CN117706933A - Multi-target complementary robust control method of piezoelectric positioning system - Google Patents
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Abstract
The invention relates to a multi-target complementary robust control method of a piezoelectric positioning system, which comprises the following steps: modeling the piezoelectric positioning system by adopting a Hammerstein model structure, and compensating hysteresis nonlinearity of the piezoelectric positioning system model based on a PI model to obtain a compensated system to be identified; dynamically modeling the compensated system based on an improved Hankel matrix system identification method to obtain the relevant characteristics of the system, namely an identified piezoelectric actuator model; and acquiring a standard feedback control model, further acquiring a generalized controlled system model, designing an LQG controller based on Kalman filtering, acquiring an augmented system model containing LQG control, inputting the identified model into the augmented system model containing LQG control, and acquiring a multi-target complementary robust control result. The invention can effectively inhibit external interference and system parameter change, and simultaneously improve steady-state precision, dynamic performance, anti-interference capability and robustness of the closed-loop system.
Description
Technical Field
The invention relates to the technical field of robust control, in particular to a multi-target complementary robust control method of a piezoelectric positioning system.
Background
The core component of the piezoelectric positioning system is a piezoelectric actuator (Piezoelectric actuators, PEA), is a device for controlling the actuating displacement and the output force through the applied voltage, has the advantages of high transduction coefficient, less heating, no electromagnetic interference and the like, and is generally suitable for the field of intelligent control. However, inherent nonlinear characteristics such as hysteresis and creep of the piezoelectric material directly affect the performance of the piezoelectric material in the micro-nano positioning system, so that the precision of the control system is reduced, the control system is unstable even, and great challenges are brought to the solution of problems such as modeling and accurate control. The piezoelectric positioning platform has the problems of disturbance, measurement noise, model uncertainty and the like in the application field, and has great influence on the control research of the piezoelectric actuator. Therefore, it is very difficult to build a high-precision model and design a robust and efficient control strategy to overcome these problems and improve the controller precision while ensuring reliable operation of the system.
For a complex uncertainty system such as a piezoelectric positioning system, accurate modeling is a necessary precondition for controller design. In order to solve the rate-dependent hysteresis characteristics of the piezoelectric actuator, an effective method is to adopt a Hammerstein model to realize the separation of static hysteresis nonlinearity and rate-dependent dynamic characteristics thereof, and the model divides a nonlinear system into a static nonlinear module which only considers the hysteresis characteristics of the system and a dynamic linear module which only considers the rate-dependent characteristics, and describes the nonlinearity problem through the series connection of the two modules. The static hysteresis nonlinear module adopts a PI model and performs inverse compensation control to realize approximate linearization of a nonlinear system, so that modeling and control problems are converted into linear system problems. For the linear link, a Hankel matrix identification method based on a state space model is provided for modeling, so that the interference of measurement noise can be effectively eliminated, the system order can be intuitively determined, the accurate identification of a complex uncertain PEA system is realized, and a solid foundation is laid for the further design of a system controller. The method is innovatively improved by considering the correlation problem of the driving signals, so that the identification accuracy of the method is further improved.
Under the background of the nanometer high-precision positioning work, the PEA high-precision control problem with external disturbance, model uncertainty and other factors is of great significance in many practical applications. For many years, many control strategies, in particular robust H, have been studied ∞ The control technology is widely studied, the method deeply insights the influence of modeling uncertainty on control performance, however, the robustness control thought needs to fully consider the stability problem of the system in the worst case, so that the robustness control method has conservation in the aspect of the compromise of control performance. In addition, linear quadratic gaussian control (Linear quadratic gaussian control, LQG) has also been extensively studied in the control of piezoelectric actuators, which provides optimal quadratic performance for white noise in systems and measurements, but does not guarantee robustness due to missing uncertainty disturbances in its framework that affect the integrity of the system model. Since the design controller needs to consider both robustness and optimal performance, hybrid H 2 /H ∞ An extended multi-objective approach to control and linear matrix inequality (Linear matrix lnequality, LMI) and the like is proposed. However, these multi-objective control methods inevitably result in inherent conflicts between robustness and optimal performance within the same level, because they all build multi-performance objectives on a single entity implementation structure of the Youla parameterization of all stable controllers, and thus, these methods cannot avoid worst-case and/or trade-off designs. Aiming at the problem, the invention provides a novel multi-target complementary control structure based on the Youla parameterization idea to solve the problem of compromise. The structure includes an optimal performance controller designed for a nominal object model, the feedback control system being controlled solely by the model uncertainty and external disturbances when not present, and the multi-target complementary control structure being estimated and compensated by the young's parameters when model uncertainty and external disturbances are present. At the same time consider H ∞ The invention uses Kalman filtering algorithm to transmit the value of state estimation as feedback quantity to the controller, thereby furtherOptimizing the control performance.
