CN105807634A - Nonlinear system event trigger controller designing method based on extended state observer - Google Patents

Abstract

The invention belongs to the field of controller design, aims at decreasing sampling points of an extended state observer, effectively saving network bandwidth resources, reducing network communication pressure and meanwhile ensuring the performance and stability of a closed-loop system adopting active disturbance rejection control, applies an event trigger mechanism to controller design and provides a nonlinear system event trigger controller designing method based on the extended state observer.The nonlinear system event trigger controller designing method specifically comprises the following steps that step 1, a single-input single-output nonlinear system model (shown in the description) is established; step 2, for estimating a state of the system and an extended state defined in the step (2), the extended state observer is designed; step 3, a sampling moment ti is determined; step 4, the state, sampled at the sampling moment ti, of the extended state observer is transmitted to the controller.The nonlinear system event trigger controller designing method is mainly applied to controller design occasions.

Description

Based on extended state observer nonlinear system event trigger controller method for designing

Technical field

The invention belongs to controller design field, specifically, relate to a kind of state utilizing event trigger theory sampling extended state observer, and utilize the method that this sample states carrys out design con-trol device.

Background technology

Many physical objecies in reality, due to the interference of inside and out, cause that it has very big uncertainty.The development of robust control and Self Adaptive Control solves many such problems.But, above-mentioned control method is likely to make designed controller become more to guard, and then Han Jing is clear et al. proposes Auto Disturbances Rejection Control Technique.Auto Disturbances Rejection Control Technique is a kind of unconventional control strategy, and its design principle is similar to outer modulus principle.Under the framework of Active Disturbance Rejection Control, the dynamic characteristic that controlled device is unknown is taken as the expansion state of object, and can be estimated by design extended state observer.Guo Baozhu et al. proves theoretically, utilizes the controller of auto-disturbance rejection technology and extended state observer design that closed loop system can be made to reach stable.

Current Auto-disturbance-rejection Controller Design method, is essentially all in the state of moment point up-sampling extended state observer of fixed cycle and is sent to controller.Above-mentioned periodically sample and to send the method for data simple, easily operated, but sometimes can cause the waste of mass communication resource.Along with embedded microcontroller is in the extensive use of many occasions, how making full use of resource extremely limited in flush bonding processor becomes vital problem.For resource-constrained embedded control system, periodically sampling is transmitted and performed control task not only wastes mass communication bandwidth, increase network transmission pressure, cause network delay and packet loss, also take up the calculating resource of CPU, the control task having requirement of real-time is adversely affected, consequently, it is possible to have influence on the performance even stability of whole control system.

Compared to the Time Triggered strategy of periodic samples, event trigger policy has only to just carry out sampling transmission when a certain event condition set in advance occurs, and the performance controlling system is similar to the systematic function under Time Triggered.By selecting suitable event condition, event trigger policy significantly reduces sampled point, thus effectively having saved network bandwidth resources.Although the existing a large amount of research work about event trigger policy, but up to the present, but without how, event trigger policy is applied to the discussion in Auto Disturbances Rejection Control Technique.Therefore, designing a kind of automatic disturbance rejection controller adopting event trigger policy is have very strong theory and realistic meaning.

Summary of the invention

In order to reduce the sampled point of extended state observer, effectively save network bandwidth resources, reduce network communication pressure, ensure to adopt performance and the stability of the closed loop system of Active Disturbance Rejection Control simultaneously, event trigger mechanism is applied in the design of controller by the present invention, based on extended state observer nonlinear system event trigger controller method for designing, specifically include following steps:

Step 1: set up the nonlinear system model of following single-input single-output:

$y^{\left(n\right)}=f(y,\stackrel{\·}{y},\mathrm{...},y^{(n-1)})+bu---\left(1\right)$

Wherein y is controlled output, and u is for controlling input, and b is given constant,For the first derivative of y, y⁽ⁱ⁾The i order derivative of expression y, i=2 ..., n-1, n, willBeing defined as f, f is single order continuously differentiable function, represents the Nonlinear Dynamic of controlled device, it addition, f should meet f (0)=0；

