CN102645895A - Optimization method of control performance of structure-tuned mass dampers (TMD)-H8 system - Google Patents

Abstract

The invention discloses an optimization method of control performance of a structure-tuned mass dampers (TMD)-H8 system. The method comprises establishing a second-order mass-damper-spring structure system (original structure), and researching the stability and performance of the structure under the action of load; optimizing the original structure by using the H8 robust algorithm, and researching the robust stability and performance robustness of a structure-H8 system; mounting a TMD device on the basis of the original structure, and researching the stability and performance of the structure-TMD system; and combining H8 and TMD control methods, i.e., further performing robust H8 optimization on the basis of primary TMD optimization by using maximal power amplification factor, and researching the robust stability and performance robustness of the structure-TMD-H8 system. The invention has the innovation that TMD control research is carried out by using the H8 robust algorithm, and has the advantage that the stability of the structure-TMD system is improved.

Description

Structure-TMD-H ∞The optimization method of system control performance

Technical field

The present invention relates to a kind of building engineering structure vibration control optimization method, a kind of specifically based on Robust Controller Design retuning mass damper structural system (structure-Tuned Mass Dampers-H8, the optimization method of control performance of structure-TMD-H8).

Background technology

The research of vibration control of civil engineering structure and application are considered to the important breakthrough of structures under wind and earthquake research.It has broken through the traditional structural design method, promptly develops into by structure-wind resistance antidetonation vibration control system dynamic response of control structure on one's own initiative from the method that only relies on change structure self performance to resist environmental load.Meanwhile, be applicable to that the device of vibration control of civil engineering structure has also had suitable development, yet control device exists still at present a lot of problems to overcome, tuned mass damper (TMD) for example, in case imbalance, its control validity will obviously descend.Generally speaking, certain specific vibration shape reaction of TMD control structure is quite effective.But for some external excitation, the high vibration shape possibly become the control vibration shape.Therefore, a plurality of TMD of the several vibration shapes of control structure possibly need to install simultaneously respectively.But this scheme will cause vibration shape pollution problem, promptly control the first vibration mode reaction (being equivalent to reduce the validity of control first vibration mode) of the TMD meeting structure for amplifying of the high vibration shape.For the structure under these excitations, introduce the ACTIVE CONTROL algorithm, for example H ₈The TMD Control Study of carrying out the nonspecific vibration shape is of great value.

Summary of the invention

1 original structure

Set up a second order, quality-damping-spring structure system model (original structure), its kinetic equation can be expressed as:

$m\stackrel{\·\·}{X}+c\stackrel{\·}{X}+\mathrm{kX}=F\left(t\right)---\left(1\right)$

In the formula:

X representes acceleration, speed and the displacement of system; M; C; K representes quality, damping and the rigidity of structure, and F (t) expression acts on structural external excitation.

Consider the uncertainty of two physical parameter c and k, that is:

$c=\stackrel{\‾}{c}(1+{\mathrm{\δ}}_c{p}_c),p_v=0.2,{\mathrm{\δ}}_c\∈[-\mathrm{1,1}]---\left(2\right)$

$k=\stackrel{\‾}{k}(1+{\mathrm{\δ}}_k{p}_k),p_k=0.3,{\mathrm{\δ}}_k\∈[-\mathrm{1,1}]---\left(3\right)$

In the formula:

Be c, the nominal value of k, p _c, p _k, δ _c, δ _kThe relative perturbation of representation parameter.

