CN103019098A - Robust controller of automotive chassis integrated system and construction method - Google Patents

Abstract

The invention provides a robust controller of an automotive chassis integrated system and a construction method. The controller is composed of an internal model controller and a support vector machine generalized inverse system, wherein the internal model controller is composed of a side slip angle internal model controller body, a yaw velocity internal model controller body and a vehicle roll angle internal model controller body, the support vector machine generalized inverse system is in series connection with the automotive chassis integrated system to form a generalized pseudo-linear system, the support vector machine generalized inverse system is composed of a support vector machine and four linear links, the generalized pseudo-linear system comprises a side slip angle linear subsystem, a yaw velocity linear subsystem and a vehicle roll angle linear subsystem, and the automotive chassis integrated system is composed of an active front steering subsystem, a direct yaw moment control subsystem and an active suspension subsystem. According to the robust controller, the defects of control methods of existing automotive chassis integrated systems are eliminated, and the decoupling control among the side direction, the longitudinal direction and the vertical direction of the automotive chassis system can be achieved.

Description

Automobile chassis integrated system robust controller and building method

Technical field

The present invention relates to the Vehicle Engineering equipment technical field, relate in particular to a kind of automobile chassis integrated system robust controller and building method.

Background technology

Along with making constant progress of science and technology, people are also more and more higher to the requirement of automotive performance, wish that automobile can have good maneuverability, ride performance and active safety etc. simultaneously.Therefore, various automobile chassis active control systems all sequential use in Hyundai Motor, such as electric boosting steering system, active suspension system, electronic stabilizing control system, anti-lock braking system etc.In general, these kinds Vehicle Chassis Electronic Control System all designs for improving a certain performance index, when they use separately, do not take into full account them to the impact of other subsystem in the auto kinetic control system, subsystems has certain limitation, but in order to give full play to the relevance between the control subsystem, to optimize and further to improve the car load combination property, synthesize and coordinate various control subsystem, research automobile chassis integrated system and control thereof have become the study hotspot of current automobile chassis system control technology and one of from now on development trend.

Yet the research of automobile chassis integrated system is easy thing by no means, exists various machineries, the Non-linear coupling factor between electric between the automobile chassis subsystems, realize the high performance control of automobile chassis integrated system, and the design of control algolithm is very crucial.Therefore, for the characteristic of automobile chassis integrated system multivariate, non-linear, strong coupling, design performance is good, antijamming capability is strong, and the robust controller tool with Decoupling Characteristics is of great significance.

In view of this, be necessary to propose a kind of automobile chassis integrated system robust controller and building method.

Summary of the invention

The object of the present invention is to provide a kind of automobile chassis integrated system robust controller and building method, the deficiency that it has overcome existing automobile chassis integrated system control method can realize the side direction of automobile chassis system, vertical and vertical decoupling zero control.

A kind of automobile chassis integrated system robust controller of the present invention, described controller consists of the support vector machine generalized inverse internal mode controller that combines and form by internal mode controller and support vector machine Generalized Inverse System, wherein:

Described internal mode controller is composed in parallel by side slip angle internal mode controller, yaw velocity internal mode controller and vehicle roll angle internal mode controller, the side slip angle internal mode controller is made of side slip angle internal model and regulator, the yaw velocity internal mode controller is made of yaw velocity internal model and regulator, and the vehicle roll angle internal mode controller is made of vehicle roll angle internal model and regulator;

Described support vector machine Generalized Inverse System is connected with the automobile chassis integrated system and is consisted of the broad sense pseudo-linear system, described support vector machine Generalized Inverse System is comprised of support vector machine and 4 linear elements with 7 inputs nodes, 3 output nodes, and the broad sense pseudo-linear system comprises side slip angle linear subsystem, the yaw velocity linear subsystem vehicle roll angle linear subsystem of unifying; Described automobile chassis integrated system turns to subsystem, direct yaw moment control subsystem and Active suspension subsystem to consist of by active front.