The model that formula (1) represents is rewritten as following state space form:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ .\\ .\\ .\\ {\stackrel{\·}{x}}_{n-1}=x_n\\ {\stackrel{\·}{x}}_n=x_{n+1}+bu\\ {\stackrel{\·}{x}}_{n+1}=h\\ y=x_1\end{array}\right.---\left(2\right)$

Wherein x_iIt is defined as the state of system, and with x=[x₁,x₂,...,x_n]^TThe state vector of expression system, x_n+1=f is the expansion state of system,Represent x_iFirst derivative, and defineNamely h is the first derivative of f；

Step 2: for state and (2) middle expansion state defined of estimating system, design following extended state observer:

$\left\{\begin{array}{c}{\stackrel{\·}{\hat{x}}}_1={\hat{x}}_2+{\mathrm{\ϵ}}^{n-1}{g}_1\left(\frac{x_1-{\hat{x}}_1}{{\mathrm{\ϵ}}^n}\right)\\ {\stackrel{\·}{\hat{x}}}_2={\hat{x}}_3+{\mathrm{\ϵ}}^{n-2}{g}_2\left(\frac{x_1-{\hat{x}}_1}{{\mathrm{\ϵ}}^n}\right)\\ .\\ .\\ .\\ {\stackrel{\·}{\hat{x}}}_n={\hat{x}}_{n+1}+g_n\left(\frac{x_1-{\hat{x}}_1}{{\mathrm{\ϵ}}^n}\right)+bu\\ {\stackrel{\·}{\hat{x}}}_{n+1}={\mathrm{\ϵ}}^{-1}{g}_{n+1}\left(\frac{x_1-{\hat{x}}_1}{{\mathrm{\ϵ}}^n}\right)\end{array}\right.---\left(3\right)$

WhereinRepresent system mode x_iEstimated value,Expression system expansion state x_n+1Estimated value, definitionFor the state vector of extended state observer, ε is a normal number, nonlinear function g_iChoose needs ensure when ε → 0, the state of extended state observer converges on the state of system；

Step 3: determine sampling instant t_i, definition sampling error e (t) is:

$e\left(t\right)=C_0\left|\right|\hat{x}\left(t\right)-\hat{x}\left(t_i\right)\left|\right|+\left|\right|{\hat{x}}_{n+1}\left(t\right)-{\hat{x}}_{n+1}\left(t_i\right)\left|\right|$

Wherein C₀To choose the controller designed to following step 4 relevant,Represent t system mode x_iEstimated value, by choosing suitable activation threshold value γ (t), and assume that first sampling instant is t₀, obtain a series of trigger instants:

t_i+1=inf{t > t_i|e(t)≥γ(t)}

Wherein i is natural number, and inf{} represents infimum；

Step 4: will at sampling instant t_iThe state of the extended state observer that sampling obtains sends controller to, and controller end utilizes this sampled value to calculate the output of executor, i.e. the control input of controlled system:

WhereinBeing Li Puxizi function, its Li Puxizi constant is C₀,Choose and need to meetAnd ensure undisturbed systemIt is asymptotically stable.

Bigger activation threshold value is taken so that do not cause frequent triggering because the change of observer state is too fast at initial time, thus saving communication bandwidth when just starting；And system basicly stable after, the change of sampling error e (t) along with observer state stable and slack-off, now choose less activation threshold value so that system mode converge to from initial point closer to region in；It is to say, activation threshold value to be chosen for the function successively decreased gradually.

Adopt the function of exponential decrease as activation threshold value function.

Compared with the prior art, the technical characterstic of the present invention and effect:

Event triggering method proposed by the invention have only to extended state observer part to currency observation with on once sampled value compare, only when sampling error exceedes previously given activation threshold value, observer just needs current sample values to be sent to controller.Controller then utilizes this sampled value to calculate and updates the output of executor, and within all the other times being not received by the sampled value that observer is sent, the output of executor remains unchanged.

