CN105204332B - Dead band and sluggish compound Sandwich system method for estimating state are contained based on Non-smooth surface observer - Google Patents

Abstract

The invention discloses a non-smooth observer-based method for estimating the state of a compound sandwich system with dead zone and hysteresis. Firstly, the principle of separation of key items and switching functions are used to learn from the established dead zone and gap. and the non-smooth state-space equation of the sandwich system with hysteresis, and construct the non-smooth state-space equation that can accurately describe the composite sandwich system containing dead zone and hysteresis; secondly, according to the constructed composite non-smooth sandwich state-space equation, when the system satisfies the existence of the observer When the non-smooth condition is met, a non-smooth state estimation observer that can automatically switch with the change of the working range of the compound non-smooth sandwich system is constructed. The advantages of this method are: by introducing a switching function to describe this type of system more accurately; the disclosed compound non-smooth observer can automatically switch with the switching of the system working range; compared with the traditional observer, the observer using this method can Estimate the state value of the system more accurately.

Description

State of Compound Sandwich System with Dead Zone and Hysteresis Based on Nonsmooth Observer estimation method

technical field

The invention belongs to the field of state estimation of nonlinear systems, in particular to a method for state estimation of a composite sandwich system with dead zones and hysteresis based on non-smooth observers.

Background technique

For any control system, whether it is to realize the closed-loop pole configuration of the system or to realize the decoupling of the system, it is inseparable from the state feedback. At the same time, accurately estimating the state of the system is also an important basis for realizing the optimal control and fault diagnosis of the system. However, not all states of the system are easily detectable directly, and some state variables cannot even be detected at all. Therefore, researchers often solve the above problems by constructing state observers. However, for a specific system, if the observer is not constructed according to the actual situation of the system for state estimation, it will inevitably lead to inaccurate state estimation. Accurate control of the system cannot be realized based on inaccurate state estimation. In fault diagnosis, inaccurate state estimation will also lead to false positives and false negatives of faults. Therefore, it is particularly important to construct a state estimation observer that can accurately estimate a specific system state and a fault prediction observer that can realize accurate fault prediction.

Dead zone, gap and hysteresis are the most common non-smooth nonlinear characteristics. Dead zones widely exist in DC motors, mechanical transmission systems, hydraulic transmission systems, mechatronics systems and power amplifier circuit systems. Backlash widely exists in gear mechanical transmission systems, electric valves, digital circuits, sensors and hydraulic systems. In recent years, intelligent actuators using memory alloys and piezoelectric ceramics have been widely used in the positioning of precision equipment such as precision machine tools, flexible robotic arms of space shuttles, and astronomical telescopes due to their advantages such as high positioning accuracy, large driving force, and fast response. In the system, but there is a non-smooth nonlinearity such as hysteresis in memory alloys and piezoelectric ceramics. Therefore, hysteresis widely exists in the positioning systems of precision machine tools, flexible robotic arms of space shuttles, and astronomical telescopes. In addition, radio frequency circuits and phase-locked loops commonly used in communication systems also contain hysteresis sandwich systems. In engineering practice, the dead zone, gap and hysteresis nonlinear characteristics are often not isolated, but connected with other traditional links and sandwiched between two linear dynamic links, which is called a non-smooth sandwich system. The non-smooth sandwich system with dead zone, gap and hysteresis can be described by the structural block diagram shown in Figure 1.

However, in actual engineering, many systems are not the typical non-smooth sandwich system shown in Figure 1. For example, a flexible robotic arm is composed of an amplifier circuit, a DC motor, piezoelectric ceramics, and the actuator of the robot arm. The circuit can be regarded as a linear link L1, the dead zone of the DC motor can be regarded as a dead zone link DZ, the hysteresis characteristic of piezoelectric ceramics can be regarded as a hysteresis link HS, and finally, the actuator of the robot arm can be regarded as a linear link L2. Therefore, the whole system can be expressed as the structure shown in Figure 2, and its non-smooth link is composed of a dead zone and a hysteresis in series.

For this reason, the present invention defines a new type of non-smooth sandwich system: if there is more than one intermediate link in the non-smooth sandwich system, but a plurality of non-smooth links connected in series, such a non-smooth sandwich system is called a composite non-smooth sandwich system system, the system shown in Figure 2 is a compound sandwich system containing dead zone and hysteresis.