Correspondingly, a kind of building method of automobile chassis integrated system robust controller, described method comprises:

S1, turn to subsystem, direct yaw moment control subsystem and Active suspension subsystem to make as a whole formation automobile chassis integrated system active front;

S2, being connected with the support vector machine with 7 inputs nodes, 3 output nodes by 4 linear elements consists of the support vector machine Generalized Inverse System, adopt the support vector machine Generalized Inverse System formation broad sense pseudo-linear system of connecting with the automobile chassis integrated system, the linearized decoupling zero of broad sense pseudo-linear system is side slip angle linear subsystem, the yaw velocity linear subsystem vehicle roll angle linear subsystem of unifying;

S3, side slip angle linear subsystem, the yaw velocity linear subsystem vehicle roll angle linear subsystem of unifying is introduced respectively side slip angle internal mode controller, yaw velocity internal mode controller and vehicle roll angle internal mode controller structure internal mode controller, with the internal mode controller composition robust controller that combines with the support vector machine Generalized Inverse System, control automobile chassis integrated system.

As a further improvement on the present invention, vectorial coefficient and the threshold value determination method of support vector machine is specially among the described step S2:

With front-wheel steer offset angle δ _c, yaw control moment T _ZWith suspension roll moment T _φStep excitation signal { δ _c, T _Z, T _φBe added to the input end of automobile chassis integrated system, gather side slip angle β, yaw velocity γ and the vehicle roll angle φ of automobile chassis integrated system;

Side slip angle β, yaw velocity γ off-line are asked respectively its first order derivative, and then obtained

With

Vehicle roll angle φ is asked its single order, second derivative, and then obtain

Signal is done standardization processing, form the training sample set of support vector machine

And choose gaussian kernel function as the kernel function of support vector machine, and choose nuclear width and regularization parameter support vector machine is trained, thus vectorial coefficient and the threshold value of definite support vector machine;

Wherein, β is that side slip angle, γ are that yaw velocity, φ are vehicle roll angle, a ₁₀, a ₁₁, a ₂₀, a ₂₁, a ₃₀, a ₃₁And a ₃₂Parameter for Generalized Inverse System.

As a further improvement on the present invention, among the described step S2 " connected with support vector machine with 7 input nodes, 3 output nodes by 4 linear elements consist of the support vector machine Generalized Inverse System " are specially:

First input of support vector machine Generalized Inverse System

As first input of support vector machine, it is through first-order system Be output as β, be second input of support vector machine;

Second input of support vector machine Generalized Inverse System

As the 3rd input of support vector machine, it is through first-order system

Be output as γ, be the 4th input of support vector machine;

The 3rd input of support vector machine Generalized Inverse System

As the 5th input of support vector machine, it is through second-order system Be output as φ, be the 6th input of support vector machine, again through an integrator s ^-1Be φ, be the 7th input of support vector machine.

As a further improvement on the present invention, described step S3 also comprises:

Determine parameter a ₁₀, a ₁₁, a ₂₀, a ₂₁, a ₃₀, a ₃₁And a ₃₂, side slip angle internal model, yaw velocity internal model and the vehicle roll angle internal model of vehicle roll angle linear subsystem are respectively so that side slip angle linear subsystem, yaw velocity linear subsystem are unified:

G _1m=1(a ₁₀s+a ₁₁)=1(s+1)、

G _2m=1(a ₂₀s+a ₂₁)=1(s+1)、

G _3m=1(a ₃₀s ²+a ₃₁s+a ₃₂)=1(s ²+1.414s+1)，

Obtaining corresponding regulator is respectively:

$C_1\left(s\right)=F_1\left(s\right)G_{1m}^{-1}=(s+1)/(2s+1),$

$C_2\left(s\right)=F_2\left(s\right)G_{2m}^{-1}=(s+1)/(2s+1),$

$C_3\left(s\right)=F_3\left(s\right)G_{3m}^{-1}=({\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HDA00002593603600011.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+1.414s+1)/{(0.5s+1)}^2.$

Automobile chassis integrated system robust controller of the present invention and building method have following beneficial effect:

By design support vector machine Generalized Inverse System, be 3 single output linearity subsystems of single input with the System with Nonlinear Coupling Linearized Decoupling of integrated this multiple-input and multiple-output of original system of automobile chassis, thereby can solve complicated nonlinear Control problem with simple linear control method;

Utilize the recurrence ability of support vector machine to approach the inversion model of automobile chassis integrated system, do not need to know the mathematical models of original system and inverse system, simplified the modeling of automobile chassis integrated system, and be that the support vector function of criterion overcomes preferably based on empirical risk minimization based on structural risk minimization be the shortcoming of the neural network of criterion;

The support vector machine generalized inverse internal mode controller of design not only turns in active front, have good control effect in direct yaw moment control and the Active suspension integrated system, and is equally applicable to automobile chassis other automatically controlled subsystem or integrated systems.

