CN108415252B - Electro-hydraulic servo system model prediction control method based on extended state observer - Google Patents

Electro-hydraulic servo system model prediction control method based on extended state observer Download PDF

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CN108415252B
CN108415252B CN201810149184.6A CN201810149184A CN108415252B CN 108415252 B CN108415252 B CN 108415252B CN 201810149184 A CN201810149184 A CN 201810149184A CN 108415252 B CN108415252 B CN 108415252B
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姚建勇
顾伟伟
吴昊
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Nanjing University of Science and Technology
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Abstract

The invention discloses an electro-hydraulic servo system model prediction control method based on an extended state observer, which comprises the steps of firstly establishing a dynamic mathematical model of a hydraulic system and discretizing the dynamic mathematical model by using a first-order Euler method; then designing a model predictive controller with state constraints based on the mathematical model; and finally designing a discrete extended state observer based on a mathematical model. The invention integrates the observation thought of the extended state on the basis of the traditional model prediction controller, and controls the output compensation by skillfully designing a model prediction equation and utilizing the interference estimation of the extended state observer, so that the control performance of the system is not influenced and higher steady-state control precision is still kept under the conditions of state constraint, matching interference and the like; the method enhances the inhibition effect of the traditional model predictive control on the undetectable external interference, can simultaneously process state constraint and inhibit the influence of the undetectable external interference on the system control performance, and obtains good tracking performance.

Description

Electro-hydraulic servo system model prediction control method based on extended state observer
Technical Field
The invention relates to the technical field of electro-hydraulic servo control, in particular to an electro-hydraulic servo system model prediction control method based on an extended state observer.
Background
The application of the electro-hydraulic servo system has a history of nearly one hundred years, and the electro-hydraulic servo system has the advantages of light weight, small size, quick response, high load rigidity and the like, so the electro-hydraulic servo system is widely applied to national defense equipment and civil industry, such as: the constant frequency and constant speed regulation of the launching platform, the human body induction device and the engine are all controlled by hydraulic pressure. The electro-hydraulic servo system is a typical nonlinear system and has a plurality of nonlinear characteristics and model uncertainty; the nonlinear characteristics mainly comprise friction nonlinearity, pressure flow nonlinearity and the like, and the model uncertainty can be divided into parameter uncertainty and uncertain nonlinearity, wherein the parameter uncertainty mainly comprises a viscous friction coefficient of an actuator, an elastic modulus of hydraulic oil, a leakage coefficient and the like, and the uncertain nonlinearity mainly comprises unmodeled friction dynamics, external interference, system high-order dynamics and the like. These non-linear characteristics exist in all hydraulic systems and influence the development of electro-hydraulic servo systems towards high precision and high frequency response. Meanwhile, the control performance of the electro-hydraulic servo system is also influenced by the existence of actuator saturation. With the development of society, the control performance requirements of the industry community on the electro-hydraulic servo system are higher, and the traditional classical control theory is difficult to meet the requirements, so that the research of more advanced nonlinear control theory aiming at the nonlinear characteristics in the electro-hydraulic servo system is urgent.
Many methods have been proposed in succession to address the problem of nonlinear control of electro-hydraulic servo systems. The self-adaptive control method is an effective method for processing the problem of uncertainty of parameters, and can obtain the steady-state performance of asymptotic tracking. But the self-adaptive control method is not satisfactory to the problems of uncertainty nonlinearity and state constraint, and the physical performance of the actual electro-hydraulic servo system is limited and has uncertainty nonlinearity, so that the self-adaptive control method cannot obtain high-precision control performance in actual application; as a robust control method, the classical sliding mode control can effectively process model uncertainty and external interference and obtain the steady-state performance of asymptotic tracking, but a discontinuous controller designed by the classical sliding mode control easily causes the flutter problem of a sliding mode surface, so that the control performance of a system is influenced. In order to solve the problems of parameter uncertainty and uncertainty nonlinearity simultaneously, an adaptive robust control method is provided, and the control method can enable a system to obtain better transient and steady-state performance under the condition that the parameter uncertainty and external interference exist simultaneously. But the method is still stranded when dealing with state constraints; later researchers combined the obstacle Lyapunov function with adaptive robust control to solve the problems of state constraint and model uncertainty, and although the method can constrain state variables, the method puts higher requirements on initial values of the system.
The traditional Model Prediction (MPC) control method can effectively deal with the constraint problem existing in the controlled system. MPC is widely applied in the chemical field and the slow time-varying industrial field at the early stage; with the progress of science and technology, the performance of computers is improved, and the computers are gradually applied to the fields of motors, robots, unmanned driving and the like; the MPC mainly derives a model prediction equation based on a state equation; then, the quadratic equation is solved by considering input constraint or state variable constraint, so that the advanced prediction and output planning of future output are achieved. The process constraint problem is a big advantage of Model Predictive (MPC) control; however, the traditional model prediction control cannot well solve the problem of non-measurable time-varying interference in system control, so that the control effect is poor when the system is in a severe external environment.