Disclosure of Invention
In order to solve the technical problems that how to build a high-precision model and design a robust and efficient control strategy to overcome inherent hysteresis, creep and other nonlinear characteristics of a piezoelectric material and the wide occurrence of disturbance, measurement noise, model uncertainty and the like in an application field of a piezoelectric positioning platform, the invention provides a multi-target complementary robust control method of a piezoelectric positioning system. Meanwhile, on the basis of realizing dynamic modeling of a system, a multi-target complementary control robust tracking scheme based on LQG control is provided.
In order to achieve the above object, the present invention provides the following solutions:
a multi-target complementary robust control method for a piezoelectric positioning system, comprising:
modeling the piezoelectric positioning system by adopting a Hammerstein model structure, and compensating hysteresis nonlinearity of the piezoelectric positioning system model based on a PI model to obtain a compensated system to be identified; dynamically modeling the compensated system to be identified based on an improved Hankel matrix system identification method to obtain a discrete transfer function and a continuous transmission function of a piezoelectric actuation model;
obtaining a standard feedback control model, obtaining a generalized controlled system model according to the standard feedback control model, designing an LQG controller based on Kalman filtering, obtaining an augmented system model containing LQG control, inputting the continuous transmission function into the augmented system model containing LQG control, and obtaining a multi-target complementary robust control result.
Optionally, obtaining the discrete transfer function of the piezoelectric actuation model includes:
acquiring the compensated input and output signals of the system to be identified, determining an autocorrelation function and a cross correlation function of the input and output signals, and acquiring an input and output relation according to the autocorrelation function and the cross correlation function;
performing correlation analysis on the impulse response and the correlation function based on the input-output relationship, performing impulse response estimation, obtaining a pulse sequence to construct a Hankel matrix, and obtaining a second relationship between the Hankel matrix and the state space description of the system model to be identified through the first relationship between the impulse response and the state equation;
singular value decomposition is carried out on the second relation, the system order of a system model to be identified is obtained, secondary decomposition is carried out on the system order, a decomposition result is obtained, system state space model parameters A, B and C are obtained according to the decomposition result, and a discrete transfer function G of the piezoelectric actuation model is obtained in combination with a meeting condition D of the system model 0 (z) and continuous transmission function G 0 (s)。
Optionally, obtaining the generalized controlled system model includes:
changing the input information position of a controller in the standard feedback control model, acquiring a multi-target complementary control structure, designing the multi-target complementary control structure as a state space model, acquiring the control input of the state space model, and designing the control input as H ∞ And the controller acquires the generalized controlled system model.
Optionally, the method for obtaining the generalized controlled system model includes:
z＝C 1 x+D 11 w+D 12 u
y＝C 2 x+D 20 w 0 +D 21 w+D 22 u
wherein w is modeling uncertainty and disturbance signal, w 0 Is white noise signal, u is control input, y is measured output or object output, z is controlledThe output of the process is obtained,for the speed of the system, A is the state transition matrix of the system, x is the state vector of the system, B 0 System matrix for white noise signal, B 1 To model the system matrix of uncertainty and disturbance signals, B 2 For input matrix, C 1 To control the output matrix, D 11 D for modeling uncertainty and direct transfer matrix of disturbance signal 12 To control the input direct transfer matrix, C 2 To be measured output matrix D 20 Is a white noise signal direct transfer matrix, D 21 To model the uncertainty and the direct transfer matrix of the disturbance signal, D 22 A direct transfer matrix for control inputs.
Optionally, obtaining the augmentation system model comprising LQG control comprises:
converting the discrete transfer function of the piezoelectric actuation model into a continuous state space form, combining measurement noise, designing a Kalman filter for estimating the actual state, defining a filter error variance matrix Li Kadi model, removing the target time of the filter error variance matrix Li Kadi model, and obtaining the gain K of the steady state value of the Kalman filter error variance k Based on the function leq (), a symmetric positive definite matrix of the Li Kadi model and the gain K are obtained k Based on state feedback F and gain K k Substituting the generalized controlled system model to obtain the enhanced system model containing LQG control.