Description of drawings

Fig. 1 is the theory diagram of automobile chassis integrated system robust controller of the present invention;

Fig. 2 is that the present invention is by synoptic diagram and the isoboles thereof of support vector machine Generalized Inverse System and the compound broad sense pseudo-linear system that consists of of automobile chassis integrated system;

To be the present invention turn to the synoptic diagram of the automobile chassis integrated system that subsystem, direct yaw moment control subsystem and Active suspension subsystem form by active front to Fig. 3.

Embodiment

Describe the present invention below with reference to each embodiment shown in the drawings.But these embodiments do not limit the present invention, and the conversion on the structure that those of ordinary skill in the art makes according to these embodiments, method or the function all is included in protection scope of the present invention.

Join shown in Figure 1, a kind of automobile chassis integrated system of the present invention robust controller, the support vector machine generalized inverse internal mode controller 5 that this controller is combined and formed by internal mode controller 4 and 2 one-tenth of support vector machine Generalized Inverse System structures, wherein:

Internal mode controller 4 is composed in parallel by side slip angle internal mode controller 41, yaw velocity internal mode controller 42 and vehicle roll angle internal mode controller 43, side slip angle internal mode controller 41 is made of side slip angle internal model 411 and regulator 412, yaw velocity internal mode controller 42 is made of yaw velocity internal model 421 and regulator 422, and vehicle roll angle internal mode controller 43 is made of vehicle roll angle internal model 431 and regulator 432;

Join shown in Figure 2, support vector machine Generalized Inverse System 2 is connected with automobile chassis integrated system 1 and is consisted of broad sense pseudo-linear system 3, support vector machine Generalized Inverse System 2 is comprised of support vector machine 21 and 4 linear elements with 7 input nodes, 3 output nodes, and broad sense pseudo-linear system 3 comprises side slip angle linear subsystem 31, yaw velocity linear subsystem 32 and vehicle roll angle linear subsystem 33; Join shown in Figure 3ly, automobile chassis integrated system 1 turns to subsystem 11, direct yaw moment control subsystem 12 and Active suspension subsystem 13 to consist of by active front.

Correspondingly, the building method of a kind of automobile chassis integrated system robust controller of the present invention may further comprise the steps:

S1, turn to subsystem 11, direct yaw moment control subsystem 12 and Active suspension subsystem 13 to make as a whole formation automobile chassis integrated system 1 active front;

S2, being connected with the support vector machine 21 with 7 inputs nodes, 3 output nodes by 4 linear elements consists of support vector machine Generalized Inverse System 2, adopt support vector machine Generalized Inverse System 2 to connect with automobile chassis integrated system 1 and consist of broad sense pseudo-linear system 3, the 3 linearized decoupling zeros of broad sense pseudo-linear system are side slip angle linear subsystem 31, yaw velocity linear subsystem 32 and vehicle roll angle linear subsystem 33;

S3, side slip angle linear subsystem 31, yaw velocity linear subsystem 32 and vehicle roll angle linear subsystem 33 are introduced respectively side slip angle internal mode controller 41, yaw velocity internal mode controller 42 and vehicle roll angle internal mode controller 43 structure internal mode controller 4, with the internal mode controller 4 composition robust controller that combines with support vector machine Generalized Inverse System 2, control automobile chassis integrated system 1.

In the embodiment of the present invention, building method specifically comprises:

S1, formation automobile chassis integrated system 1.Active front by automobile chassis turns to subsystem 11, direct yaw moment control subsystem 12 and Active suspension subsystem 13 to make the as a whole automobile chassis integrated system 1 shown in Figure 3 that forms;

S2, by analyze, equivalence and derivation, be the structure of support vector machine Generalized Inverse System 2 and the basis on the learning training supplying method.