Disclosure of Invention
The invention aims to provide a model prediction control method of an electro-hydraulic servo system with strong robustness, anti-saturation and high tracking performance, and skillfully combines an extended state observer to solve the problems of undetectable state and interference of the electro-hydraulic servo system in practical application so as to realize high-precision control of the electro-hydraulic servo system.
In order to realize the scheme, the technical scheme adopted by the invention is as follows:
an electro-hydraulic servo system model prediction control method based on an extended state observer comprises the following steps:
step 1, establishing a dynamic mathematical model of a hydraulic system and discretizing the dynamic mathematical model by using a first-order Euler method;
step 2, designing a model predictive controller with state constraint based on a mathematical model;
and 3, designing a discrete extended state observer based on the mathematical model.
Further, step 1, a dynamic mathematical model of the hydraulic system is established and discretized by using a first-order euler method, which specifically comprises the following steps:
step 1-1, for a typical electro-hydraulic servo system, driving an inertial load by a valve-controlled hydraulic actuator; therefore, according to newton's second law, the system's equation of motion is:
Figure BDA0001579529960000021
in formula (1): m is the inertial load mass; pLThe pressure difference of the two hydraulic cavities is shown, and B is a viscous friction coefficient; a is the effective piston area of the hydraulic cylinder; f (t)) Other unmodeled friction and interference; y is the displacement of the inertial load; t is a time variable; neglecting the leakage of the hydraulic system, the pressure dynamic equation of the two cavities of the hydraulic actuator is as follows:
Figure BDA0001579529960000022
in formula (2): vtIs the sum of the two cavity volumes of the actuator; beta is aeIs the effective elastic modulus of the hydraulic oil; ctIs the internal leakage coefficient; qLIs the load flow of the system; q (t) is the model error; since a highly responsive servo valve is used, it is assumed here that the control input is proportional to the spool displacement of the servo valve, i.e. xv＝kiu; thus QLIt can be calculated by the following equation:
Figure BDA0001579529960000031
in formula (3): k is a radical oftThe total flow gain; psSupplying oil pressure to the system; prIs the system return pressure; cdIs the flow coefficient; omega is the valve core area gradient; ρ is the density of the oil; k is a radical ofiIs a proportionality coefficient; where sign (u) is defined as:
Figure BDA0001579529960000032
(2.2) defining state variables:
Figure BDA0001579529960000033
equation of motion (1) is converted to an equation of state:
Figure BDA0001579529960000034
in formula (5):
Figure BDA0001579529960000035
Figure BDA0001579529960000036
d (t) is the total interference and model error of the system; the method can obtain the following steps by adopting a first-order Euler discrete method:
Figure BDA0001579529960000037
in formula (6): t issTo sample time, Ad＝I3×3+TsA，Bud＝TsBu，Bdn＝TsBd，Cd＝C；I3×3A unit vector of third order;
for the controller design, assume the following:
assume that 1: the total disturbance in the system can be estimated by an observer, wherein the estimation error of the observer is not considered in the design of the controller, namely:
Figure BDA0001579529960000041
in formula (7):
Figure BDA0001579529960000042
is an estimated disturbance of the observer; w (k) is observer interference estimation error;
assume 2: according to the principle of model prediction, the latest measured value is required to be used as an input initial value, and the future time is predicted through a certain time; therefore, the prediction time domain is set to Np, the control time domain of the system is Nc, and Nc is less than or equal to Np; for the subsequent design of the controller it is assumed that:
Figure BDA0001579529960000043
further, the step 2 of designing a model predictive controller with state constraints based on the mathematical model comprises the following steps:
defining: Δ x (k) ═ x (k) — x (k-1) is the increment of the two time states, and similarly, Δ u (k) ═ u (k) — u (k-1) and Δ d (k) ═ d (k) — d (k-1) are the two time increments of input and interference, respectively; the discretized model of equation (6) can be derived:
Figure BDA0001579529960000044
defining: Δ x (k +1| k) represents the prediction Δ x (k +1) at time Δ x (k) and y (k +1| k) represents the prediction y (k +1) at time Δ x (k) and thus the following equation can be obtained:
Figure BDA0001579529960000045
by substituting equation (10) for equation (9), a predicted value of output y can be obtained, that is:
Figure BDA0001579529960000051
defining:
Ye(k+1|k)＝[y(k+1|k),y(k+2|k),...,y(k+Np|k)]T
ΔU(k)＝[Δu(k),Δu(k+1),...,Δu(k+Nc-1)]T
the state prediction equation is therefore:
Ye(k+1|k)＝HxΔx(k)+HIy(k)+HdnΔd(k)+HuΔU(k) (12)
in formula (12): hI、Hx、HuAnd HdnCan be derived from formula (11); due to system state x2And x3The estimation is carried out by a subsequent state observer, so that an estimation error exists; defining:
Figure BDA0001579529960000052
Figure BDA0001579529960000053
for the state increment of the state estimation,
Figure BDA0001579529960000054
estimating the state increment error, defining:
Figure BDA0001579529960000055
considering the estimation error approximation as interference, where γ is the gain term; from the formula (12), the following formula can be obtained:
Figure BDA0001579529960000056
to track target instruction xdDefining an objective function to reflect the control performance of the system, and making next control prediction input according to the objective function; the invention defines the objective function as follows:
Figure BDA0001579529960000057
in formula (14): r and Q are each NpOrder tracking error sum NcDetermining an expected target by adjusting a diagonal weight matrix of the order control increment; further, Xd(k +1) is the reference instruction target. In order to solve the quadratic form of the formula (13), it is necessary to substitute the formula (13) into the formula (14) and reduce the formula to a standard form by collation, that is:
J＝ΔU(k)THΔU(k)-G(k+1|k)TΔU(k) (15)
in formula (15):
H＝Hu TRTRHu+QTQ
G(k+1|k)＝2Hu TRTREp(k+1|k)
Figure BDA0001579529960000061
considering the state constraints:
Xmin≤Xi≤Xmax i＝1,2,3
wherein:
Xi＝[xij,xij,..,xij]T j＝1,2,...,Np
Xmin＝[xmin,xmin,..,xmin]T
Xmax＝[xmax,xmax,..,xmax]T
Xd(k+1)＝[xd,xd,...,xd]T
the reasoning method according to model prediction can obtain:
Figure BDA0001579529960000062
in formula (16): hix,Hidn,HiuCan also be prepared fromd、BudAnd BdnExpressed, substituting it into the state constraint described above yields:
Figure BDA0001579529960000063
can obtain the following products by arrangement:
EΔU(k)≤F (17)
in formula (17):
Figure BDA0001579529960000064
considering the state constraint by taking J as an optimization target based on a Hildreth quadratic solving method, and solving a delta U control increment sequence as follows:
Figure BDA0001579529960000065
Figure BDA0001579529960000066
in formula (18): h isijRepresentation matrix E (2H)-1ETThe ith and j elements of (1); n is matrix E (2H)-1ETThe number of columns; k is a radical ofiIs a vector (F + EH)-1G) The ith element of (1); n is a natural number; in formula (19): u (k) is the control law at the current time.
Further, the step 3 of designing the discrete extended state observer based on the mathematical model specifically includes:
the following formula (5) can be obtained, and the specific design is as follows:
Figure BDA0001579529960000071
in equation (20), h (t) is the derivative of the interference d (t), and is defined as:
Figure BDA0001579529960000072
Figure BDA0001579529960000073
X＝[x1,x2,x3,x4]T(ii) a Therefore, it can be obtained from the formula (20):
Figure BDA0001579529960000074
the discretized model obtained by the first-order euler method is:
X(k+1)＝AodX(k)+G(u,x)du(k)+Δd(k) (22)
in formula (22):
Figure BDA0001579529960000075
the observer is therefore designed as follows:
Figure BDA0001579529960000076
in the formula (23)
Figure BDA0001579529960000077
i is 1,2,3,4 is a design parameter, Ho is a gain of the observer, and ω o>0 is the observer bandwidth; defining:
Figure BDA0001579529960000078
subtracting equation (23) from equation (22) yields the following equation:
Figure BDA0001579529960000081
defining:
Figure BDA0001579529960000082
representing the estimation error, the following equation is obtained from equation (24):
Figure BDA0001579529960000083
in formula (25):
Figure BDA0001579529960000084
selecting
Figure BDA0001579529960000085
i is 1,2,3,4 such that AooSatisfying the Helverz matrix, there must be a matrix P that satisfies the Lyapunov equation, i.e.:
Figure BDA0001579529960000086
in the formula (26), I4×4Is an identity matrix of order 4.
Compared with the prior art, the invention has the following remarkable advantages: the invention provides an electro-hydraulic servo system model prediction control method (ESOMPC) based on an extended state observer, which integrates the idea of the Extended State Observer (ESO) on the traditional model prediction control Method (MPC), and enables the control performance of the system to be unaffected and still maintain higher steady-state control precision under the conditions of state constraint, matching interference and the like of the system by designing a model prediction equation and utilizing the interference estimation of the extended state observer as control output compensation.
Drawings
FIG. 1 is a schematic diagram of an electro-hydraulic servo control system.
FIG. 2 is a schematic diagram of a model predictive control method of an electro-hydraulic servo system in an extended state observer.
FIG. 3 shows that the system interference is d (t) sin (2t) [1-exp (-0.01 t)3)]In time, the tracking process schematic diagram of the expected instruction is output by the system under the action of the ESOMPC controller designed by the invention.
FIG. 4 shows that the system interference is d (t) sin (2t) [1-exp (-0.01 t)3)]In time, the comparison curve graph of the change of the tracking error of the system along with time under the control of the ESOMPC controller and the PID is designed.