Compared with the Auto-disturbance-rejection Control of tradition periodic sampling, the method that event triggers have only to currency with on the error of a moment sampled value exceed certain limit, just carry out transmission of sampling when namely system state change is bigger, therefore can be significantly reduced sampling number.Especially for permanent value control system, after system stability, the value that periodic samples obtains is essentially identical, and the output of controller is also held essentially constant, and now carrying out periodically sampling transmission obviously will waste substantial amounts of Internet resources.Except saving network bandwidth resources, it is calculated owing to controller end has only to when receiving sampled value and updates executor's output, therefore taking controller end cpu resource is decreased, the system of improve processes the real-time of other tasks, reduce the renewal frequency of executor simultaneously, contribute to reducing executor's abrasion, improve executor's life-span.

It should be noted that, event proposed by the invention is used to trigger Active Disturbance Rejection Control, closed loop system finally can only be stabilized to a bounded domain, as long as but activation threshold value choose sufficiently small, and choose suitable extended state observer parameter, system mode just can be allowed to be stabilized in an arbitrarily small region, and such zonule is stably usually acceptable in practice.It addition, observer part compares, it is necessary to take fractional hardware or software resource, but for the limited system of the communication resource, save the limited communication resource even more important, and event triggering method also mitigates the burden of controller end.

Accompanying drawing explanation

Fig. 1 is closed-loop control system structure chart.

Fig. 2 is event trigger element workflow diagram.

Fig. 3 is event trigger element fundamental diagram.

Fig. 4 is controller design flow diagram.

Detailed description of the invention

Transmit the sampled value of now extended state observer when controller design of the present invention is to utilize trigger conditions to reach to controller, then pass through controller end and calculate the output updating executor, so that whole closed loop system reaches stable.Save network bandwidth resources so on the one hand, also mitigate the computation burden of controller on the other hand and reduce the renewal frequency of executor.Specific implementation is: initially set up the nonlinear model of system, and this model is integration chain form, and is translated into state-space model, then designs corresponding extended state observer；Designing suitable trigger condition on this basis, only just sample when trigger condition meets and send observer state, controller calculates by this sampled value and updates executor's output, thus the control task of completion system.

In order to be illustrated more clearly that the purpose of the present invention, technical scheme and advantage, setting up from model below, design principle, the present invention is further explained explanation by several aspects such as method for designing.Should be appreciated that specific design method described herein is only in order to explain the present invention, is not intended to limit the present invention.

Event based on a nonlinear systems of extended state observer triggers control, comprises the following steps that.

Step 1: set up the nonlinear system model of following single-input single-output:

$y=f(y,\stackrel{\·}{y},\mathrm{...},y^{(n-1)})+bu---\left(1\right)$

Wherein y is controlled output, and u is for controlling input, and b is given constantFor the first derivative of y, y⁽ⁱ⁾(i=2 ..., n-1, n) represent the i order derivative of y.We simply willBeing defined as f, it is single order continuously differentiable function, represents the Nonlinear Dynamic of controlled device, and it is possible to be unknown.It is to say, controlled system necessarily can be expressed as the system of integration chain form, and f needs to meet f (0)=0.

The model that formula (1) represents can be rewritten as following state space form:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ .\\ .\\ .\\ {\stackrel{\·}{x}}_{n-1}=x_n\\ {\stackrel{\·}{x}}_n=x_{n+1}+bu\\ {\stackrel{\·}{x}}_{n+1}=h\\ y=x_1\end{array}\right.---\left(2\right)$

Wherein x_i(i=1,2 ..., n) it is defined as the state of system, and with x=[x₁,x₂,...,x_n]^TThe state vector of expression system, x_n+1=f is the expansion state of system,Represent x_iFirst derivative, and defineNamely h is the first derivative of f.