Linear fraction transformation in the employing can obtain the following expression formula about c and k.

c＝F _U(M _c，δ _c)(4)

k＝F _U(M _k，δ _k)(5)

In the formula: $M_c=\left[\begin{array}{cc}0& \stackrel{\‾}{c}\\ p_c& \stackrel{\‾}{c}\end{array}\right],$ $M_k=\left[\begin{array}{cc}0& \stackrel{\‾}{k}\\ p_k& \stackrel{\‾}{k}\end{array}\right].$

Use y _c, y _kAnd u _c, u _kRepresent parameter perturbation δ _c, δ _kInput/output variable the time, the equation of motion of structure can be expressed as:

${\stackrel{.}{x}}_1=x_2$

$x_2=\frac{1}{m}(u-v_c-v_k)$

$y_c={\stackrel{\‾}{c}x}_2$

$y_k={\stackrel{\‾}{k}x}_1$

$v_c=p_c{\stackrel{\‾}{c}u}_c+{\stackrel{\‾}{c}x}_2---\left(6\right)$

$v_k=p_k{\stackrel{\‾}{k}u}_x+{\stackrel{\‾}{k}x}_1$

y＝x ₁

u _c＝δ _cy _c

u _k＝δ _ky _k

Formula (6) is carried out comprehensively:

$\left[\begin{array}{c}{\stackrel{\·}{x}}_1\\ {\stackrel{\·}{x}}_2\\ y_c\\ y_k\\ y\end{array}\right]=\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ -\frac{\stackrel{\‾}{k}}{m}& -\frac{\stackrel{\‾}{c}}{m}& -\frac{p_c\stackrel{\‾}{c}}{m}& -\frac{p_k\stackrel{\‾}{k}}{m}& -\frac{1}{m}\\ 0& \stackrel{\‾}{c}& 0& 0& 0\\ \stackrel{\‾}{k}& 0& 0& 0& 0\\ 1& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ u_c\\ u_k\\ u\end{array}\right]---\left(7\right)$

$\left[\begin{array}{c}{u}_c\\ u_k\end{array}\right]=\left[\begin{array}{cc}{\mathrm{\&delta;}}_{\mathrm{<figure-callout\; id="0"\; label="c"\; filenames="HSA00000706381900022.png,HSA00000706381900071.png"\; state="\{\{state\}\}">c</figure-callout>}}& 0\\ 0& {\mathrm{\&delta;}}_k\end{array}\right]\left[\begin{array}{c}{y}_c\\ y_k\end{array}\right]---\left(8\right)$

The uncertainty of uncertain matrix Δ=diag in the formula (8) (δ c, δ k) representative structure.

Before being optimized, confirm that the parameter of original structure is following: the quality m=17500kg of structure, the damping ratio ξ of structure=0.02, the frequency f=3Hz of structure can obtain the ratio of rigidity and the damping ratio of main structure with this.Adopt MATLAB to carry out systematic analysis, obtain performance map.

2 structures-H _∞System

2.1 the designing requirement of closed-loop system

Equipment is carried out designing requirement to be through finding the solution the nominal performance and stability, robust stability and robust performance of a linear o controller u (s)=K (s) y (s) to guarantee closed-loop system.

(1) nominal performance and stability

To the structural design controller, so that stable in the closed-loop system; Name device model G _MdsAlso should reach the expected performance of closed-loop system.Among the design, adopt the performance criteria of mixed sensitivity, that is: as closed-loop system

$\left|\right|\left[\begin{array}{c}{W}_{p}S\left(G_{\mathrm{mds}}\right)\\ W_u\mathrm{KS}\left(G_{\mathrm{mds}}\right)\end{array}\right]\left|\right|<1---\left(9\right)$

In the formula, S (G _Mds)=(1+G _MdsK) ^-1Be the output sensitivity function of nominal system, W _p, W _uFor being used for representing outside (output), weighting function disturbs frequecy characteristic and the performance requirement level of d.If system can satisfy above-mentioned inequality, that just representes that closed-loop system can successfully be reduced to a gratifying level with disturbing effect, and has reached the performance that requires.The transport function that on behalf of fiducial error, sensitivity function S follow the tracks of.

(2) robust stability

If for any device model G=F _U(G _Mds, Δ), closed-loop system is stable in all guaranteeing, and closed-loop system just can reach robust stability so.In conjunction with this concrete design, robust stability can reduce for any-0.3≤Δ K≤0.3 ,-0.2≤Δ C≤0.2, and structure still can keep steady state (SS).