FIG. 5 shows that the system interference is d (t) sin (2t) [1-exp (-0.01 t)3)]The invention designs an estimation situation chart of the state and the interference under the action of the ESOMPC controller.
FIG. 6 shows the system speed state constraint and system disturbance d (t) sin (2t) [1-exp (-0.01 t)3)]Then (c) is performed. The state variable x of the system under the action of the ESOMPC controller designed by the invention2Graph over time.
FIG. 7 illustrates the system speed state constraint and system disturbance d (t) sin (2t) [1-exp (-0.01 t)3)]The invention also discloses a curve graph of the change of the control output of the system along with time under the action of the ESOMPC controller.
FIG. 8 illustrates the system speed state constraint and system disturbance d (t) sin (2t) [1-exp (-0.01 t)3)]The comparison curve graph of the tracking error of the system along with the change of time under the control of the ESOMPC controller and the PID is designed.
Fig. 9 shows the system interference d (t) t4sin(4t)[1-exp(-0.01t3)]Times ESOMPC controllerGraph of the tracking error of the system over time under influence.
Fig. 10 shows system interference as d (t) ═ t4sin(4t)[1-exp(-0.01t3)]The tracking error of the system is compared with a curve graph under the action of the ESOMPC controller, the MPC controller and the PID controller designed by the invention.
Detailed Description
With reference to fig. 1-2, the method for model predictive control of the electro-hydraulic servo system based on the extended state observer comprises the following steps:
step 1, establishing a dynamic mathematical model of a hydraulic system and discretizing the dynamic mathematical model by using a first-order Euler method;
(1.1) FIG. 1 is a typical electro-hydraulic servo system in which inertial loads are driven by a valve-controlled hydraulic actuator; therefore, according to newton's second law, the equation of motion of the hydraulic system is:
Figure BDA0001579529960000091
in formula (1): m is an inertial load parameter; pLThe pressure difference of the two hydraulic cavities is shown, and B is a viscous friction coefficient; a is the effective piston area; (t) other unmodeled friction and interference; y is the displacement of the inertial load; t is a time variable.
Neglecting the leakage of the hydraulic actuator, the pressure dynamic equation of the two cavities of the hydraulic actuator is:
Figure BDA0001579529960000092
in formula (2): vtIs the sum of the two cavity volumes of the actuator; beta is aeIs the effective elastic modulus of the hydraulic oil; ctIs the internal leakage coefficient; qLIs the load flow of the system; q (t) is the model error; since a highly responsive servo valve is used, it is assumed here that the control input is proportional to the spool displacement of the servo valve, i.e. xv＝kiu; thus, QLIt can be calculated by the following equation:
Figure BDA0001579529960000101
in formula (3): k is a radical oftThe total flow gain; psSupplying oil pressure to the system; prIs the system return pressure; cdIs the flow coefficient; omega is the valve core area gradient; ρ is the density of the oil; k is a radical ofiIs a proportionality coefficient; where sign (u) is defined as:
Figure BDA0001579529960000102
(1.2) defining state variables:
Figure BDA0001579529960000103
the equation of motion of equation (1) is converted into a state equation:
Figure BDA0001579529960000104
in formula (5):
Figure BDA0001579529960000105
Figure BDA0001579529960000106
d (t) is the total disturbance of the system, including disturbance due to external load, unmodeled friction, unmodeled dynamics, and deviation of the actual parameters of the system from the modeled parameters; the method can obtain the following steps by adopting a first-order Euler discrete method:
Figure BDA0001579529960000107
in formula (6): t issTo sample time, Ad＝I3×3+TsA，Bud＝TsBu，Bdn＝TsBd，Cd＝C；I3×3A unit vector of third order;
for ease of controller design, assume the following:
assume that 1: the total disturbance in the system can be estimated by an observer, wherein the estimation error of the observer is not considered in the design of the controller, namely:
Figure BDA0001579529960000111
in formula (7):
Figure BDA0001579529960000112
is an estimated disturbance of the observer; w (k) is observer interference estimation error;
assume 2: according to the principle of model prediction, the latest measured value is required to be used as an input initial value, and the future time is predicted through a certain time; therefore, the prediction time domain is set to Np, the control time domain of the system is Nc, and Nc is less than or equal to Np; for the subsequent design of the controller it is assumed that:
Figure BDA0001579529960000113
step 2, designing a model predictive controller with state constraint based on a mathematical model, and comprising the following steps:
defining: Δ x (k) ═ x (k) — x (k-1) is the increment of the two time states, and similarly, Δ u (k) ═ u (k) — u (k-1) and Δ d (k) ═ d (k) — d (k-1) are the two time increments of input and interference, respectively; the discretized model of equation (6) can be derived:
Figure BDA0001579529960000114
defining Δ x (k +1| k) to represent the prediction of Δ x (k +1) at time Δ x (k) at time k, and similarly y (k +1| k) to represent the prediction of y (k +1) at time Δ x (k) at time k; the recurrence equation can thus be found as follows:
Figure BDA0001579529960000115
by substituting equation (10) for equation (9), a predicted value of output y can be obtained, that is:
Figure BDA0001579529960000121
defining:
Ye(k+1|k)＝[y(k+1|k),y(k+2|k),...,y(k+Np|k)]T
ΔU(k)＝[Δu(k),Δu(k+1),...,Δu(k+Nc-1)]T
the state prediction equation is therefore:
Ye(k+1|k)＝HxΔx(k)+HIy(k)+HdnΔd(k)+HuΔU(k) (12)
in formula (12): hI、Hx、HuAnd HdnCan be derived from formula (11); due to system state x2And x3The estimation is carried out by a subsequent state observer, so that an estimation error exists; defining:
Figure BDA0001579529960000122
Figure BDA0001579529960000123
for the state increment of the state estimation,
Figure BDA0001579529960000124
estimating the state increment error, defining:
Figure BDA0001579529960000125
considering the estimation error approximation as interference; wherein γ is a gain term; from the formula (12), the following formula can be obtained:
Figure BDA0001579529960000126
to track target instruction xdDefining an objective function to inverseMapping the system control performance, and making the next control prediction input according to the objective function; the invention defines the objective function as follows:
Figure BDA0001579529960000127
in formula (14): r and Q are each NpOrder tracking error sum NcDetermining an expected target by adjusting a diagonal weight matrix of the order control increment; in addition Xd(k +1) is the reference instruction target. In order to solve the quadratic form of expression (14), it is necessary to substitute expression (13) into expression (14) and reduce it to a standard form by collation, that is:
J＝ΔU(k)THΔU(k)-G(k+1|k)TΔU(k) (15)
in formula (15):
H＝Hu TRTRHu+QTQ
G(k+1|k)＝2Hu TRTREp(k+1|k)
Figure BDA0001579529960000131
considering the state constraints:
Xmin≤Xi≤Xmax i＝1,2,3
wherein:
Xi＝[xij,xij,..,xij]T j＝1,2,...,Np
Xmin＝[xmin,xmin,..,xmin]T
Xmax＝[xmax,xmax,..,xmax]T
Xd(k+1)＝[xd,xd,...,xd]T
according to the reasoning method of the state prediction equation, the following can be obtained:
Figure BDA0001579529960000132
in formula (16): hix,Hidn,HiuCan also be prepared fromd、BudAnd BdnExpressed, substituting it into the above constraints can result in:
Figure BDA0001579529960000133
can obtain the following products by arrangement:
EΔU(k)≤F (17)
in formula (17):
Figure BDA0001579529960000134
solving a delta U control increment sequence by using a Hildreth quadratic solution method, namely taking J as an optimization target in consideration of the state constraint condition of the solution, namely
Figure BDA0001579529960000135
Figure BDA0001579529960000136
In formula (18): h isijRepresentation matrix E (2H)-1ETIn the ith row and jth column, n is the matrix E (2H)-1ETNumber of columns, kiIs a vector (F + EH)-1G) N is a natural number; in formula (19): u (k) is the control law at the current time.
Step 3, designing a discrete extended state observer based on a mathematical model, specifically as follows:
because the system may have the conditions that the state is not measurable and the external interference cannot be measured, the system object model is required to be combined to design Extended State Observation (ESO), and the interference estimation is used as the output compensation of the controller output so as to inhibit the influence of the external interference on the control precision; according to the formula (5), the concrete design is as follows:
Figure BDA0001579529960000141
in equation (20), h (t) is the derivative of the interference d (t), and is defined as:
Figure BDA0001579529960000142
Figure BDA0001579529960000143
X＝[x1,x2,x3,x4]T(ii) a Thus, from equation (20):
Figure BDA0001579529960000144
the first-order euler dispersion method can be obtained from equation (21):
X(k+1)＝AodX(k)+G(u,x)du(k)+Δd(k) (22)
in formula (22):
Figure BDA0001579529960000145
the observer is therefore designed as follows:
Figure BDA0001579529960000146
in the formula (22)
Figure BDA0001579529960000151
i is 1,2,3,4 is a design parameter, Ho is a gain of the observer, and ω o>0 is the observer bandwidth; defining:
Figure BDA0001579529960000152
4, subtracting formula (23) from formula (22) yields the following formula:
Figure BDA0001579529960000153
defining:
Figure BDA0001579529960000154
representing the estimation error, which can be derived from equation (24):
Figure BDA0001579529960000155
in formula (25):
Figure BDA0001579529960000156
selecting
Figure BDA0001579529960000157
i is 1,2,3,4 such that AooSatisfying the Helverz matrix, there must be a matrix P that satisfies the Lyapunov equation, i.e.:
Figure BDA0001579529960000158
in formula (26): i is4×4Is a fourth order identity matrix.
The present invention will be described in detail with reference to the following examples and drawings.
