CN105204332B - Dead band and sluggish compound Sandwich system method for estimating state are contained based on Non-smooth surface observer - Google Patents
Dead band and sluggish compound Sandwich system method for estimating state are contained based on Non-smooth surface observer Download PDFInfo
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Abstract
The invention discloses a kind of compound Sandwich system method for estimating state for containing dead band and sluggishness based on Non-smooth surface observer, first with key item separation principle and switching function, from simple to complex, use for reference the band dead band built, band gap and with sluggish Sandwich system Non-smooth surface state space equation, build the Non-smooth surface state space equation that energy accurate description contains dead band and sluggish compound Sandwich system；Secondly according to the compound Non-smooth surface sandwich state space equation of structure, when system meets the existence consition of observer, the Non-smooth surface state estimation observer that construction can change and automatically switch with compound Non-smooth surface Sandwich system operation interval.This method have the advantage that：Such system is more accurately described by introducing switching function；Disclosed compound Non-smooth surface observer can automatically switch with the switching between system work area；Compared with traditional observer, using this method observer can more accurately estimating system state value.
Description
Technical Field
The invention belongs to the field of state estimation of a nonlinear system, and particularly relates to a state estimation method of a composite sandwich system containing a dead zone and hysteresis based on a non-smooth observer.
Background
For any control system, no matter the closed-loop pole configuration of the system is realized, or the decoupling of the system is realized, the state feedback cannot be separated. Meanwhile, the accurate estimation of the state of the system is also an important basis for realizing the optimal control and fault diagnosis of the system. However, the state of the system is not always directly detectable, and some state variables may not even be detectable at all. Therefore, researchers often solve the above problem by constructing a state observer. However, for a specific system, if the observer is not constructed for state estimation according to the actual condition of the system, the state estimation is inevitably inaccurate. The system cannot be accurately controlled according to the inaccurate estimation state, and the inaccurate state estimation can also cause false alarm and false failure in fault diagnosis. Therefore, it is important to construct a state estimation observer capable of accurately estimating a specific system state and a fault prediction observer capable of accurately predicting a fault.
Dead zones, gaps and hysteresis are the most common non-smooth non-linear characteristics. The dead zone is widely existed in a direct current motor, a mechanical transmission system, a hydraulic transmission system, an electromechanical integrated system and a power amplifier circuit system. Backlash is widely present in gear mechanical transmission systems, electrically operated valves, digital circuits, sensors and hydraulic systems. In recent years, an intelligent execution device using memory alloy and piezoelectric ceramic has the advantages of high positioning accuracy, large driving force, fast response and the like, and is widely applied to positioning systems of precision equipment such as a precision machining machine tool, a flexible mechanical arm of a space shuttle, an astronomical telescope and the like. In addition, the rf circuit and the pll commonly used in the communication system also include a hysteresis sandwich system. In engineering practice, the non-linear characteristics of the dead zone, the gap and the hysteresis are not isolated, but are connected with other traditional links and clamped between two linear dynamic links, so that the system is called a non-smooth sandwich system. A non-smooth sandwich system with dead zones, gaps and hysteresis can be generally described by the structural block diagram shown in fig. 1.
However, in practical engineering, many systems are not typical non-smooth sandwich systems as shown in fig. 1, for example, a flexible robot arm is composed of an amplifying circuit, a dc motor, piezoelectric ceramics and an actuator of the robot arm, the amplifying circuit can be regarded as a linear element L1, the dc motor with a dead zone can be regarded as a dead zone element DZ, the piezoelectric ceramics with a hysteresis characteristic can be regarded as a hysteresis element HS, and finally the actuator of the robot arm can be regarded as a linear element L2. Thus, the entire system can be represented as the structure shown in fig. 2, with the non-smooth element consisting of a dead zone in series with a hysteresis.
To this end, the present invention defines a new class of non-smooth sandwich systems: if there is more than one intermediate link in a non-smooth sandwich system, but a plurality of non-smooth links are connected in series, such a non-smooth sandwich system is called a composite non-smooth sandwich system, and the system shown in FIG. 2 is a composite sandwich system with dead zones and hysteresis.