(3) robust performance

Except structure need satisfy the robust stability, for all G=F _U(G _Mds, Δ) closed-loop system must satisfy following performance criteria.

${\left|\right|\left[\begin{array}{c}{W}_p{(1+\mathrm{GK})}^{-1}\\ W_{u}K{(1+\mathrm{GK})}^{-1}\end{array}\right]\left|\right|}_{\&infin;}<1---\left(10\right)$

Δ is represented the uncertainty of structural model, G _MdsBe the nominal model of structural system, Δ and G _MdsFormed transfer function matrix G jointly.In general, the matrix Δ is a transfer function matrix, and is assumed to stable.Δ is unknown, but must meet || Δ | _∞＜1.Variable d is the interference of structure output, and its expression formula is suc as formula shown in (3.11).

$\left[\begin{array}{c}{e}_p\\ e_u\end{array}\right]=\left[\begin{array}{c}{W}_p{(I+\mathrm{GK})}^{-1}\\ W_{u}K{(I+\mathrm{GK})}^{-1}\end{array}\right]d---\left(11\right)$

Can find out that from following formula performance criteria can be expressed as for all possible uncertain transfer function matrix Δ, from d to e _pAnd e _uIts norm of transport function as far as possible little.Adopt weighting matrix W _p, W _uBe illustrated in the interior performance requirement of different frequency scope to system.W _pSelection is shown below.

$W_p\left(s\right)=0.6\frac{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HSA00000706381900052.png,HSA00000706381900061.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+7.9s+2.01}{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HSA00000706381900052.png,HSA00000706381900061.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+8.0s+2}---\left(12\right)$

This formula has guaranteed system's better anti-interference and transient response.Control weighting function W _uBe decided to be 10 ^-5

In order to reach the interference free performance of expectation, need satisfy inequality || W _p(I+GK) ^-1|| _∞＜1.In the design, W _pBe a scalar function, above-mentioned inequality can be expressed as sensitivity function (i.e. (I+GK) ^-1) the singular value curve must be (promptly at weighting function ) below of singular value curve, promptly

||W _p(I+GK) ^-1|| _∞＜1(13)

And if only if σ [(I+GK) ^-1(j ω)]＜| 1/W _p(j ω) |.

2.2 suboptimum H _∞Design of Controller

The controller of this design is a H ₈The suboptimal control device through stability controller K, minimizes F _L(P, infinitely great norm K).Because F _L(P K) is the transport function of nominal closed-loop system, and it is input as and disturbs dist, is output as error e, wherein $e=\left[\begin{array}{c}{e}_p\\ e_u\end{array}\right],$ Therefore we extract corresponding transfer function matrix P from system, and design corresponding suboptimum H to the open cycle system P that draws ₈Controller.Through finding the solution, the controller K matrix that draws.

2.3 the comprehensive evaluation of controller and system

This paper has calculated the singular value curve that adopts the closed-loop system of controller K, and the sensitivity function of closed-loop system and the inverse of performance weighting function are compared closed-loop system H ₈Less than 1, satisfy equation (13).

Understand the double optimization controller performance through basic frequency-domain analysis and time domain performance analysis, adopt the robust stability and the performance robustness of the next further understanding of singular value μ analysis system.

3 structures-TMD system

For structure-TMD system analysis model, its kinetic equation can be expressed as:

$m_s{\stackrel{\&CenterDot;\&CenterDot;}{X}}_s+c_s{\stackrel{\&CenterDot;}{X}}_s+k_s{X}_s=m_s{F}_a\left(t\right)---\left(14\right)$

$m_d{\stackrel{\&CenterDot;\&CenterDot;}{x}}_d+c_d{\stackrel{\&CenterDot;}{x}}_d+k_d{x}_d=m_d{F}_a\left(t\right)---\left(15\right)$

In the formula:

X _sAcceleration, speed and the displacement of expression main system (structure), m _s, c _s, k _sQuality, damping and the rigidity of expression main system;

x _dAcceleration, speed and the displacement of expression mass, m _d, c _d, k _dQuality, damping and the rigidity of expression mass; F _a(t) expression acts on the external excitation on the main system, and this example China and foreign countries are actuated to seismic acceleration.The situation that perturbation takes place the main system parameter will be discussed in this model.Consider two physical parameter c, the uncertainty of k now.