Optionally, the method for obtaining the augmentation system model containing the LQG control comprises the following steps:
z＝C 1 x+D 11 w+D 12 u l +D 12 u f
y＝C 2 x+D 21 w+D 22 u l +D 22 u f
wherein w is modeling uncertainty and disturbance signal, w 0 For a white noise signal, y is the measured output or object output, z is the controlled output,is the speed vector of the system, A is the state transition matrix of the system, x is the state vector of the system, B 0 System matrix for white noise signal, B 1 To model the system matrix of uncertainty and disturbance signals, B 2 For input matrix, u l For nominal control input, u f For additional adjustment signals->K is the speed estimate k For gain, F is the LQG control designed state feedback gain, C 1 To control the output matrix, D 11 D for modeling uncertainty and direct transfer matrix of disturbance signal 12 To control the input direct transfer matrix, C 2 To be measured output matrix D 20 Is a white noise signal direct transfer matrix, D 21 To model the uncertainty and the direct transfer matrix of the disturbance signal, D 22 For a direct transfer matrix of control inputs, +.>Is a control input.
Optionally, obtaining the multi-target complementary robust control result includes:
introducing the augmentation system model containing LQG control into a parameter e for equivalent conversion, obtaining a conversion model, inputting the continuous transmission function into the conversion model, and obtaining an output result;
acquiring a static state feedback control model in the augmentation system model containing LQG control, simplifying the augmentation system model containing LQG control according to the static state feedback control model, and acquiring a corresponding system model;
determining a closed loop transfer function T according to the corresponding system model zw The satisfaction condition of (t) and H ∞ Obtaining a matrix inequality model according to charge conditions of a controller, obtaining a feasible solution model according to the matrix inequality model, substituting the output result into the matrix inequality model and the feasible solution model, and obtaining a multi-target complementary robust control result
The beneficial effects of the invention are as follows:
compared with other identification methods, the method has the advantages that the method not only can effectively eliminate the interference of measurement noise, but also can intuitively determine the system order and realize the accurate identification of the complex uncertain PEA system, thereby laying a solid foundation for the further design of a system controller. Compared with other robust control methods, the control structure based on Kalman filtering algorithm, youla parameterization and generalized internal model principle effectively solves the problem of conflict between system robustness and optimal performance in the same level, and further optimizes the performance of a control system.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions of the prior art, the drawings that are needed in the embodiments will be briefly described below, it being obvious that the drawings in the following description are only some embodiments of the present invention, and that other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a standard feedback block diagram of an embodiment of the present invention;
FIG. 2 is a diagram of a Youla controller parameterization architecture in an embodiment of the present invention;
FIG. 3 is a block diagram of a multi-target complementary control architecture of an embodiment of the present invention;
FIG. 4 is a multi-target complementary control block diagram of a piezoelectric positioning system of an embodiment of the invention;
FIG. 5 is a block diagram of LQG based on Kalman filtering according to an embodiment of the invention;
FIG. 6 is a standard H of an embodiment of the invention ∞ A control block diagram;
FIG. 7 is a graph of the input-output method frequency response of a piezoelectric positioning system according to an embodiment of the present invention;
FIG. 8 is a graph of the input pulse method frequency response of a piezoelectric positioning system according to an embodiment of the present invention;
FIG. 9 is a graph of the spectral frequency response of a piezoelectric positioning system according to an embodiment of the present invention;
FIG. 10 is a Kalman filtered estimation error graph of an embodiment of the present invention;
FIG. 11 is a graph of tracking control of a single frequency versus a composite frequency for a sinusoidal signal in accordance with an embodiment of the present invention; wherein, FIG. 11 (a), FIG. 11 (b), FIG. 11 (c) and FIG. 11 (d) are comparative graphs of tracking control experiment curves of different control schemes under a single frequency sinusoidal signal with reference input amplitude of 1 μm and frequency of 5Hz and 60Hz and a composite frequency sinusoidal signal with 20/40Hz and 20/40/60/80Hz respectively;
FIG. 12 is a graph of tracking control of single frequency versus complex frequency for a square wave signal according to an embodiment of the present invention; wherein, fig. 12 (a), fig. 12 (b), fig. 12 (c) and fig. 12 (d) are comparative graphs of tracking control experiment curves of different control schemes under a single frequency square wave signal with reference input amplitude of 1 μm and frequency of 5Hz and 55Hz and a composite frequency square wave signal with reference input amplitude of 20/40Hz and composite frequency of 10/20/40Hz respectively.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
In order that the above-recited objects, features and advantages of the present invention will become more readily apparent, a more particular description of the invention will be rendered by reference to the appended drawings and appended detailed description.