Step 2: for state and (2) middle expansion state defined of estimating system, we design following extended state observer:

$\left\{\begin{array}{c}{\stackrel{\·}{\hat{x}}}_1={\hat{x}}_2+{\mathrm{\ϵ}}^{n-1}{g}_1\left(\frac{x_1-{\hat{x}}_1}{{\mathrm{\ϵ}}^n}\right)\\ {\stackrel{\·}{\hat{x}}}_2={\hat{x}}_3+{\mathrm{\ϵ}}^{n-2}{g}_2\left(\frac{x_1-{\hat{x}}_1}{{\mathrm{\ϵ}}^n}\right)\\ .\\ .\\ .\\ {\stackrel{\·}{\hat{x}}}_n={\hat{x}}_{n+1}+g_n\left(\frac{x_1-{\hat{x}}_1}{{\mathrm{\ϵ}}^n}\right)+bu\\ {\stackrel{\·}{\hat{x}}}_{n+1}={\mathrm{\ϵ}}^{-1}{g}_{n+1}\left(\frac{x_1-{\hat{x}}_1}{{\mathrm{\ϵ}}^n}\right)\end{array}\right.---\left(3\right)$

WhereinRepresent status system x_iEstimated value,Expression system expansion state x_n+1Estimated value, definitionFor the state vector of extended state observer, ε is a normal number, nonlinear function g_iNeeds condition A2 below).

Definition extended state observer observes the error of state estimation and the system mode obtained:

$\begin{array}{cc}{\mathrm{\η}}_i=\frac{x_i-{\hat{x}}_i}{{\mathrm{\ϵ}}^{n+1-i}},& i=1,2,\mathrm{...},n+1\end{array}$

Following error system equation can be obtained by system (2) and extended state observer (3):

$\left\{\begin{array}{c}{\stackrel{\·}{\mathrm{\η}}}_1=\frac{{\mathrm{\η}}_2-g_1\left({\mathrm{\η}}_1\right)}{\mathrm{\ϵ}}\\ .\\ .\\ .\\ {\stackrel{\·}{\mathrm{\η}}}_n=\frac{{\mathrm{\η}}_{n+1}-g_n\left({\mathrm{\η}}_1\right)}{\mathrm{\ϵ}}\\ {\stackrel{\·}{\mathrm{\η}}}_{n+1}=\frac{\mathrm{\ϵ}h-g_{n+1}\left({\mathrm{\η}}_1\right)}{\mathrm{\ϵ}}\end{array}\right.---\left(4\right)$

For above-mentioned error system, if there is the continuously differentiable function V of positive definite₁And W₁, and normal number λ₁、λ₂、λ₃、λ₄And β₁So that condition below meets:

A1)λ₁||η||²≤V₁(η)≤λ₂||η||²,λ₃||η||²≤W₁(η)≤λ₄||η||²

A2)

A3)

And there is positive integer k and normal number a_j(j=0,1,2 ..., n) so that continuously differentiable function f meets with lower inequality:

B)

Then may certify that when system mode bounded (always meeting in reality), for the boundary σ ' > 0 that any one is given, total exist a positive number ε₀So that as long as we choose ε ∈ (0, ε₀), error state can converge in σ ' the neighborhood of initial point, i.e. | | η | |≤σ '.In short, when ε → 0, the state of extended state observer converges on the state of system.Can choose liapunov function in concrete proof procedure is V₁(η), by A1) known V₁(η) it is positive definite and successively decreases, then to liapunov function V₁(η) derivation and utilize A1)-A3) and B) may certify that, V₁(η) converging in a boundary being directly proportional to ε, same | | η | | also will converge in a boundary being directly proportional to ε, as long as thus being obtained by ε sufficiently small, | | η | | just can converge in the region being sufficiently close to 0.Therefore, the state that can be considered as extended state observer in practice converges on the state of system.

Assume B) it is that nature meets when input, disturbance bounded, therefore the design key of extended state observer is in that to choose suitable g_iMake to assume A1)-A3) meet, for sufficiently small positive number ε, so the extended state observer of design enables to its state and converges on the state of system.

Step 3: determine sampling instant t_i.Definition sampling error e (t) is:

$e\left(t\right)=C_0\left|\right|\hat{x}\left(t\right)-\hat{x}\left(t_i\right)\left|\right|+\left|\right|{\hat{x}}_{n+1}\left(t\right)-{\hat{x}}_{n+1}\left(t_i\right)\left|\right|---\left(5\right)$

Wherein C₀Choose in following step 4 designed by controller relevant.By choosing suitable activation threshold value γ (t) (it needs the condition met also will illustrate in step 4), and assume that first sampling instant is t₀, a series of trigger instants can be obtained:

t_i+1=inf{t > t_i|e(t)≥γ(t)}⑹

Wherein i is natural number, and inf{} represents infimum.