Constructing corresponding observers for specific systems has always been a research hotspot in the field of control engineering. Since D.J. Luenberger (1971) proposed the famous Luenberger observer in the 1970s, the design theory and method for the observer of steady linear system have been mature. But it is different for nonlinear systems. First, the observability of nonlinear systems is a local characteristic; Observability is not only related to system structure, but also related to system input. Because of the complexity of nonlinear systems, it is difficult for us to find a unified observer construction method for nonlinear systems, and we often construct a specific type of observer for a certain type of nonlinear system.

In terms of observer design for special systems, Ma Kemao and Ma Ping (2003) conducted research on observer design for nonlinear systems satisfying Lipschitz conditions. Zhu Fanglai (2004) designed a reduced-dimensional observer for Lyapunov-like nonlinear systems for state estimation. Zhou Shaowu and Zhang Jing (2001) proposed an improved design method of Lipschitz nonlinear system observer. Luan Xiaoli et al. (2008) constructed an observer for a nonlinear uncertain time-delay system, and used the observer to robustly control this type of system. Han Chunyan et al. (2009) constructed a specific observer for stochastic time-delay systems for state estimation. Wu Min et al. (2008) completed the design of state estimator with unobservable transition event graph. A.Lj.Juloski. et al. (2002, 2003) designed a Luenberger-type switched observer for state estimation for piecewise linear systems. A. Alessandri et al. (2001) used the shared Lyapunov function method to solve the observer gain matrix of a piecewise linear switched system. Zhou Zupeng, Tan Yonghong (2011, 2012) designed a non-smooth observer for a single sandwich system with dead zone, gap and hysteresis for state estimation.

However, the above state estimation methods are designed for the observer for a specific system. In these methods, it is assumed that the system is completely observable or given only one non-smooth link, and the composite non-smooth sandwich system is incomplete in a certain working range. Observable and has two non-smooth links in series. Therefore, when designing state estimation observers for composite non-smooth sandwich systems, the incomplete observability of the system and the new problems after composite must be considered at the same time. Therefore, the present invention will expand the theory of state observers for complex systems and solve the problem of accurate state estimation of state variables in compound sandwich systems containing dead zones and hysteresis. The present invention has certain scientific and theoretical significance and practical application prospects.

Contents of the invention

Aiming at the deficiencies of the above technologies, the present invention discloses a state estimation method for a compound sandwich system with dead zone and hysteresis based on a non-smooth observer, compared with a single non-smooth sandwich system with dead zone, gap and hysteresis , using this method can solve complex problems that cannot be solved by a single non-smooth observer. Compared with the traditional observer, the observer using this method can estimate the state value of the system more accurately.

The technical scheme that realizes the object of the present invention is:

The state estimation method of compound sandwich system with dead zone and hysteresis based on non-smooth observer includes the following steps:

Step 1: Using the key item separation principle and switching function, from simple to complex, drawing on the non-smooth state space equations of the sandwich system with dead zone, gap and hysteresis, construct a composite sandwich that can accurately describe dead zone and hysteresis The non-smooth state-space equation of the system;

Step 2: According to the compound non-smooth sandwich state space equation constructed in step 1, when the system satisfies the existence condition of the observer, construct a non-smooth state estimation observer that can automatically switch with the change of the working range of the compound non-smooth sandwich system, and The existence conditions and convergence theorems of the corresponding nonsmooth state estimation observers are given.

Described step 1 comprises the following steps:

(1) The front-end linear link L ₁ (.) of the composite sandwich system with dead zone and hysteresis, as shown in formula (1):

The back-end linear link L ₂ (.) of the compound sandwich system with dead zone and hysteresis is shown in formula (2):

(2) Modeling of dead zone and hysteresis

The mathematical model between the input and output of the dead zone link DZ is shown in formula (3):

The mathematical model of the hysteresis input-output relationship is shown in formula (4):

in,

in, y _i ∈ R ^1×1 , u∈R ^1×1 , v _i ∈R 1× ¹ , i=1,2, x _1i and x _2i respectively represent the i-th state variable of the front-end linear link and the back-end linear link; is the state transition matrix, is the input matrix, y _i ∈ R ^1×1 is the output variable, n _i represents the dimension of the i-th linear subsystem, u∈R ^1×1 is the input variable, v ₁ ∈ R ^1×1 is the hysteresis input variable It is also the output variable of the dead zone, and v ₂ ∈ R ^1×1 is the output variable of the hysteresis. z _i (k) is the output of the i-th gap, w _i is the weight of the i-th gap in forming the hysteresis, B _i (□) is the input-output function of a single gap, n is used to construct the hysteresis gap Number; m _1b is the slope of the gap, m _1d is the slope of the dead zone; D _1b is the width of the gap, D _1d is the width of the dead zone.