The invention discloses a multi-target complementary robust control method of a piezoelectric positioning system, which comprises the following steps: for the piezoelectric actuator with problems of hysteresis nonlinearity, model uncertainty, external interference, measurement noise and the like (the core component of the piezoelectric positioning system is the piezoelectric actuator, the piezoelectric actuator model in the embodiment is the piezoelectric positioning system), a Hammerstein model is adopted to model the piezoelectric actuator, a PI inverse model is firstly obtained according to the system characteristic and is connected in series on a system feedforward channel to compensate the hysteresis nonlinearity of the system, and then a Hankel matrix system identification method based on Markov coefficients is provided to dynamically model the whole system with the nonlinear part compensated.
Markov coefficient-based Hankel matrix system identification:
and inputting the determined input signals into a system to be identified, and collecting the input and output signals of the system. Based on the input/output sequence of the system, the correlation sequence (function) of the system is determined. Let u (k) and y (k) be the system input signal and the measurement output signal, respectively, then the autocorrelation function and the cross correlation function are:
where N is the sequence length of one period of the input pseudo-random signal, u represents the input of the system, y represents the output of the system, i is the sampling point, and T is the sampling period.
The input-output relationship of the discrete linear system is as follows:
where y (kT) is the input-output relationship, k is the sampling point, and g is the impulse response.
The functional expression between the impulse response and the correlation function is obtained as follows:
where j is the sampling point.
When N is sufficiently large, the impulse response estimate based on the correlation analysis described above is:
constructing a Hankel matrix H according to the obtained pulse sequence, wherein the Hankel matrix H is as follows:
according to the relation between the impulse response of the system and the state equation, the relation between the Hankel matrix H and the description of the system state space is as follows:
wherein C is an output matrix, A n-1 Is a state matrix, and B is an input matrix.
To determine the system architecture, it is subjected to singular value decomposition:
H＝Udiag{σ 1 …σ n }V T (8)
wherein sigma 1 ≥σ 2 ≥…≥σ r ＞＞σ r+1 ≥…≥σ n Not less than 0 and U, V is an orthogonal matrix, i.e. U T U＝I,V T V=i, T is the transposed symbol, I is the identity matrix.
According to the determined system order, it is further decomposed as follows:
H＝Udiag{σ 1 …σ n }V T ＝[U 1 U 2 ]diag{∑ 1 ,∑ 2 }[V 1 V 2 ] T ＝U 1 ∑ 1 V 1 T +U 2 ∑ 2 V 2 T ≈U 1 ∑ 1 V 1 T (9)
taking:
wherein sigma 1 1 st singular value, σ, of Hankel array r Is the r singular value of the Hankel array.
Parameters C and B can be taken as:
definition H 1 The method comprises the following steps:
the parameter a may be taken as:
wherein A is a system matrix, Σ 1 Is a singular value diagonal array of a Hankel array,transpose of orthogonal matrix in singular value decomposition, H 1 Is Hankel matrix, V 1 Is an orthogonal matrix in singular value decomposition.
The engineering system satisfies D=g (0) ≡0, and the transfer function of the identification system can be obtained according to the parameters A, B, C and D.
The discrete transfer function of the piezoelectric actuator obtained by the Hankel matrix method is as follows:
wherein G is 0 (z) is a discrete transfer function, z is an operator in discrete form, G 0 (s) is a continuous transfer function, s is a complex frequency domain operator.
Continuous transfer function after system serialization of recognition:
wherein G is 0 (s) is a continuous transmission function, e is a scientific counting method and is an operator of a complex frequency domain.
Controller design
(1) LQG-based multi-target complementary control design
Consider a standard feedback control system as shown in fig. 1, where P is the controlled system (piezoelectric actuator) and K is the feedback controller.
Lemma 1: let K be 0 Internal stabilization in the feedback system shown in FIG. 1, let K 0 And P has the following left and right intersubstance decomposition:
wherein,N、M -1 、and respectively different operator matrixes of the mutual mass decomposition.
The controller K of each internal stability feedback system as shown in fig. 1 can be written in the form of:
wherein,and respectively different operator matrixes of the mutual mass decomposition.
For some Q.epsilon.H ∞ Make (V (≡) -N-infinity) Q (≡)) +.0.
Selection in Standard Youla ParametricV and U, such that->And->In particular, select K 0 As a stable observer-based controller. However, this K 0 The choice of (2) is not always desirable in subsequent developments because the controller parameterization of the above-mentioned approach 1 does not have such constraints. Therefore, K needs to be selected 0 As a nominal controller to meet nominal design objectives.
The controller of the feedback system of equation (19) may be represented by 5 functional blocks as shown in fig. 1 or 5 functional blocks as shown in fig. 2 after the total transfer function K is obtained.
By changing the input signal position of the structure shown in fig. 2, a new control method is formed, i.e. the multi-target complementary control is shown in fig. 3, the controller structure does not change the stability inside the system, since the transfer function of the output y to the control signal u is not changed.