The mathematical model of model automobile chassis integrated system 1 (the 4 rank differential equation), its vector relatively rank is { 1,1,2, }.Can prove that through deriving this 4 rank differential equation is reversible, namely Generalized Inverse System exists, and then sets up automobile chassis integrated system 1 Generalized Inverse System model, is the foundation on support vector machine Generalized Inverse System 2 supplying methods;

3 that determine the support vector machine Generalized Inverse System are input as

With

3 outputs are respectively 3 inputs of automobile chassis integrated system 1, i.e. front-wheel steer offset angle δ _c, yaw control moment T _ZWith suspension roll moment T _φ, as shown in Figure 2;

According to the actual conditions of automobile chassis, rationally regulate Generalized Inverse System parameter a ₁₀, a ₁₁, a ₂₀, a ₂₁, a ₃₀, a ₃₁And a ₃₂, make limit reasonable disposition in complex plane of the linear subsystem of 3 Linearized Decouplings, the unsettled subsystem of integral form is become stable subsystem.

Wherein, β is that side slip angle, γ are that yaw velocity, φ are vehicle roll angle, a ₁₀, a ₁₁, a ₂₀, a ₂₁, a ₃₀, a ₃₁And a ₃₂Parameter for Generalized Inverse System.

In conjunction with shown in Figure 2, adopt support vector machine 21 and 4 linear element structure support vector machine Generalized Inverse Systems 2.Wherein the input number of nodes of support vector machine 21 is 7, and output node is several 7, and the vectorial coefficient of support vector machine 21 and threshold value will be determined in next step off-line learning.Then adopt the support vector machine 21 with 7 input nodes, 3 output nodes to add 4 linear elements and construct the support vector machine Generalized Inverse System 2 with 3 input nodes, 3 output nodes, wherein:

First input of support vector machine Generalized Inverse System 2

As first input of support vector machine 21, it is through first-order system

Be output as β, be second input of support vector machine 21;

Second input of support vector machine Generalized Inverse System 2

As the 3rd input of support vector machine 21, it is through first-order system

Be output as γ, be the 4th input of support vector machine 21;

The 3rd input of support vector machine Generalized Inverse System 2

As the 5th input of support vector machine 21, it is through second-order system Be output as

Be the 6th input of support vector machine 21, again through an integrator s ^-1Be φ, be the 7th input of support vector machine 21.

Support vector machine 21 forms support vector machine Generalized Inverse System 2 with 4 linear elements, and the output of support vector machine 21 is exactly the output of support vector machine Generalized Inverse System 2.

Adjust vectorial coefficient and the threshold value of support vector machine 21.

With front-wheel steer offset angle δ _c, yaw control moment T _ZWith suspension roll moment T _φStep excitation signal { δ _c, T _Z, T _φBe added to the input end of automobile chassis integrated system 1, gather side slip angle β, yaw velocity γ and the vehicle roll angle φ of automobile chassis integrated system 1;

Side slip angle β, yaw velocity γ off-line are asked respectively its first order derivative, and then obtained

With

Vehicle roll angle φ is asked its single order, second derivative, and then obtain

Signal is done standardization processing, form the training sample set of support vector machine 21

And choose gaussian kernel function as the kernel function of support vector machine 21, and choose suitable nuclear width and regularization parameter according to actual conditions, support vector machine 21 is trained, thus vectorial coefficient and the threshold value of definite support vector machine 21.Wherein, a ₁₀, a ₁₁, a ₂₀, a ₂₁, a ₃₀, a ₃₁And a ₃₂Parameter for Generalized Inverse System.

Broad sense pseudo-linear system 3 forms 3 linear subsystems 31,32,33.Support vector machine Generalized Inverse System 2 forms broad sense pseudo-linear system 3 with automobile chassis integrated system 1 series winding, 3 equivalences of broad sense pseudo-linear system are 1 side slip angle linear subsystem 31,1 yaw velocity linear subsystem 32 and 1 vehicle roll angle linear subsystem 33, as shown in Figure 2.

S3, design internal mode controller 4.Internal mode controller 4 is composed in parallel by side slip angle internal mode controller 41, yaw velocity internal mode controller 42 and vehicle roll angle internal mode controller 43, side slip angle internal mode controller 41 is made of side slip angle internal model 411 and regulator 412, yaw velocity internal mode controller 42 is made of yaw velocity internal model 421 and regulator 422, and vehicle roll angle internal mode controller 43 is made of vehicle roll angle internal model 431 and regulator 432.