Described step 2 comprises the following steps:

(1) Construct the non-smooth observer of the system. According to formulas (1), (2), (3) and (4), the state observer of the compound sandwich system with dead zone and hysteresis is constructed as follows:

in,

in, is the feedback matrix;

(2) Given the existence condition and convergence theorem of the observer in formula (5):

Set the system to meet the following conditions:

Condition 1 The state variable x is bounded, ie ‖x(k)‖ _m ≤ x _b , x _b ≥ 0; where ‖‖ _m represents the norm of m, and x _b represents the maximum boundary value of the state variable;

Condition 2 The initial error of the observer is bounded, that is, ‖e(1)‖ _m ≤ e _b , e _b ≥ 0; e _b represents the maximum boundary value of the initial error of the observer;

Condition 3 The eigenvalues of the transition matrix A ₁ of the front-end linear subsystem are all within the unit circle;

Theorem: For a composite sandwich system with dead zone and hysteresis that satisfies the above three assumptions, an observer as shown in equation (5) can be constructed to estimate the state of the system. If the feedback matrix K ₂ of the observer is selected such that The eigenvalues of (A ₂ -K ₂ C ₂₂ ) are all within the unit circle, then the estimation error of the observer shown in equation (5) will eventually converge to zero, where A ₂ represents the transition matrix of the back-end linear system, K ₂ represents the feedback matrix of the back-end linear system, and C ₂₂ represents the output matrix of the back-end linear system.

The advantage of the inventive method is:

1. Fully consider the non-linear and non-smooth characteristics of the dead zone and hysteresis and the two kinds of series composite characteristics. In the composite sandwich system, the switching function is introduced to describe this type of system more accurately, and the composite function formula (4) is established. An accurate mathematical model is established to describe this type of system. Formula (4) connects the dead zone and hysteretic non-smooth properties together, and solves the problem of how to accurately describe the composite sandwich.

2. On the basis of the accurate mathematical model of the constructed system, construct a composite non-smooth state observer to estimate its state value, wherein the composite non-smooth observer can automatically switch with the switching of the system working range, especially in the formula (5) It includes the compound expression relation connecting two non-smooth links, successfully solves the problem of constructing the state observer of the compound non-smooth sandwich system, and gives the existence condition and convergence theorem of the state observer of this kind of system.

3. Compared with the single non-smooth sandwich non-smooth observer with dead zone, gap, and hysteresis, the constructed compound non-smooth sandwich system successfully solves the state estimation problem of multiple non-smooth links in series, which is the past method unsolvable.

4. Compared with the traditional linear observer, the constructed compound non-smooth state estimation method can estimate the state value of the system more accurately.

Description of drawings

Fig.1 Structure of non-smooth sandwich system;

Fig. 2 Structure of compound sandwich system with dead zone and hysteresis;

Figure 3 is the input-output relationship diagram of the dead zone link DZ;

Figure 4 hysteresis input and output characteristic diagram;

The input-output relationship diagram of the i-th gap in Fig. 5;

Fig. 6 is a state estimation result diagram of the observer of the present invention;

Fig. 7 State estimation result diagram of traditional linear observer;

Fig. 8 is a comparison diagram of the state estimation error between the observer of the present invention and the traditional linear observer.

detailed description

Below in conjunction with embodiment and accompanying drawing, content of the present invention will be further elaborated, but not limitation of the present invention:

Example

The state estimation method of compound sandwich system with dead zone and hysteresis based on non-smooth observer includes the following steps:

Step 1: Using the key item separation principle and switching function, from simple to complex, drawing on the non-smooth state space equations of the sandwich system with dead zone, gap and hysteresis, construct a composite sandwich that can accurately describe dead zone and hysteresis Nonsmooth state-space equations for the system.

Step 1 includes the following steps:

(1) Front-end linear link L ₁ of compound sandwich system with dead zone and hysteresis:

Back-end linear link L ₂ of compound sandwich system with dead zone and hysteresis:

(2) Modeling of dead zone and hysteresis

The relationship between the input and output of the dead zone DZ is shown in Figure 3, and the mathematical model between its input and output is:

Wherein, m _1d =m _2d =1, D _1d =D _2d =0.01.

The input and output characteristics of hysteresis are shown in Figure 4, and the mathematical model of its input and output relationship is:

HS: The hysteresis is formed by superimposing n=7 gaps, and the parameters of the 7 gaps are as follows:

w _i =1, m _1b =m _2b =1, D _1b =D _2b =c _i /2, (i=1,2,...,7),

c ₁ =0.14, c ₂ =0.12, c ₃ =0.1, c ₄ =0.08, c ₅ =0.06, c ₆ =0.04, c ₇ =0.02, where c _i represents the width of the i-th gap, and the i-th gap The input-output relationship is shown in Figure 5.