Considering the control design problem of the piezoelectric positioning system in the present description based on the multi-target complementary control structure described in fig. 3, the control structure can be designed as a state space model as shown in fig. 4, specifically: based on the multi-target complementary control structure described in fig. 3, considering the problems of external disturbance, model uncertainty, measurement noise and the like in the piezoelectric positioning system, the optimal controller in the multi-target complementary control structure is designed into LQG control based on kalman filtering, thereby forming a shape as shown in fig. 4A state space model structure; wherein the control input u is split into two components: u (u) l Andrepresenting the nominal control input and the additional regulation signal, respectively. The nominal controller K(s) is designed as an LQG control structure based on Kalman filtering for performance control of the nominal system of the piezoelectric actuator.
Will beI.e. < ->Designed as H ∞ The controller solves the uncertainty disturbance, and the generalized controlled system (a model which is obtained by a system model after absorbing model uncertainty, external interference, noise and the like) is as follows:
z＝C 1 x+D 11 w+D 12 u (21)
y＝C 2 x+D 20 w 0 +D 21 w+D 22 u (22)
wherein w is modeling uncertainty and disturbance signal, w 0 Is a white noise signal, u is a control input, y is a measured output or object output, z is a controlled output,for the speed of the system, A is the state transition matrix of the system, x is the state vector (displacement) of the system, B 0 System matrix for white noise signal, B 1 To model the system matrix of uncertainty and disturbance signals, B 2 For input matrix, C 1 To control the output matrix, D 11 D for modeling uncertainty and direct transfer matrix of disturbance signal 12 To control the input direct transfer matrix, C 2 To be measured output matrix D 20 Is a white noise signal direct transfer matrix,D 21 to model the uncertainty and the direct transfer matrix of the disturbance signal, D 22 A direct transfer matrix for control inputs.
Wherein:
C 2 ＝[-167.4758 0.0306]
C 1 ＝[1 0.45]，D＝0
w represents modeling uncertainty and/or disturbance signal, w 0 The white noise signal u is a control input, y is a measured output or an object output, z is a controlled output, and the system evaluation signal.
LQG controller design based on Kalman filtering
Converting the PEA system identified by the Hankel matrix method based on Markov coefficients in the formula (16) into a continuous state space form and considering measurement noise existing in the system, designing a Kalman filter to estimate the actual state, wherein a state space model is as follows:
y(t)＝Cx(t)+v(t)＝[-167.4758 0.0306]x(t)+v(t) (24)
wherein w (t) and v (t) represent the process noise and the measurement noise of the system, respectively. Assuming they are gaussian white noise independent of each other, subject to normal distribution, and both are independent of the initial state x (0), the covariance of the noise is defined as:
Q＝cov(w)＝E{ww T } (25)
R＝cov(v)＝E{vv T } (26)
wherein E is the mean value.
Optimal filter equation:
filter gain equation:
K k (t)＝P(t)C T (t)R -1 (t) (28)
filtering error variance matrix licark lifting equation:
when the estimation process has reached stability, the Li Kadi differential equation is independent of time, and its differential is zero, then equation (29) can be converted to:
0＝AP+PA T +JQJ T -PC T R -1 CP (30)
wherein P is covariance matrix, A T Transpose of the state matrix, J is the process noise matrix, Q is the covariance of the process noise, J T For transpose of process noise matrix, C T For transpose of output matrix, R -1 The inverse of the covariance of the measured noise, C, is the output matrix.
Its symmetrical positive definite solution P ∞ I.e. the steady state value of the variance of the Kalman filtering error, the gain is:
K k ＝P ∞ C T R -1 (31)
the filter equation is:
where Bu (t) is the input to the system.
Solving equation (30), the symmetric positive definite matrix P and Kalman filter gain K of Li-Kalman equation can be obtained by using function leq k Obtaining
Defining an estimation error:
from equations (23) and (32), the time derivative of the estimated error x can be calculated as:
set F and K k The state feedback and Kalman observer gain for the LQG control design of the nominal system with only white noise in (20) - (22), respectively, are shown in fig. 5.