$C_s={\stackrel{\&OverBar;}{C}}_s(1+{\mathrm{\&delta;}}_c{p}_c),p_c=0.2,{\mathrm{\&delta;}}_c\&Element;[-\mathrm{1,1}]---\left(16\right)$

$K_s={\stackrel{\&OverBar;}{K}}_s(1+{\mathrm{\&delta;}}_k{\mathrm{\&rho;}}_k),p_k=0.3,{\mathrm{\&delta;}}_k\&Element;[-\mathrm{1,1}]---\left(17\right)$

In the formula:

Be C _s, K _sNominal value.p _c, p _k, δ _c, δ _kThe relative perturbation of representation parameter.The same last chapter of other abbreviation.Use y _c, y _kAnd u _c, u _kRepresent parameter perturbation δ _c, δ _kInput/output variable, then the equation of motion of structure-TMD system can be expressed as:

$\left[\begin{array}{c}{\stackrel{\&CenterDot;}{X}}_s\\ {\stackrel{\&CenterDot;}{X}}_d\\ {\stackrel{\&CenterDot;\&CenterDot;}{X}}_s\\ {\stackrel{\&CenterDot;\&CenterDot;}{X}}_d\\ Y_k\\ {\mathrm{<figure-callout\; id="0"\; label="Y\; c\; Z"\; filenames="HSA00000706381900022.png,HSA00000706381900071.png"\; state="\{\{state\}\}">Y</figure-callout>}}_c& \\ Z\end{array}\right]\left[\begin{array}{ccccccc}0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ -\frac{{\stackrel{\&OverBar;}{K}}_s+K_d}{M_s}& \frac{K_d}{M_s}& -\frac{{\stackrel{\&OverBar;}{C}}_s+C_d}{M_s}& \frac{C_d}{M_s}& \frac{P_k\&times;{\stackrel{\&OverBar;}{K}}_s}{M_s}& \frac{P_c\&times;{\stackrel{\&OverBar;}{C}}_s}{M_s}& \frac{1}{M_s}\\ \frac{K_d}{m_d}& -\frac{K_d}{m_d}& \frac{C_d}{m_d}& -\frac{C_d}{{\mathrm{<figure-callout\; id="0"\; label="m\; d"\; filenames="HSA00000706381900022.png,HSA00000706381900071.png"\; state="\{\{state\}\}">m</figure-callout>}}_d}& 0& 0& 0\\ -1& 0& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0\end{array}\right]\&times;\left[\begin{array}{c}{X}_s\\ X_d\\ {\stackrel{\&CenterDot;}{X}}_s\\ {\stackrel{\&CenterDot;}{X}}_d\\ U_k\\ U_c\\ u\end{array}\right]---\left(18\right)$

$\left[\begin{array}{c}{U}_k\\ U_c\end{array}\right]=\left[\begin{array}{cc}{\mathrm{\&delta;}}_{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="HSA00000706381900022.png,HSA00000706381900071.png"\; state="\{\{state\}\}">k</figure-callout>}}& 0\\ 0& {\mathrm{\&delta;}}_c\end{array}\right]\left[\begin{array}{c}{Y}_k\\ Y_c\end{array}\right]---\left(19\right)$

Uncertain matrix Δ=diag (δ in the formula (19) _k, δ _c) uncertainty of representative structure.This chapter is the basis with the original structure, and the TMD control device is installed on original structure, and the correlation parameter of structure is seen the original structure model.

The stability of structure-TMD system that adopted frequency-domain analysis methods such as Bode diagram, singular value curve is understood the structural behaviour performance through the rank response that jumps, and with original structure relatively.