Step 2: According to the compound non-smooth sandwich state space equation constructed in step 1, when the system satisfies the existence condition of the observer, construct a non-smooth state estimation observer that can automatically switch with the change of the working range of the compound non-smooth sandwich system, and The existence conditions and convergence theorems of the corresponding nonsmooth state estimation observers are given.

Step 2 includes the following steps:

(1) Construct the non-smooth state observer as shown in formula (5) proposed in the summary of the invention as follows:

in,

c ₁ =0.14, c ₂ =0.12, c ₃ =0.1, c ₄ =0.08, c ₅ =0.06, c ₆ =0.04, c ₇ =0.02,

Wherein, K ₂ =[0.1 0.1] ^T ∈ R ^2×1 is the feedback matrix.

(2) Analyze the convergence of the non-smooth observer formula (6):

The embodiment satisfies the three conditions of the convergence theorem:

Condition 1: The state variable x of the embodiment is bounded, i.e. ‖x(k)‖ ₁ ≤ 100, where ‖‖ ₁ means 1 norm;

Condition 2: the initial error of the observer in the embodiment is bounded, that is, ‖e( ₁ )‖ 1≤20;

Condition ₃ : The eigenvalues of the transition matrix A1 of the front-end linear subsystem of the embodiment are [0.8,0.45] and ^T is all within the unit circle;

In the embodiment, select K ₂ =[0.1 0.1] ^T , then the eigenvalues of (A ₂ -K ₂ C ₂ ) [0.8000+0.0316i,0.8000-0.0316i] ^T are all within the unit circle, according to step 2 The convergence theorem shows that the estimated value of the state of the system will eventually converge to the true value.

In order to illustrate the effectiveness of the state estimation method and the non-smooth state observer of the present invention, this embodiment also constructs a traditional linear observer to estimate the state of the same system, and the linear observer is shown in formula (7):

in,

Among them, A represents the transition matrix, B represents the input matrix, K _l represents the gain matrix, the gain matrix K _l of the linear observer is set =[0 0 0.1 0.1] ^T , and the initial state value of the given observer is The initial values of the simulation system are all x(0)=[0,0,0,0] ^T , the sampling period is 0.01 second, and the simulation time is 3 seconds.

It can be seen from FIG. 6 that the state estimation method of the present invention can track the actual state value of the system well and obtain accurate state estimation value. It can be seen from Fig. 7 that the traditional linear observer cannot accurately estimate the two state values of x ₂₁ (k) and x ₂₂ (k) of the system. It can be seen from Fig. 8 that for the two states of x ₂₁ (k) and x ₂₂ (k), the estimation error of the traditional linear observer estimation method is much larger than the estimation error of the state estimation method of the present invention.

Claims (1)

1. State estimation method for compound sandwich system with dead zone and hysteresis based on non-smooth observer, including the use of key item separation principle and switching function, from simple to complex, drawing on the established sandwich with dead zone, gap and hysteresis The non-smooth state-space equation of the system is a step of constructing a non-smooth state-space equation that can accurately describe the compound sandwich system containing dead zone and hysteresis, and is characterized in that: it also includes According to the constructed compound non-smooth sandwich state space equation, when the system satisfies the existence condition of the observer, construct a non-smooth state estimation observer that can automatically switch with the change of the working range of the compound non-smooth sandwich system, and give the corresponding non-smooth The steps of the existence condition and convergence theorem of the state estimation observer are as follows: (1) Construct the non-smooth observer of the system, and construct the state observer of the composite sandwich system with dead zone and hysteresis as follows: in, is the feedback matrix; (2) Given equation (5) the existence condition and convergence theorem of the observer: Set the system to meet the following conditions: condition 1 state variable is bounded, that is, in Indicates the m-norm, Indicates the maximum boundary value of the state variable; Condition 2 The initial error of the observer is bounded, namely Indicates the maximum boundary value of the initial error of the observer; Condition 3 Transition matrix of front-end linear subsystem The eigenvalues of are all within the unit circle; Theorem: For a compound sandwich system with dead zone and hysteresis that satisfies the above three assumptions, an observer as shown in equation (5) can be constructed to estimate the state of the system. If the feedback matrix of the observer is selected , making The eigenvalues of are all within the unit circle, then the estimation error of the observer shown in equation (5) will eventually converge to zero, where, represents the transition matrix of the back-end linear system, Represents the feedback matrix of the back-end linear system, Represents the output matrix of the backend linear system.