The model of the augmentation system including LQG control (after the performance controller is designed as an LQG controller based on Kalman filtering in multi-target complementary control, the system model of the whole multi-target complementary control structure) is:
z＝C 1 x+D 11 w+D 12 u l +D 12 u f (38)
y＝C 2 x+D 21 w+D 22 u l +D 22 u f (39)
wherein w is modeling uncertainty and disturbance signal, w 0 For a white noise signal, y is the measured output or object output, z is the controlled output,is the speed vector of the system, A is the state transition matrix of the system, x is the state vector (displacement) of the system, B 0 System matrix for white noise signal, B 1 To model the system matrix of uncertainty and disturbance signals, B 2 For input matrix, u l For nominal control input, u f For additional adjustment signals->K is the speed estimate k For gain, F is the LQG control designed state feedback gain, C 1 To control the output matrix, D 11 D for modeling uncertainty and direct transfer matrix of disturbance signal 12 To control the input direct transfer matrix, C 2 To be measured output matrix D 20 Is a white noise signal direct transfer matrix, D 21 To model the uncertainty and the direct transfer matrix of the disturbance signal, D 22 A direct transfer matrix for control inputs.
For an augmentation system, due to white noise w 0 The disturbance w is filtered by Kalman filtering, so that only the uncertainty disturbance w of the model is considered, and an additional adjusting signal is required to be designedTo solve the problem of model uncertainty disturbance and recover the LQG performance in the case of modeling mismatch.
Introduction into piezoelectric actuator models (36) - (40)The system can then be equivalently converted to the following:
wherein:
this control problem can be converted into standard H ∞ A robust control architecture is shown in fig. 6.
(3) Robust controller solution
For robust H as shown in FIG. 6 ∞ In the control standard model, assuming that the augmentation control object is G, the transfer function from W to Z may be expressed as:
wherein U is f (s) is a control input to which,is a robust controller of a piezoelectric positioning system, F(s) is a residual signal, G 21 (s) is a generalized controlled object element, G 22 (s) is a generalized controlled object element.
For the standard robust control block diagram shown in fig. 6, a true real physical controller is foundStabilizing the inside of the closed loop control system and minimizing the closed loop transfer function matrix T zw H of(s) ∞ Norms, namely:
substituting the piezoelectric actuator system model into equations (41) to (43) to obtain:
assuming that the state of the system is directly measurable, a static state feedback controller is designed:
so that the corresponding system
Is progressively stable.
Assuming its closed loop transfer function T zw (t) satisfies:
then there is a status feedback H ∞ The conditions for the controller are: if and only if there is a symmetric positive definite matrixAnd a matrix W such that the following matrix inequality->
Hold, and if there is a feasible solution to the equation (57) matrix inequalityW * Then
Is a state feedback H of the system ∞ The controller uses a solver feasp to solve, substitutes the formula (50) and the formula (51) into the formula (57) and the formula (58), and obtains the following programming:
fig. 7, 8 and 9 are respectively system frequency response comparison diagrams obtained by adopting a system numerical solution obtained by an input/output method, an impulse response method and a frequency spectrum relation method to a piezoelectric positioning system and identifying by a subspace identification method (double-dashed line), a least square method (dotted line) and a Hankel matrix system identification method (dot-dash line) based on Markov coefficients. As can be seen from fig. 7, 8 and 9, the system frequency response obtained by the Hankel matrix system identification method based on the Markov coefficient has the best fitting degree with the obtained numerical solution, and the subspace identification method identifies the system frequency response obtained by the subspace identification method has the second fitting degree with the obtained numerical solution. Therefore, the system obtained by identification based on the Markov coefficient Hankel matrix system identification method can accurately express the frequency characteristic of the actual data of the system.
FIG. 10 shows the estimated error of Kalman filtering, and it can be seen from the equation (35) and FIG. 10 that A-K is calculated at t → ≡ k C stabilizes and the estimation error converges to 0, so that the stability of the Kalman filter is ensured.
Fig. 11 (a), 11 (b), 11 (c) and 11 (d) are comparative graphs of tracking control experiments for different control schemes with a single frequency sinusoidal signal with reference input amplitude of 1 μm and frequency of 5Hz, 60Hz and a composite frequency sinusoidal signal with 20/40Hz, 20/40/60/80Hz, respectively. As can be seen from the tracking curve local amplification effect diagrams of different control schemes under the sinusoidal signal interference of FIG. 11, the LQG control-based multi-target complementary control method (straight line) provided by the invention is compared with H ∞ Robust control (dash-dot line) and LQG control (double-dash line) based on Kalman filtering algorithm, the displacement of which is closest to the desired displacement (dashed line), and H ∞ The robust control is inferior, so the tracking effect of the method provided by the invention is best. This is because the present invention considers H ∞ On the basis that the robustness of the system cannot be guaranteed due to the fact that the system robustness is lost due to uncertainty interference in the LQG framework and the control performance is compromised in the robust control, a novel control structure for balancing the system robustness and the optimal control performance is innovatively provided, and the robustness and the system control performance are improved simultaneously. The Kalman filtering algorithm is utilized to further optimize the control performance of the system under the condition of limiting the uncertainty of the system, and a better tracking precision is obtained on the basis, so that the effectiveness of the method provided by the invention is fully demonstrated, and the method is a strip with gradually increased single frequency and gradually complicated frequency compounding degreeThe more pronounced the effect is. Thus, in combination, the method of the invention is the most effective, which is based on the original H ∞ The control and LQG control based on Kalman filtering algorithm achieve obvious performance improvement. In addition, as can be seen from the partial amplification diagrams of the tracking errors of the frequencies, compared with the other two methods, the control method provided by the invention has the advantages that the fluctuation amplitude of the tracking errors is minimum, and the absolute value of the errors is minimum, so that the control stability is best, and the effect is more obvious along with the gradual increase of a single frequency and the gradual complexity of the frequency.
