospheric airship analytical model prediction path tracking control method according to claim 1, characterized in that: the guidance law calculation described in step three is as follows:
firstly, defining an inertia coordinate system { I }, a speed coordinate system { W } and a path coordinate system { F }; defining the projection of the distance of the airship centroid relative to a reference point on the path on a path coordinate system as { p }F}F＝[xF,yF,zF]T(ii) a Obtaining p according to the geometrical relationship of the three coordinate systemsI＝pd(l)+pF(ii) a Deriving both sides to obtain:
the kinematic equation for the position error is then obtained as:
in the above formulaTracking a movement speed of a reference point on the desired path; omegaF/IThe projection of the path coordinate system on the path coordinate system relative to the angular velocity of the inertia system; the above formula relative path coordinate system is developed to obtain:
in order to derive a posture error equation, firstly, an auxiliary coordinate system { D } needs to be defined, wherein the coordinate system is used for describing the posture of the airship approaching the target path; the auxiliary coordinate system has its origin at the center of mass of the airship and is composed of three mutually orthogonal vectors To represent; the three vectors are defined as follows:
where d > 0 is a constant, a very important design parameter; calculating a transformation matrix from the auxiliary coordinate system to the path coordinate system by the three vectors
Definition ofIs a transformation matrix from the velocity coordinate system to the auxiliary coordinate system, thenExpressed as:
definition ofThe calculation formula is as follows:
wherein, is composed ofAn element of a first row and a first column of the matrix;
the derivation is carried out on two sides of the above formula to obtain:
wherein,for attitude error, { omega }W/D}WIs the representation of the angular velocity of the velocity coordinate system relative to the auxiliary coordinate system under the velocity coordinate system; they are calculated by the following formulas, respectively:
substituting (14) into (13) results in the attitude error equation as:
definition ofIs a generalized error vector; we want to design the conductivity such that xpf(ii) progressively converge to zero; the desired angular velocity calculation formula is as follows:
wherein,is a normal number; the moving speed of the reference point on the desired path is calculated by:
wherein k islIs a normal number; due to qcAnd rcThe angular velocity is expected under a velocity coordinate system, so the angular velocity is also required to be converted into the expected angular velocity under a body coordinate system; definition of ωdThe expected angular velocity of the airship under the body coordinate system is obtained; the conversion formula is as follows:
4. the stratospheric airship analytical model prediction path tracking control method according to claim 1, characterized in that: the control law solved in the fourth step is calculated as follows:
writing the six-degree-of-freedom dynamic model obtained in the step two into the following matrix expression form:
wherein x ═ u, v, w, p, q, r]TIs a state quantity, [ mu (t) ]x,Fz,δR,δEL,δER]TThe system input is the thrust of the airship propeller and the deflection angles of the rudder and the elevator, and y is [ u, w, p, q, r ═]TThe output quantity of the system can be controlled;
f(x)＝[f1(x),f2(x),f3(x),f4(x),f5(x),f6(x)]Tand h (x) ═ h1(x),h2(x),h3(x),h4(x),h5(x)]TIs a non-linear function vector with respect to the state quantity x, B ═ B1,b2,b3,b4,b5,b6]TA constant matrix with 6 rows and 5 columns;
firstly, defining the performance index of the rolling time domain as follows:
whereinIn order to output the predicted value,to output a predicted value as desired, T is the predicted time domain length; the model predictive control algorithm is mainly used for finding out predictive inputMaking the performance index obtain the minimum value; similar to other rolling horizon controls, the actual control input is the optimal control inputWhen τ is equal to 0A value of (d);
the output equation is derived as:
the relative order rho of the system is 1 according to the equation; assume a control order of κ; obtained by Taylor expansion
Obtaining system input quantity according to an analytic model predictive control algorithm as follows:
whereinAndis the derivative of lie, yd(t) is a reference output at the current time;denotes yd(t) taking the i-th derivative over time; where M isρIs given by:
where K is a matrixThe first 5 rows, and the two matrices are calculated by:
wherein
Where the relative order is 1; the control order is a variable parameter, the larger the control order is, the better the control precision is, but reasonable selection can ensure the closed-loop system index convergence; the expected output at each moment is obtained by calculation in the steps 1 and 3, wherein the expected values of the forward speed and the longitudinal speed are not changed into constant values, the expected value of the angular speed is obtained by calculation through an outer loop guidance law at each moment, and the expected output is assumed to be unchanged in a prediction time domain; the desired output is represented as: y isd＝[ud,wd,pd,qd,rd]T＝[C,0,pd,qd,rd]T；
Assuming that the derivative of the desired output is zero, we get the final system inputs as:
at each moment, the state quantity and the measurable output are measured by the combined inertial navigation sensor, and the system input quantity at the current moment is calculated through the measured values; and (4) directly acting the calculated input quantity on the airscrew, the rudder and the elevator of the airship, and finishing the path tracking control of the airship.
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