4 structures-TMD-H _∞System

Carry out a basic enterprising step of initial optimization at TMD and carry out robust H ₈Optimize the selection isostructure-H of weighting function ₈System, the controller K that calculates.Understand the double optimization controller performance through basic frequency-domain analysis and time domain performance analysis, adopt the robust stability and the performance robustness of the next further understanding of singular value μ analysis system.

Description of drawings

The analytic process figure of Fig. 1 structure-TMD-H8 system control performance optimization method

Fig. 2 simplifies the original structure illustraton of model

The BODE figure of Fig. 3 building structure

The singular value curve of Fig. 4 building structure

The rank response that jumps of Fig. 5 building structure

Fig. 6 structure-H ∞ system robust stability analysis

Fig. 7 structure-H ∞ system robust performance curve

The BODE figure of Fig. 8 structure-H ∞ system

The singular value curve of Fig. 9 structure-H ∞ system

The jump rank response of Figure 10 structure-H ∞ system

The analytical model of Figure 11 structure-TMD system

The BODE figure of Figure 12 structure-TMD system

The rank response that jumps of Figure 13 structure-TMD system

The singular value curve of Figure 14 structure-TMD system

The The Robust Stability Analysis of Figure 15 structure-TMD-H ∞ system

The name and the robust performance of Figure 16 structure-TMD-H ∞ system

The BODE figure of Figure 17 structure-TMD-H ∞ system

The singular value curve of Figure 18 structure-TMD-H ∞ system

The rank response that jumps of Figure 19 structure-TMD-H ∞ system

Embodiment

The first step: set up a second order, quality-damping-spring system (original structure), like Fig. 2.The stability of original structure that adopted frequency-domain analysis methods such as Bode diagram, singular value curve is understood the structural behaviour performance through the rank response that jumps, like Fig. 3 to shown in Figure 5.

As can beappreciated from fig. 3, at amplitude versus frequency characte 20log|G (j ω) H (j ω) | in the frequency range of＞0dB, phase-frequency characteristic G (j ω) H (j ω) with-there be not any passing through in the π line, so system is stable.Know that by Fig. 4 the singular value curve of structure is not too mild, harmonic peak is higher, the less stable of description architecture.Can find out that from Fig. 5 structure is the overdamping state, and the peak response time be 8s, the time is longer.Therefore, the target of this system optimization control can be divided into following two aspects, and robust designs should improve controller performance and guarantee the stability that system is preferable.

Second step: structure is adopted H8 robust algorithm optimization, on the basis of original structure kinetic equation, the situation of structurally associated parameter perturbation is considered, transform equation, choose weighting function the Robust Optimal problem has been changed into the Mixed Sensitivity problem.Understand the double optimization controller performance through basic frequency-domain analysis and time domain performance analysis, adopt the robust stability and the performance robustness of the next further understanding of singular value μ analysis system, extremely shown in Figure 10 like Fig. 6.

Find out that from Fig. 6 singular value in [15.2,21.54] rad/s scope inner structure greater than 1, is illustrated in this scope, if there is the destructuring perturbation, robust stability can't be guaranteed so.The peak value of closed-loop system robust performance curve shown in Figure 7 is (a long dotted line peak value) 0.60002, explains that The controller can satisfy certain performance index to each controlled device in [0.1,100] rad/s, and this has explained H to a certain extent ₈The stronger characteristics of robustness that control method itself has.As can beappreciated from fig. 8 the magnitude margin of system is 2.1590e+006dB, and phase margin is infinitely great (being Inf), the good stability of illustrative system, and also the singular value curve of closed-loop system is gently in open cycle system, and the stability of illustrative system is improved.As can beappreciated from fig. 10, the steady-state value of system is 1, and the system peak response time is similar to 0.22s, is lower than the peak response time of open cycle system, shows that response of structure speed is improved.

The 3rd step: a TMD device is installed on the basis of original structure, shown in figure 11.The stability of original structure that adopted frequency-domain analysis methods such as Bode diagram, singular value curve is understood the structural behaviour performance through the rank response that jumps, like Figure 12 to shown in Figure 14.