Step 4: will at sampling instant t_iThe state of the extended state observer that sampling obtains sends controller to, and controller end utilizes this sampled value to calculate the output of executor, i.e. the control input of controlled system:

WhereinBeing Li Puxizi function, its Li Puxizi constant is C₀, constant in the error function being namely previously mentioned in step 3.Choose and need to makeAnd meet the continuously differentiable function V that there is positive definite₂And W₂, and normal number λ₂₁、λ₂₂、λ₂₃、λ₂₄And β₂So that condition below is set up:

C1)λ₂₁||x||²≤V₂(x)≤λ₂₂||x||²,λ₂₃||x||²≤W₂(x)≤λ₂₄||x||²

C2)

C3)

Wherein assume C1) and C2) show undisturbed systemBe asymptotically stable (can pass through choose V₂X () proves for liapunov function).

By choosing the controller meeting above-mentioned hypothesis, the state of closed loop system is by certain neighborhood near initial point of half global convergence, thus obtaining following theorem.

Theorem 1: for nonlinear system (1), design extended state observer (3) and event trigger controller (7), make to assume A1)-A3), B) and C1)-C3) all set up, the definition of sampling error and sampling instant is as shown in (5) and (6), then for any given positive number σ and original state x₀, certainly exist a positive number ε₁, for arbitrary ε ∈ (0, ε₁), as long as the supremum ρ of activation threshold value γ (t) meetsThen system mode can converge in the σ neighborhood of initial point, i.e. | | x | |≤σ.As long as in short, ε and ρ selects sufficiently small, system mode is by the region converging to sufficiently close together initial point.

It is V that theorem 1 can pass through to choose the liapunov function of closed loop system₂X () proves.Utilize and assume C1)-C3) and step 2 in the observer state that obtains converge on the conclusion of system mode, obtain V₂X the derivative of () is in the conclusion of the outer strictly decreasing of σ neighborhood of initial point, thus proof system state can converge in the σ neighborhood of initial point, i.e. and | | x | |≤σ.

If the supremum ρ of activation threshold value γ (t) is chosen for 0, i.e. γ (t)≤0, theorem 1 deteriorates to the result of ideally continuous sampling, such controller also can complete task of making system mode restrain, but continuous sampling is generally adopted the form of time sampling at equal intervals in practice, communication bandwidth is taken the control strategy of event activation pattern being generally far longer than in the present invention to adopt by it.

When just starting, owing to the state of extended state observer changes ratio comparatively fast, now the change of sampling error e (t) also ratio is comparatively fast, if choosing less activation threshold value, frequent triggering will be caused, thus taking substantial amounts of communication bandwidth and making executor's frequent updating, disagree with the purpose of the present invention.Therefore, bigger activation threshold value should be chosen when just starting.And system basicly stable after, the change of sampling error e (t) also with observer state stable and slack-off, now can choose less activation threshold value so that system mode converge to from initial point closer to region in.In sum, activation threshold value is chosen for the function successively decreased gradually and controls effect better, such as adopts the function of exponential decrease can meet requirement as activation threshold value function.

Above-described it is embodied as step; the purpose of the present invention, technical scheme and beneficial effect have been further described; it is it should be understood that; the foregoing is only the general step of the present invention; it is not limited to the present invention; all within the spirit and principles in the present invention, any amendment of making, equivalent replacement, improvement etc., should be included within protection scope of the present invention.