Examples
In order to evaluate the performance of the designed controller, the following parameters are taken in Matlab simulation to model the electro-hydraulic servo system:
the inertial load parameter M is 30 kg; the viscous friction coefficient B is 4000 N.m.s/rad; effective piston area a 9.0478 × 10-4m2(ii) a Sum of two cavity products Vt＝7.962×10-5(ii) a Pressure P of fuel supplys12 Mpa; oil return pressure Pr＝0Mpa；
Figure BDA0001579529960000159
Effective modulus of elasticity beta of hydraulic oile＝7×108Pa; coefficient of leakage Ct＝4×10- 11m3and/s/Pa. The expected instruction for a given system is x1d＝10sin(2t)[1-exp(-0.01t3)]mm。
According to three different system working conditions, the simulation process is divided into three parts: the following controllers were taken for comparison:
a PID controller: the controller is a system controller commonly used in industry and mainly comprises a proportional term, an integral term and a differential term; the PID controller parameter selection steps are as follows: firstly, the proportional term is adjusted to stabilize the system, then the integral term is adjusted to improve the control precision, and finally, parameters are comprehensively fine-adjusted to enable the system to obtain the best tracking performance. The selected controller parameters are: k is a radical ofP＝-5000,kI＝1000,kD＝0。
MPC controller: the controller is widely applied to the slow time-varying industrial fields such as chemical engineering and the like, can predict the output in advance, can process the system control problem under the condition of state constraint, and has the following parameters: predicting time domain Np(ii) 5; control time domain Nc2; the weight coefficient is: r ═ diag {100,100,500,200,100} Q ═ diag {0.1,0.1 }.
The ESOMPC controller is designed, and compared with the traditional controller, the ESOMPC controller can give consideration to both state constraint and time-varying matching interference. Taking parameters of the controller: predicting time domain Np(ii) 5; control time domain Nc2; the interference feedback gain term γ is 200; the parameters of the observer are as follows: h ═ 0.8,64,800,4 × 107]The weight coefficient is: r ═ diag {100,100,500,200,100}, Q ═ diag {0.1,0.1 }.
Time-varying interference d (t) sin (2t) [1-exp (-0.01 t)3)]No constraint condition;
the tracking of the system output to the expected command under the action of the ESOMPC controller, the system tracking error comparison curve of the ESOMPC controller and the PID controller, and the estimation conditions of the ESOMPC controller to the system state and the interference are respectively shown in FIG. 3, FIG. 4 and FIG. 5. As can be seen from FIGS. 3 and 4, the steady-state tracking error of the system under the control of the ESOMPC controller is within about 0.05mm, while the steady-state error of the control accuracy of the conventional PID control is within about 0.1 mm; in contrast, the designed controller tracking performance is better than that of a PID controller; from fig. 5, it can be seen that the designed ESOMPC controller can accurately estimate the system status and the non-measurable interference.
Working condition (c) time-varying interference d (t) sin (2t) [1-exp (-0.01 t)3)]，x2∈[-19,19]State constraint of mm/s;
the ESOMPC controller is suitable for the situation possibly met in the actual electro-hydraulic servo control, namely the electro-hydraulic servo control problem under the condition of state constraint is considered, the ESOMPC controller is characterized by being capable of well processing the state constraint and the input constraint problem, the ESOMPC controller is also the reason for introducing Model Predictive Control (MPC) into the electro-hydraulic servo control, the ESOMPC controller can be used for predicting the system output and planning control input sequence in advance, and the electro-hydraulic servo system can be effectively controlled under the condition of state constraint.
FIG. 6 is a time-dependent output speed curve of the system under the control of an ESOMPC controller, and FIG. 7 is a control output condition of the controller under the working condition; FIG. 8 is a comparison curve of the position tracking error of the ESOMPC and the PID controller under the working condition; as can be seen from fig. 6, the ESOMPC controller can accurately restrict the speed within the specified range while ensuring the accuracy; FIG. 7 is a control output of the controller; as can be seen from fig. 8, under the same constraint, the steady-state control accuracy of the ESOMPC is higher than that of the PID controller.
Operating mode c time-varying interference d (t) t4sin(4t)[1-exp(-0.01t3)]No constraint condition;
if the designed control method can be well adapted to the extreme working condition, the designed control method can be widely applied to various working conditions in engineering practice. According to the expression of the time-varying interference, the interference value and the first derivative value of the interference to the time are increased along with the time, so that the extreme condition is considered because the effectiveness of the designed controller for dealing with the extreme condition on the electro-hydraulic servo system is fully shown, and the advantage of the ESOMPC in dealing with the non-measurable time-varying interference compared with the traditional MPC is fully highlighted, and the advantage of the ESOMPC in dealing with the problem of the time-varying interference of the system is fully demonstrated compared with a typical PID controller.
FIG. 9 is a graph of the position tracking error of the ESOMPC controller under this condition; the amplitude of the interference is increased along with the increase of the time, but the control precision of the system can still be kept at 0.1mm under the control of the ESOMPC; FIG. 10 is a comparison of tracking errors of a system under the control of three different controllers, and it can be seen that the PID controller cannot suppress time-varying interference at all, and the tracking error of the system increases with time; under the control of the MPC, the control precision of the system is always maintained within 0.2 mm; compared with the traditional MPC controller, the control precision of the ESOMPC controller is improved by two times.