Select appropriate parameter a ₁₀, a ₁₁, a ₂₀, a ₂₁, a ₃₀, a ₃₁And a ₃₂, so that the side slip angle internal model 411 of side slip angle linear subsystem 31, yaw velocity linear subsystem 32 and vehicle roll angle linear subsystem 33, yaw velocity internal model 421 and vehicle roll angle internal model 431 are respectively:

G _1m=1(a ₁₀s+a ₁₁)=1/(s+1)、

G _2m=1(a ₂₀s+a ₂₁)=1(s+1)、

G _3m=1(a ₃₀s ²+a ₃₁s+a ₃₂)=1(s ²+1414s+1)，

Obtaining corresponding regulator 412,422,432 is respectively:

$C_1\left(s\right)=F_1\left(s\right)G_{1m}^{-1}=(s+1)/(2s+1),$

$C_2\left(s\right)=F_2\left(s\right)G_{2m}^{-1}=(s+1)/(2s+1),$

$C_3\left(s\right)=F_3\left(s\right)G_{3m}^{-1}=({\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HDA00002593603600011.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+1.414s+1)/{(0.5s+1)}^2.$

Wherein, a ₁₀, a ₁₁, a ₂₀, a ₂₁, a ₃₀And a ₃₂Value be taken as 1, a ₃₁Value be taken as 1.414, F ₁(s)=F ₂(s)=1 (0.5s+1) ², F ₃(s)=1 (2s+1) is corresponding regulator 412, a type low-pass filter of 422,432.

In conjunction with above method, at last with internal mode controller 4, the support vector machine Generalized Inverse System 2 common robust controllers 5 that form.

Can be found out that by technique scheme automobile chassis integrated system robust controller of the present invention and building method have following beneficial effect:

By design support vector machine Generalized Inverse System, be 3 single output linearity subsystems of single input with the System with Nonlinear Coupling Linearized Decoupling of integrated this multiple-input and multiple-output of original system of automobile chassis, thereby can solve complicated nonlinear Control problem with simple linear control method;

Utilize the recurrence ability of support vector machine to approach the inversion model of automobile chassis integrated system, do not need to know the mathematical models of original system and inverse system, simplified the modeling of automobile chassis integrated system, and be that the support vector function of criterion overcomes preferably based on empirical risk minimization based on structural risk minimization be the shortcoming of the neural network of criterion;

The support vector machine generalized inverse internal mode controller of design not only turns in active front, have good control effect in direct yaw moment control and the Active suspension integrated system, and is equally applicable to automobile chassis other automatically controlled subsystem or integrated systems.

Be to be understood that, although this instructions is described according to embodiment, but be not that each embodiment only comprises an independently technical scheme, this narrating mode of instructions only is for clarity sake, those skilled in the art should make instructions as a whole, technical scheme in each embodiment also can through appropriate combination, form other embodiments that it will be appreciated by those skilled in the art that.

Above listed a series of detailed description only is specifying for feasibility embodiment of the present invention; they are not to limit protection scope of the present invention, allly do not break away from equivalent embodiment or the change that skill spirit of the present invention does and all should be included within protection scope of the present invention.

Claims (5)

1. an automobile chassis integrated system robust controller is characterized in that, described controller consists of the support vector machine generalized inverse internal mode controller that combines and form by internal mode controller and support vector machine Generalized Inverse System, wherein:

Described internal mode controller is composed in parallel by side slip angle internal mode controller, yaw velocity internal mode controller and vehicle roll angle internal mode controller, the side slip angle internal mode controller is made of side slip angle internal model and regulator, the yaw velocity internal mode controller is made of yaw velocity internal model and regulator, and the vehicle roll angle internal mode controller is made of vehicle roll angle internal model and regulator;

Described support vector machine Generalized Inverse System is connected with the automobile chassis integrated system and is consisted of the broad sense pseudo-linear system, described support vector machine Generalized Inverse System is comprised of support vector machine and 4 linear elements with 7 inputs nodes, 3 output nodes, and the broad sense pseudo-linear system comprises side slip angle linear subsystem, the yaw velocity linear subsystem vehicle roll angle linear subsystem of unifying; Described automobile chassis integrated system turns to subsystem, direct yaw moment control subsystem and Active suspension subsystem to consist of by active front.