Claims (3)

1. based on an extended state observer nonlinear system event trigger controller method for designing, it is characterized in that, step is as follows:

Step 1: set up the nonlinear system model of following single-input single-output:

$y^{\left(n\right)}=f(y,\stackrel{\·}{y},...,y^{(n-1)})+bu---\left(1\right)$

Wherein y is controlled output, and u is for controlling input, and b is given constant,For the first derivative of y, y⁽ⁱ⁾The i order derivative of expression y, i=2 ..., n-1, n, willBeing defined as f, f is single order continuously differentiable function, represents the Nonlinear Dynamic of controlled device, it addition, f should meet f (0)=0；

The model that formula (1) represents is rewritten as following state space form:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ .\\ .\\ .\\ {\stackrel{\·}{x}}_{n-1}=x_n\\ {\stackrel{\·}{x}}_n=x_{n+1}+bu\\ {\stackrel{\·}{x}}_{n+1}=h\\ y=x_1\end{array}\right.---\left(2\right)$

Wherein x_iIt is defined as the state of system, and with x=[x₁,x₂,...,x_n]^TThe state vector of expression system, x_n+1=f is the expansion state of system,Represent x_iFirst derivative, and defineNamely h is the first derivative of f；

Step 2: for state and (2) middle expansion state defined of estimating system, design following extended state observer:

$\left\{\begin{array}{c}{\stackrel{\·}{\hat{x}}}_1={\hat{x}}_2+{\mathrm{\ϵ}}^{n-1}{g}_1\left(\frac{x_1-{\hat{x}}_1}{{\mathrm{\ϵ}}^n}\right)\\ {\stackrel{\·}{\hat{x}}}_2={\hat{x}}_3+{\mathrm{\ϵ}}^{n-2}{g}_2\left(\frac{x_1-{\hat{x}}_1}{{\mathrm{\ϵ}}^n}\right)\\ .\\ .\\ .\\ {\stackrel{\·}{\hat{x}}}_n={\hat{x}}_{n+1}+g_n\left(\frac{x_1-{\hat{x}}_1}{{\mathrm{\ϵ}}^n}\right)+bu\\ {\stackrel{\·}{\hat{x}}}_{n+1}={\mathrm{\ϵ}}^{-1}{g}_{n+1}\left(\frac{x_1-{\hat{x}}_1}{{\mathrm{\ϵ}}^n}\right)\end{array}\right.---\left(3\right)$

WhereinRepresent system mode x_iEstimated value,Expression system expansion state x_n+1Estimated value, definitionFor the state vector of extended state observer, ε is a normal number, nonlinear function g_iChoose needs ensure when ε → 0, the state of extended state observer converges on the state of system；

Step 3: determine sampling instant t_i, definition sampling error e (t) is:

$e\left(t\right)=C_0\left|\right|\hat{x}\left(t\right)-\hat{x}\left(t_i\right)\left|\right|+\left|\right|{\hat{x}}_{n+1}\left(t\right)-{\hat{x}}_{n+1}\left(t_i\right)\left|\right|$

Wherein C₀To choose the controller designed to following step 4 relevant,Represent t system mode x_iEstimated value, by choosing suitable activation threshold value γ (t), and assume that first sampling instant is t₀, obtain a series of trigger instants:

t_i+1=inf{t > t_i|e(t)≥γ(t)}

Wherein i is natural number, and inf{} represents infimum；

Step 4: will at sampling instant t_iThe state of the extended state observer that sampling obtains sends controller to, and controller end utilizes this sampled value to calculate the output of executor, i.e. the control input of controlled system:

WhereinBeing Li Puxizi function, its Li Puxizi constant is C₀,Choose and need to meetAnd ensure undisturbed systemIt is asymptotically stable.

2. as claimed in claim 1 based on extended state observer nonlinear system event trigger controller method for designing, it is characterized in that, bigger activation threshold value is taken so that do not cause frequent triggering because the change of observer state is too fast at initial time, thus saving communication bandwidth when just starting；And system basicly stable after, the change of sampling error e (t) along with observer state stable and slack-off, now choose less activation threshold value so that system mode converge to from initial point closer to region in；It is to say, activation threshold value to be chosen for the function successively decreased gradually.

3. as claimed in claim 1 based on extended state observer nonlinear system event trigger controller method for designing, it is characterized in that, adopt the function of exponential decrease as activation threshold value function.