Fig. 12 (a), 12 (b), 12 (c) and 12 (d) are graphs comparing tracking control experiments of different control schemes under a single frequency square wave signal with reference input amplitude of 1 μm and frequency of 5Hz and 55Hz and a composite frequency square wave signal with reference input amplitude of 20/40Hz and reference input amplitude of 10/20/40Hz, respectively. Similarly, as can be seen from the local amplification effect diagram of the tracking curves of the different control schemes of the single frequency and the composite frequency under the interference of the square wave signal in fig. 12, the multi-target complementary control method (straight line) based on the LQG control provided by the invention can be more close to the expected displacement (double-scribing) no matter what frequency is changed, but the standard H ∞ The robust control (scribing) and LQG control (dotted line) based on Kalman filtering algorithm gradually deviate from the expected displacement along with the gradual increase of single frequency and the gradual complexity of the frequency, so compared with other two control methods, the control method provided by the invention has better tracking effect and better control stability. In addition, as can be seen from the partial amplification diagrams of the tracking errors of the frequencies, compared with the other two methods, the tracking errors of the control method provided by the invention have smaller oscillation amplitude and smaller absolute value, so that the control stability is better.
On the basis of modeling a piezoelectric actuator dynamic hysteresis nonlinear system with model uncertainty and external interference by adopting a Hammerstein model, the invention provides a Markov coefficient-based Hankel matrix system identification method for dynamically modeling the whole system with nonlinear parts compensated, so that the system can embody high-frequency resonance characteristics, thereby better describing the reality of the systemThe characteristics of the system are realized. Meanwhile, in order to balance the optimal control performance and robustness of the piezoelectric positioning system, the invention provides a multi-target complementary control robust tracking scheme based on LQG control on the basis of realizing dynamic modeling of the system. An uncertainty model of the piezoelectric actuator is determined through experiments, and H based on a multi-target complementary control structure is provided ∞ The robust tracking controller enhances the robust anti-interference capability of the system while ensuring the tracking performance of the system. Experimental results show that based on the proposed tracking control strategy, good tracking of single-frequency and composite-frequency reference signals in a given frequency range can be realized, model uncertainty, interference and noise can be processed to a certain extent, PEA has better robustness and stability, and feasibility and effectiveness of the proposed method are proved. Overall, LQG control and robust H compared to Kalman filter algorithm based ∞ The control method provided by the invention has better tracking effect on the piezoelectric positioning system.
The foregoing is merely a preferred embodiment of the present application, but the scope of the present application is not limited thereto, and any changes or substitutions easily conceivable by those skilled in the art within the technical scope of the present application should be covered in the scope of the present application. Therefore, the protection scope of the present application shall be subject to the protection scope of the claims.
The above embodiments are merely illustrative of the preferred embodiments of the present invention, and the scope of the present invention is not limited thereto, but various modifications and improvements made by those skilled in the art to which the present invention pertains are made without departing from the spirit of the present invention, and all modifications and improvements fall within the scope of the present invention as defined in the appended claims.
Claims (7)
1. A multi-target complementary robust control method for a piezoelectric positioning system, comprising:
modeling the piezoelectric positioning system by adopting a Hammerstein model structure, and compensating hysteresis nonlinearity of the piezoelectric positioning system model based on a PI model to obtain a compensated system to be identified; dynamically modeling the compensated system to be identified based on an improved Hankel matrix system identification method to obtain a discrete transfer function and a continuous transmission function of a piezoelectric actuation model;
obtaining a standard feedback control model, obtaining a generalized controlled system model according to the standard feedback control model, designing an LQG controller based on Kalman filtering, obtaining an augmented system model containing LQG control, inputting the continuous transmission function into the augmented system model containing LQG control, and obtaining a multi-target complementary robust control result.