Constructing a corresponding observer for a particular system has always been a research hotspot in the field of control engineering. The design theory and methodology for a stationary linear system observer has matured since the well-known Luenberger observer was proposed in the 70 s of this century by d.j. Luenberger (1971). But is different for the nonlinear system, firstly, the visibility of the nonlinear system is a local characteristic; second, the visibility of a linear system is independent of the system inputs and depends only on the configuration of the system itself, while the visibility of a non-linear system is not only related to the system configuration but also to the system inputs. Due to the complexity of the nonlinear system, it is difficult to find a unified observer constructing method for the nonlinear system, and a specific observer is often constructed for a certain type of nonlinear system.
In terms of special system observer design, makroot, lemna (2003) conducted observer design studies on nonlinear systems that meet the Lipschitz condition. A dimension reduction observer is designed for a Lyapunov-like nonlinear system by Zhufanlai (2004) to carry out state estimation. Zhou Shaowu, Colosseum (2001) proposed a design method for an improved Lipschitz nonlinear System observer. Koelreuteri et al (2008) construct an observer of a nonlinear uncertain time-lag system, and use the observer to perform robust control on the system. Hanchun et al (2009) build a specific observer for a stochastic time lag system for state estimation. Wumin et al (2008) completed the state estimator design with an unobservable transition event graph. Lj. juloski et al (2002, 2003) design a Luenberger type switching observer for piecewise linear systems for state estimation. Alessandri et al (2001) solved the observer gain matrix of the piecewise linear switching system using a common Lyapunov function method. Zhou Zu Peng, Tan Yong hong (2011, 2012) designed a non-smooth observer for a single sandwich system with dead zones, gaps and hysteresis to perform state estimation.
However, the above state estimation methods are designed for observer development for specific systems, and in these methods, it is assumed that the system is completely observable or only one non-smooth link is given, whereas the composite non-smooth sandwich system is not completely observable in a certain working interval and has two series non-smooth links. Therefore, when designing a state estimation observer for a composite non-smooth sandwich system, the incomplete observation of the system and new problems after the composition must be considered at the same time. Therefore, the method expands the theory of the state observer of the complex system, solves the problem of accurate state estimation of the state variable of the complex sandwich system containing dead zones and hysteresis, and has certain scientific theoretical significance and practical application prospect.
Disclosure of Invention
In order to overcome the technical defects, the invention discloses a state estimation method for a composite sandwich system with a dead zone and hysteresis based on a non-smooth observer, which can solve the composite problem which cannot be solved by the single non-smooth observer compared with the single non-smooth sandwich system with the dead zone, the gap and the hysteresis.
The technical scheme for realizing the purpose of the invention is as follows:
the method for estimating the state of the composite sandwich system containing the dead zone and the hysteresis based on the non-smooth observer comprises the following steps:
step 1: constructing a non-smooth state space equation capable of accurately describing a composite sandwich system containing dead zones and hysteresis by using a key item separation principle and a switching function from simple to complex and by using the constructed non-smooth state space equation with the dead zones, the band gaps and the hysteresis sandwich system for reference;
step 2: according to the composite non-smooth sandwich state space equation constructed in the step 1, when the system meets the existence condition of the observer, a non-smooth state estimation observer capable of automatically switching along with the change of the working interval of the composite non-smooth sandwich system is constructed, and the existence condition and the convergence theorem of the corresponding non-smooth state estimation observer are given.
The step 1 comprises the following steps:
(1) front-end linear link L of composite sandwich system with dead zone and hysteresis1(.), as shown in equation (1):
rear-end linear link L of composite sandwich system with dead zone and hysteresis2(.), as shown in equation (2):
(2) modeling of dead zones and lags
A mathematical model between the input and the output of the dead zone link DZ is shown as a formula (3):
the mathematical model of the input-output relationship of the hysteresis is shown in equation (4):
wherein,
wherein,yi∈R1×1，u∈R1×1，vi∈R1×1，i＝1,2， x1iand x2iThe ith state variable respectively represents a front-end linear link and a rear-end linear link;is a matrix of state transitions that is,is an input matrix, yi∈R1×1Is an output variable, niDimension representing the ith Linear subsystem, u ∈ R1×1Is an input variable, v1∈R1×1Is an input variable of hysteresis and is also an output variable of dead zone, v2∈R1×1Is the output variable of the lag. z is a radical ofi(k) Is the output of the ith gap, wiIs the weight that the ith gap accounts for in the constituent lag, Bi(□) is a single roomThe input and output functions of the gaps, n is used for constructing the number of the delayed gaps; m is1bIs the slope of the gap, m1dIs the slope of the dead zone; d1bIs the width of the gap, D1dIs the width of the dead zone.