Claims (3)

1. The model predictive control method of the electro-hydraulic servo system based on the extended state observer is characterized by comprising the following steps of:
step 1, establishing a dynamic mathematical model of a hydraulic system and discretizing the dynamic mathematical model by using a first-order Euler method; the method comprises the following specific steps:
step 1-1, for an electro-hydraulic servo system, driving an inertial load through a valve-controlled hydraulic actuator; therefore, according to newton's second law, the system's equation of motion is:
Figure FDA0002785297280000011
in formula (1): m is the inertial load mass; pLThe pressure difference of two cavities of the hydraulic cylinder; b is a viscous friction coefficient; a is the effective piston area of the hydraulic cylinder; (t) other unmodeled friction and interference; y is the displacement of the inertial load; t is a time variable; neglecting the leakage of the hydraulic motor, the pressure dynamic equation of the hydraulic actuator is:
Figure FDA0002785297280000012
in formula (2):Vtis the sum of the two cavity volumes of the actuator; beta is aeIs the effective elastic modulus of the hydraulic oil; ctIs the internal leakage coefficient; qLIs the load flow; q (t) is the model error; since a highly responsive servo valve is used, it is assumed here that the control input is proportional to the spool displacement of the servo valve, i.e. xv＝kiu; thus QLThe relationship to the control input is:
Figure FDA0002785297280000013
in formula (3): k is a radical oftThe total flow gain; psSupplying oil pressure to the system; cdIs the flow coefficient; omega is the valve core area gradient; ρ is the density of the oil; k is a radical ofiIs a proportionality coefficient; where sign (u) is defined as:
Figure FDA0002785297280000014
step 1-2, defining state variables:
Figure FDA0002785297280000015
the state equation of the system is:
Figure FDA0002785297280000016
in formula (5):
Figure FDA0002785297280000021
C＝[1 0 0]，
Figure FDA0002785297280000022
Figure FDA0002785297280000023
d (t) is the total interference of the system; the method can obtain the following steps by adopting a first-order Euler discrete method:
Figure FDA0002785297280000024
in formula (6): t issTo sample time, Ad＝I3×3+TsA，Bud＝TsBu，Bdn＝TsBd，Cd＝C；I3×3A unit vector of third order;
the following is assumed:
assume that 1: the total disturbance in the system can be estimated by an observer, wherein the estimation error of the observer is not considered in the design of the controller, namely:
Figure FDA0002785297280000025
in formula (7):
Figure FDA0002785297280000026
is an estimated disturbance of the observer; w (k) is observer interference estimation error;
assume 2: according to the principle of model prediction, the latest measured value is required to be used as an input initial value, and the future time is predicted through a certain time; thus setting the prediction time domain to NpThe control time domain of the system is NcAnd N isc≤Np(ii) a For the subsequent design of the controller it is assumed that:
Figure FDA0002785297280000027
step 2, designing a model predictive controller with state constraint based on a mathematical model;
and 3, designing a discrete extended state observer based on the mathematical model.
2. The method for model predictive control of an electro-hydraulic servo system based on an extended state observer according to claim 1, wherein the step 2 of designing a model predictive controller with state constraints based on a mathematical model comprises the following steps:
defining: Δ x (k) ═ x (k) — x (k-1) is the increment of the two time states, and similarly, Δ u (k) ═ u (k) — u (k-1) and Δ d (k) ═ d (k) — d (k-1) are the two time increments of input and interference, respectively; from the discretized model of equation (6), an incremental equation of state can be obtained as:
Figure FDA0002785297280000031
defining Δ x (k +1| k) to represent the prediction of Δ x (k +1) at time Δ x (k) at time k, and similarly y (k +1| k) to represent the prediction of y (k +1) at time Δ x (k) at time k; the recurrence equation can thus be found as follows:
Figure FDA0002785297280000032
by substituting equation (10) for equation (9), a predicted value of output y can be obtained, that is:
Figure FDA0002785297280000033
defining:
Ye(k+1|k)＝[y(k+1|k),y(k+2|k),...,y(k+Np|k)]T
ΔU(k)＝[Δu(k),Δu(k+1),...,Δu(k+Nc-1)]T
the state prediction equation is therefore:
Ye(k+1|k)＝HxΔx(k)+HIy(k)+HdnΔd(k)+HuΔU(k) (12)
in formula (12): hI、Hx、HuAnd HdnCan be obtained from formula (11); due to the fact thatSystem state x2And x3The estimation is carried out by a subsequent state observer, so that an estimation error exists; defining:
Figure FDA0002785297280000041
Figure FDA0002785297280000042
for the state increment of the state estimation,
Figure FDA0002785297280000043
estimating the state increment error, defining:
Figure FDA0002785297280000044