2. The method of claim 1, wherein obtaining a discrete transfer function of the piezoelectric actuation model comprises:
acquiring the compensated input and output signals of the system to be identified, determining an autocorrelation function and a cross correlation function of the input and output signals, and acquiring an input and output relation according to the autocorrelation function and the cross correlation function;
performing correlation analysis on the impulse response and the correlation function based on the input-output relationship, performing impulse response estimation, obtaining a pulse sequence to construct a Hankel matrix, and obtaining a second relationship between the Hankel matrix and the state space description of the system model to be identified through the first relationship between the impulse response and the state equation;
singular value decomposition is carried out on the second relation, the system order of a system model to be identified is obtained, secondary decomposition is carried out on the system order, a decomposition result is obtained, system state space model parameters A, B and C are obtained according to the decomposition result, and a discrete transfer function G of the piezoelectric actuation model is obtained in combination with a meeting condition D of the system model 0 (z) and continuous transmission function G 0 (s)。
3. The method of claim 1, wherein obtaining the generalized controlled system model comprises:
changing the standardThe method comprises the steps of feeding back an input information position of a controller in a control model, obtaining a multi-target complementary control structure, designing the multi-target complementary control structure into a state space model, obtaining a control input of the state space model, and designing the control input into H ∞ And the controller acquires the generalized controlled system model.
4. The multi-target complementary robust control method of a piezoelectric positioning system according to claim 1, wherein the method for obtaining the generalized controlled system model is as follows:
z＝C 1 x+D 11 w+D 12 u
y＝C 2 x+D 20 w 0 +D 21 w+D 22 u
wherein w is modeling uncertainty and disturbance signal, w 0 Is a white noise signal, u is a control input, y is a measured output or object output, z is a controlled output,for the speed of the system, A is the state transition matrix of the system, x is the state vector of the system, B 0 System matrix for white noise signal, B 1 To model the system matrix of uncertainty and disturbance signals, B 2 For input matrix, C 1 To control the output matrix, D 11 D for modeling uncertainty and direct transfer matrix of disturbance signal 12 To control the input direct transfer matrix, C 2 To be measured output matrix D 20 Is a white noise signal direct transfer matrix, D 21 To model the uncertainty and the direct transfer matrix of the disturbance signal, D 22 A direct transfer matrix for control inputs.
5. The method of claim 1, wherein obtaining the augmented system model including LQG control comprises:
converting the discrete transfer function of the piezoelectric actuation model into a continuous state space form, combining measurement noise, designing a Kalman filter for estimating the actual state, defining a filter error variance matrix Li Kadi model, removing the target time of the filter error variance matrix Li Kadi model, and obtaining the gain K of the steady state value of the Kalman filter error variance k Based on the function leq (), a symmetric positive definite matrix of the Li Kadi model and the gain K are obtained k Based on state feedback F and gain K k Substituting the generalized controlled system model to obtain the enhanced system model containing LQG control.
6. The method of claim 1, wherein the method for obtaining the model of the augmentation system comprising LQG control is:
z＝C 1 x+D 11 w+D 12 u l +D 12 u f
y＝C 2 x+D 21 w+D 22 u l +D 22 u f
wherein w is modeling uncertainty and disturbance signal, w 0 For a white noise signal, y is the measured output or object output, z is the controlled output,is the speed vector of the system, A is the state transition matrix of the system, x is the state vector of the system, B 0 System matrix for white noise signal, B 1 To model the system matrix of uncertainty and disturbance signals, B 2 For input matrix, u l For nominal control input, u f For additional adjustment signals->K is the speed estimate k For gain, F is the LQG control designed state feedback gain, C 1 To control the output matrix, D 11 D for modeling uncertainty and direct transfer matrix of disturbance signal 12 To control the input direct transfer matrix, C 2 To be measured output matrix D 20 Is a white noise signal direct transfer matrix, D 21 To model the uncertainty and the direct transfer matrix of the disturbance signal, D 22 For a direct transfer matrix of control inputs, +.>Is a control input.
7. The method of claim 1, wherein obtaining the multi-target complementary robust control result comprises:
introducing the augmentation system model containing LQG control into a parameter e for equivalent conversion, obtaining a conversion model, inputting the continuous transmission function into the conversion model, and obtaining an output result;
acquiring a static state feedback control model in the augmentation system model containing LQG control, simplifying the augmentation system model containing LQG control according to the static state feedback control model, and acquiring a corresponding system model;
determining a closed loop transfer function T according to the corresponding system model zw The satisfaction condition of (t) and H ∞ Obtaining the charge condition of the controllerThe matrix inequality model is used for acquiring a feasible solution model according to the matrix inequality model, substituting the output result into the matrix inequality model and the feasible solution model to acquire a multi-target complementary robust control result
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