The step 2 comprises the following steps:
(1) the non-smooth observer of the system is constructed, according to equations (1), (2), (3) and (4), as follows:
wherein,
wherein,is a feedback matrix;
(2) given the theorem of existence and convergence of the observer of equation (5):
the system is set to meet the following conditions:
condition 1 state variable x is bounded, i.e.‖x(k)‖m≤xb，xbNot less than 0; wherein |)mDenotes m norm, xbRepresents the maximum boundary value of the state variable;
the initial error of the conditional 2 observer is bounded, i.e., | e (1) |m≤eb，eb≥0；ebA maximum boundary value representing an initial error of the observer;
condition 3 transfer matrix A of a front-end linear subsystem1All within the unit circle;
theorem: for a composite sandwich system containing dead zone and hysteresis and satisfying the above three assumed conditions, an observer shown in formula (5) can be constructed to estimate the state of the system, if a feedback matrix K of the observer is selected2So that (A)2-K2C22) Are within the unit circle, the estimation error of the observer shown in equation (5) eventually converges to zero, where a2Transfer matrix, K, representing a back-end linear system2Feedback matrix representing the back-end linear system, C22The output matrix of the back-end linear system is represented.
The method has the advantages that:
1. the nonlinear and non-smooth characteristics of the dead zone and the hysteresis and the two series composite characteristics are fully considered, the system is more accurately described by introducing a switching function into the composite sandwich system, an accurate mathematical model is established by a composite function formula (4) to describe the system, the dead zone and the hysteresis non-smooth characteristics are connected together by the formula (4), and the problem of accurately describing the composite sandwich is solved.
2. On the basis of an accurate mathematical model of a constructed system, a composite non-smooth state observer is constructed to estimate a state value of the composite non-smooth state observer, wherein the composite non-smooth observer can be automatically switched along with the switching of a system working interval, particularly, a formula (5) comprises a composite expression relation connecting two non-smooth links, the difficult problem of constructing the composite non-smooth sandwich system state observer is successfully solved, and the existing condition and convergence theorem of the system state observer is given.
3. Compared with a single unsmooth sandwich unsmooth observer with a dead zone, a gap and hysteresis, the constructed composite unsmooth sandwich system successfully solves the problem of state estimation of a plurality of unsmooth links in series, which cannot be solved by the traditional method.
4. Compared with the traditional linear observer, the constructed composite non-smooth state estimation method can estimate the state value of the system more accurately.
Drawings
FIG. 1 shows the structure of a non-smooth sandwich system;
FIG. 2 is a structure of a composite sandwich system containing dead zones and hysteresis;
FIG. 3 is a diagram of the input/output relationship of a dead zone link DZ;
FIG. 4 is a graph of the input-output characteristics of the lag;
FIG. 5 is a graph of the I-th gap input/output relationship;
FIG. 6 is a state estimation result diagram of an observer of the present invention;
FIG. 7 is a graph of state estimation results for a conventional linear observer;
FIG. 8 is a graph comparing state estimation errors for an observer of the present invention and a conventional linear observer.
Detailed Description
The invention will be further elucidated with reference to the following examples and drawings, without however being limited thereto:
examples
The method for estimating the state of the composite sandwich system containing the dead zone and the hysteresis based on the non-smooth observer comprises the following steps:
step 1: by utilizing the key item separation principle and the switching function, the non-smooth state space equation of the composite sandwich system which can accurately describe the dead zone and the hysteresis is constructed by using the constructed non-smooth state space equation of the sandwich system with the dead zone, the band gap and the hysteresis from simple to complex.
The step 1 comprises the following steps:
(1) front-end linear link L of composite sandwich system with dead zone and hysteresis1：
Rear-end linear link L of composite sandwich system with dead zone and hysteresis2：
(2) Modeling of dead zones and lags
The input and output relationship of the dead zone link DZ is shown in FIG. 3, and the mathematical model between the input and the output is as follows:
wherein m is1d＝m2d＝1,D1d＝D2d＝0.01。
The input-output characteristics of the hysteresis are shown in fig. 4, and the mathematical model of the input-output relationship is:
HS: the hysteresis is formed by superposing n-7 gaps, and the parameters of the 7 gaps are as follows:
wi＝1,m1b＝m2b＝1,D1b＝D2b＝ci/2,(i＝1,2,...,7)，
c1＝0.14,c2＝0.12,c3＝0.1,c4＝0.08,c5＝0.06,c6＝0.04,c70.02, wherein ciThe width of the ith gap is shown, and the input-output relationship of the ith gap is shown in fig. 5.