considering the estimation error approximation as interference, where γ is the gain term; from the formula (12), the following formula can be obtained:
Figure FDA0002785297280000045
to track target instruction xdDefining an objective function to reflect the control performance of the system, and making next control prediction input according to the objective function; the objective function is defined as follows:
Figure FDA0002785297280000046
in formula (14): r and Q are each NpOrder tracking error sum NcDetermining an expected target by adjusting a diagonal weight matrix of the order control increment; xd(k +1) is a reference instruction target; in order to solve the quadratic form of the formula (14), the formula (13) is substituted into the formula (14), and the quadratic form is reduced to a standard form by arrangement, that is:
J＝ΔU(k)THΔU(k)-G(k+1|k)TΔU(k) (15)
in formula (15):
H＝Hu TRTRHu+QTQ
G(k+1|k)＝2Hu TRTREp(k+1|k)
Figure FDA0002785297280000048
considering the state constraints:
Xmin≤Xi≤Xmax i＝1,2,3
wherein:
Xi＝[xij,xij,..,xij]T j＝1,2,...,Np
Xmin＝[xmin,xmin,..,xmin]T
Xmax＝[xmax,xmax,..,xmax]T
Xd(k+1)＝[xd,xd,...,xd]T
according to the reasoning method of the state prediction equation, the following can be obtained:
Figure FDA0002785297280000047
in formula (16): hix,Hidn,HiuCan also be prepared fromd、BudAnd BdnExpressed, substituting it into the state constraint described above yields:
Figure FDA0002785297280000051
can obtain the following products by arrangement:
EΔU(k)≤F (17)
in formula (17):
Figure FDA0002785297280000052
solving a delta U control increment sequence by using a Hildreth quadratic solution method, namely taking J as an optimization target in consideration of the state constraint condition of the solution, namely
Figure FDA0002785297280000053
Figure FDA0002785297280000054
In formula (18): h isijRepresentation matrix E (2H)-1ETThe ith and j elements of (1); n is matrix E (2H)-1ETThe number of columns; k is a radical ofiIs a vector (F + EH)-1G) The ith element of (1); n is a natural number; in formula (19): u (k) is the control law at the current time.
3. The method for model predictive control of the electro-hydraulic servo system based on the extended state observer according to claim 2, wherein step 3 is to design the discrete extended state observer based on a mathematical model, and specifically comprises the following steps:
according to the formula (5), the concrete design is as follows:
Figure FDA0002785297280000055
in equation (20), h (t) is the derivative of the interference d (t), and is defined as:
Figure FDA0002785297280000056
Figure FDA0002785297280000061
X＝[x1,x2,x3,x4]T(ii) a Thus, from equation (20):
Figure FDA0002785297280000062
through the first-order Euler method dispersion, the following can be obtained:
X(k+1)＝AodX(k)+Gd(u,x)u(k)+Δd(k) (22)
in formula (22):
Figure FDA0002785297280000063
the observer was designed as follows:
Figure FDA0002785297280000064
in the formula (23)
Figure FDA0002785297280000065
To design the parameters, HoIs the gain of the observer; omegaoObserver bandwidth > 0, defined:
Figure FDA0002785297280000066
subtracting equation (23) from equation (22) yields the following equation:
Figure FDA0002785297280000067
defining:
Figure FDA0002785297280000068
representing the estimation error, the following equation is obtained from equation (24):
Figure FDA0002785297280000069
in formula (25):
Figure FDA00027852972800000610
selecting
Figure FDA00027852972800000611
So that A isooSatisfying the Helverz matrix, there must be a matrix P that satisfies the Lyapunov equation, i.e.:
Figure FDA0002785297280000071
in the formula (26) I4×4Is an identity matrix of order 4.
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Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN104635490A (en) * 2014-12-15 2015-05-20 南京理工大学 Output feedback control method for asymmetric servo cylinder positional servo system
CN106125553A (en) * 2016-08-24 2016-11-16 南京理工大学 A kind of hydraulic system self-adaptation control method considering state constraint
CN106154833A (en) * 2016-07-14 2016-11-23 南京理工大学 A kind of electro-hydraulic load simulator output feedback ontrol method

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN104635490A (en) * 2014-12-15 2015-05-20 南京理工大学 Output feedback control method for asymmetric servo cylinder positional servo system
CN106154833A (en) * 2016-07-14 2016-11-23 南京理工大学 A kind of electro-hydraulic load simulator output feedback ontrol method
CN106125553A (en) * 2016-08-24 2016-11-16 南京理工大学 A kind of hydraulic system self-adaptation control method considering state constraint

Non-Patent Citations (2)

* Cited by examiner, † Cited by third party
Title
Extended-State-Observer-Based Output Feedback Nonlinear Robust Control of Hydraulic Systems With Backstepping;Jianyong Yao等;《IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS》;20140206;第61卷(第11期);第6285-6293页 *
具有自适应增益的电液位置伺服系统超螺旋滑模控制;陈丽君等;《机床与液压》;20160630;第44卷(第11期);第69-73页 *

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