With the performance of building structure in the last chapter, the response of structure time has reduced, peak response around in 0.1s.Can find equally that by Figure 13 the time domain vibration trend of structure-TMD system is strong, the less stable of description architecture-TMD system.This characteristic also can be schemed from the BODE of building structure relatively and structure-TMD system, draw behind the singular value curve; Promptly at amplitude versus frequency characte 20log|G (j ω) H (j ω) | in the frequency range of＞0dB; There is not any passing through in phase-frequency characteristic G (j ω) H (j ω) with-π line; This description architecture-TMD system stability, but the singular value curve harmonic peak of structure-TMD system is higher, and the stability of description architecture-TMD system is not good.Therefore, the stability that how further to improve structure-TMD system is H ₈The design object of controller.

The 4th step: TMD, two kinds of control methods of H8 are mutually combined, promptly carry out robust H8 and optimize in the basic enterprising step that TMD adopts the maximum power amplification coefficient to carry out initial optimization.Understand the double optimization controller performance through basic frequency-domain analysis and time domain performance analysis, adopt the robust stability and the performance robustness of the next further understanding of singular value μ analysis system, extremely shown in Figure 19 like Figure 15.

The frequency domain response of the short dash line representative structure singular value of the top among Figure 15; Find out that from figure singular value in [14.7,22.3] rad/s scope inner structure greater than 1, is illustrated in this scope; If there is the destructuring perturbation, robust stability can't be guaranteed so.Can be known that by Figure 16 the performance of system has all reached nominal performance and robust performance in most of frequency ranges, this has explained H to a certain extent ₈The stronger characteristics of robustness that control method itself has.As can beappreciated from fig. 17 the magnitude margin of system is 1.7500e+005dB, and phase margin is Inf, description architecture-TMD-H ₈The good stability of system is through comparative structure-TMD system and structure-TMD-H ₈The singular value curve of system can find out that latter's harmonic peak is lower, and the stability of description architecture-TMD system has been enhanced.As can beappreciated from fig. 19, structure-TMD-H ₈The steady-state value of system is 1, and the system peak response time through the rank response diagram that jumps of comparative structure-TMD system, can find out that the time domain vibration has reduced widely for for 0.15s, shows that the stability of structure-TMD system strengthens.

Synthesizing map 1 to Figure 19 can be known: original structure less stable and its response speed are slower.When adopting H ₈After controller was optimized it, response speed was improved; The singular value curve of structure also becomes gently, explains that stability is improved; But the robust performance curve of structure be not in all scopes all less than 1, explain that The controller can't arbitrarily make controlled device satisfy certain performance index in the frequency separation, the robustness of structure still needs further to improve.The response speed of structure-TMD system is very fast, and there is vibration in the rank response but it jumps, and the singular value curve is not milder, explains that TMD can improve the performance of structure, but can't improve stability of structure.With chapter 3 H ₈Controller is compared, and finds H ₈Controller can improve stability of structure, but is weaker than TMD slightly for the enhancing performance of performance.Adopt H ₈After controller was optimized the TMD system, stability of structure was improved, but structure-TMD-H ₈The response time of system is longer than structure-TMD system response time, than structure-H ₈System response time is short, and H is described ₈Controller can improve stability of structure, but can't further improve response of structure speed.In addition, structure-TMD-H ₈The robust performance curve of system and robust stability peak of curve are all less than structure-H ₈System explains that the robust performance of double optimization system is improved.

Claims (2)

1. use H ₈The robust criterion is optimized structure-the TMD system, improves system stability, and thinking is:

The tectonic system model is set up system equation and H ₈The robust criterion is respectively to original structure, structure-H ₈System, structure-TMD system, structure-TMD-H ₈System analyzes, and the comparative analysis result proves that this optimization method really can improve the performance of system.

2. the tectonic system model is set up system equation; Carry out H ₈Robust designs is confirmed method of evaluating performance.