Step 2: according to the composite non-smooth sandwich state space equation constructed in the step 1, when the system meets the existence condition of the observer, a non-smooth state estimation observer capable of automatically switching along with the change of the working interval of the composite non-smooth sandwich system is constructed, and the existence condition and the convergence theorem of the corresponding non-smooth state estimation observer are given.
The step 2 comprises the following steps:
(1) the non-smooth state observer as shown in formula (5) proposed in the summary of the invention was constructed as follows:
wherein,
c1＝0.14,c2＝0.12,c3＝0.1,c4＝0.08,c5＝0.06,c6＝0.04,c7＝0.02，
wherein, K2＝[0.1 0.1]T∈R2×1Is a feedback matrix.
(2) The non-smooth observer equation (6) was analyzed for convergence:
the embodiment satisfies three conditions of the convergence theorem:
condition 1: the state variable x of an embodiment is bounded, i.e. it is‖x(k)‖1≤ 100, wherein |1Represents a norm of 1;
condition 2: the initial error of the observer in the embodiment is bounded, i.e. | e (1) |1≤20；
Condition 3: transfer matrix A of a front-end linear subsystem of an embodiment1Has a characteristic value of [0.8,0.45 ]]TAre all within the unit circle;
in the examples, K is selected2＝[0.1 0.1]TThen (A)2-K2C2) The characteristic value of (1) is [0.8000+0.0316i,0.8000-0.0316i ]]TAll within the unit circle, the estimated value of the state of the system will eventually converge to the true value according to the convergence theorem in step 2.
In order to illustrate the effectiveness of the state estimation method and the non-smooth state observer of the present invention, the present embodiment further constructs a conventional linear observer to perform state estimation on the same system, where the linear observer is shown in formula (7):
wherein,
wherein A represents a transition matrix, B represents an input matrix, KlRepresenting the gain matrix, setting the gain matrix K of the linear observerl＝[0 0 0.1 0.1]TAnd given the initial state values of the observer as allThe initial values of the simulation system are all x (0) [0,0,0,0 ]]TThe sampling period was 0.01 seconds and the simulation time was 3 seconds.
As can be seen from fig. 6, the state estimation method of the present invention can well track the actual state value of the system, and obtain an accurate state estimation value. As can be seen from FIG. 7, the conventional linear observer cannot accurately estimate x of the system21(k) And x22(k) Two state values. As can be seen from FIG. 8, for x21(k) And x22(k) For two states, the estimation error of the conventional linear observer estimation method is much larger than that of the state estimation method of the invention.
Claims (1)
1. The method for estimating the state of the composite sandwich system containing the dead zone and the hysteresis based on the non-smooth observer comprises the steps of constructing a non-smooth state space equation capable of accurately describing the composite sandwich system containing the dead zone and the hysteresis by using a key item separation principle and a switching function from simple to complex by taking reference to the constructed non-smooth state space equation of the sandwich system with the dead zone, the band gap and the hysteresis, and is characterized in that: also comprises
According to the constructed composite non-smooth sandwich state space equation, when the system meets the existence condition of the observer, a non-smooth state estimation observer capable of automatically switching along with the change of the working interval of the composite non-smooth sandwich system is constructed, and the existence condition and the convergence theorem of the corresponding non-smooth state estimation observer are given, which specifically comprises the following steps:
(1) the non-smooth observer of the system is constructed, and the state observer of the composite sandwich system containing the dead zone and the hysteresis is constructed as follows:
wherein,is a feedback matrix;
(2) given the theorem of existence and convergence of the observer of equation (5):
the system is set to meet the following conditions:
condition 1 state variableIs bounded, i.e.WhereinThe number of the m-norm is expressed,represents the maximum boundary value of the state variable;
the initial error of the condition 2 observer is bounded, i.e.A maximum boundary value representing an initial error of the observer;
condition 3 transfer matrix for a front-end linear subsystemAll within the unit circle;
theorem: for a composite sandwich system containing dead zone and hysteresis and satisfying the above three assumed conditions, an observer as shown in formula (5) can be constructed to estimate the state of the system, if a feedback matrix of the observer is selectedSo thatAre within the unit circle, the estimated error of the observer shown in equation (5) eventually converges to zero, where,a transition matrix representing the back-end linear system,a feedback matrix representing the back-end linear system,the output matrix of the back-end linear system is represented.
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