2. the building method of an automobile chassis integrated system robust controller as claimed in claim 1 is characterized in that, described method comprises:

S1, turn to subsystem, direct yaw moment control subsystem and Active suspension subsystem to make as a whole formation automobile chassis integrated system active front;

S2, being connected with the support vector machine with 7 inputs nodes, 3 output nodes by 4 linear elements consists of the support vector machine Generalized Inverse System, adopt the support vector machine Generalized Inverse System formation broad sense pseudo-linear system of connecting with the automobile chassis integrated system, the linearized decoupling zero of broad sense pseudo-linear system is side slip angle linear subsystem, the yaw velocity linear subsystem vehicle roll angle linear subsystem of unifying;

S3, side slip angle linear subsystem, the yaw velocity linear subsystem vehicle roll angle linear subsystem of unifying is introduced respectively side slip angle internal mode controller, yaw velocity internal mode controller and vehicle roll angle internal mode controller structure internal mode controller, with the internal mode controller composition robust controller that combines with the support vector machine Generalized Inverse System, control automobile chassis integrated system.

3. want the building method of 2 described automobile chassis integrated system robust controllers according to right, it is characterized in that, vectorial coefficient and the threshold value determination method of support vector machine is specially among the described step S2:

With front-wheel steer offset angle δ _c, yaw control moment T _ZWith suspension roll moment T _φStep excitation signal { δ _c, T _Z, T _φBe added to the input end of automobile chassis integrated system, gather side slip angle β, yaw velocity γ and the vehicle roll angle φ of automobile chassis integrated system;

Side slip angle β, yaw velocity γ off-line are asked respectively its first order derivative, and then obtained With

Vehicle roll angle φ is asked its single order, second derivative, and then obtain

Signal is done standardization processing, form the training sample set of support vector machine

And choose gaussian kernel function as the kernel function of support vector machine, and choose nuclear width and regularization parameter support vector machine is trained, thus vectorial coefficient and the threshold value of definite support vector machine;

Wherein, β is that side slip angle, γ are that yaw velocity, φ are vehicle roll angle, a ₁₀, a ₁₁, a ₂₀, a ₂₁, a ₃₀, a ₃₁And a ₃₂Parameter for the support vector machine Generalized Inverse System.

4. want the building method of 3 described automobile chassis integrated system robust controllers according to right, it is characterized in that " connected with support vector machine with 7 input nodes, 3 output nodes by 4 linear elements consist of the support vector machine Generalized Inverse System " are specially among the described step S2:

First input of support vector machine Generalized Inverse System

As first input of support vector machine, it is through first-order system

Be output as β, be second input of support vector machine;

Second input of support vector machine Generalized Inverse System

As the 3rd input of support vector machine, it is through first-order system Be output as γ, be the 4th input of support vector machine;

The 3rd input of support vector machine Generalized Inverse System As the 5th input of support vector machine, it is through second-order system

Be output as φ, be the 6th input of support vector machine, again through an integrator s ^-1Be φ, be the 7th input of support vector machine.

5. want the building method of 2 described automobile chassis integrated system robust controllers according to right, it is characterized in that described step S3 also comprises:

Determine parameter a ₁₀, a ₁₁, a ₂₀, a ₂₁, a ₃₀, a ₃₁And a ₃₂, side slip angle internal model, yaw velocity internal model and the vehicle roll angle internal model of vehicle roll angle linear subsystem are respectively so that side slip angle linear subsystem, yaw velocity linear subsystem are unified:

G _1m=1/(a ₁₀s+a ₁₁)=1/(s+1)、

G _2m=1/(a ₂₀s+a ₂₁)=1(s+1)、

G _3m=1(a ₃₀s ²+a ₃₁s+a ₃₂)=1(s ²+1414s+1)，

Obtaining corresponding regulator is respectively:

$C_1\left(s\right)=F_1\left(s\right)G_{1m}^{-1}=(s+1)/(2s+1),$

$C_2\left(s\right)=F_2\left(s\right)G_{2m}^{-1}=(s+1)/(2s+1),$

$C_3\left(s\right)=F_3\left(s\right)G_{3m}^{-1}=(s^2+1.414s+1)/{(0.5s+1)}^2.$

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