1) for structure-TMD system analysis model, its kinetic equation can be expressed as:

$m_s{\stackrel{\&CenterDot;\&CenterDot;}{X}}_s+c_s{\stackrel{\&CenterDot;}{X}}_s+k_s{X}_s=m_s{F}_a\left(t\right)$

$m_d{\stackrel{\&CenterDot;\&CenterDot;}{x}}_d+c_d{\stackrel{\&CenterDot;}{x}}_d+k_d{x}_d=m_d{F}_a\left(t\right)$

In the formula:

X _sAcceleration, speed and the displacement of expression main system (structure), m _s, c _s, k _sQuality, damping and the rigidity of expression main system;

x _dAcceleration, speed and the displacement of expression mass, m _d, c _d, k _dQuality, damping and the rigidity of expression mass; F _a(t) expression acts on the external excitation on the main system, considers two physical parameter c, the uncertainty of k:

$C_s={\stackrel{\&OverBar;}{C}}_s(1+{\mathrm{\&delta;}}_c{p}_c),p_c=0.2,{\mathrm{\&delta;}}_c\&Element;[-\mathrm{1,1}]$

$K_s={\stackrel{\&OverBar;}{K}}_s(1+{\mathrm{\&delta;}}_k{\mathrm{\&rho;}}_k),p_k=0.3,{\mathrm{\&delta;}}_k\&Element;[-\mathrm{1,1}]$

In the formula:

Be C _s, K _sNominal value.p _c, p _k, δ _c, δ _kThe relative perturbation of representation parameter.The same last chapter of other abbreviation.Use y _c, y _kAnd u _c, u _kRepresent parameter perturbation δ _c, δ _kInput/output variable, then the equation of motion of structure-TMD system can be expressed as:

$\left[\begin{array}{c}{\stackrel{\&CenterDot;}{X}}_s\\ {\stackrel{\&CenterDot;}{X}}_d\\ {\stackrel{\&CenterDot;\&CenterDot;}{X}}_s\\ {\stackrel{\&CenterDot;\&CenterDot;}{X}}_d\\ Y_k\\ Y_c\\ Z\end{array}\right]\left[\begin{array}{ccccccc}0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ -\frac{{\stackrel{\&OverBar;}{K}}_s+K_d}{M_s}& \frac{K_d}{M_s}& -\frac{{\stackrel{\&OverBar;}{C}}_s+C_d}{M_s}& \frac{C_d}{M_s}& \frac{P_k\&times;{\stackrel{\&OverBar;}{K}}_s}{M_s}& \frac{P_c\&times;{\stackrel{\&OverBar;}{C}}_s}{M_s}& \frac{1}{M_s}\\ \frac{K_d}{m_d}& -\frac{K_d}{m_d}& \frac{C_d}{m_d}& -\frac{C_d}{m_d}& 0& 0& 0\\ -1& 0& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0\end{array}\right]\&times;\left[\begin{array}{c}{X}_s\\ X_d\\ {\stackrel{\&CenterDot;}{X}}_s\\ {\stackrel{\&CenterDot;}{X}}_d\\ U_k\\ U_c\\ u\end{array}\right]$

$\left[\begin{array}{c}{U}_k\\ U_c\end{array}\right]=\left[\begin{array}{cc}{\mathrm{\&delta;}}_k& 0\\ 0& {\mathrm{\&delta;}}_c\end{array}\right]\left[\begin{array}{c}{Y}_k\\ Y_c\end{array}\right]$

Uncertain matrix Δ=diag (δ in the formula _k, δ _c) uncertainty of representative structure.

2) optimization requires closed-loop system H ₈Less than 1, promptly require to satisfy equation || W _p(I+GK) ^-1|| _∞＜1, in the formula, S (G)=(I+GK) ^-1Be the output sensitivity function of nominal system, W _pFor being used for representing outside (output), weighting function disturbs frequecy characteristic and the performance requirement level of d.

3) performance of understanding optimal controller through the frequency-domain analysis and the time domain performance analysis of system, the robust stability and the performance robustness of the next further understanding of employing singular value μ analysis system.

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