Điều khiển kết cấu - Chương 8 ppsx

545 Chapter 8 Dynamic Control Algorithms - Time invariant systems 8.1 Introduction This chapter extends the active control strategy introduced in the previous chapter to deal with time dependent loading. The discussion is restricted to time invariant systems, i.e., the case where the system properties and force feedback algorithm are constant over the duration of the time response. The material presented here is organized as follows. Firstly, the state-space formulation for a SDOF system is developed, and used to generate the free vibration response. This solution provides the basis for establishing a criterion for dynamic stability of a SDOF system. Secondly, linear negative feedback is introduced, and the topic of stability is revisited. The primary focus is on assessing the effect of time delay in applying the control force on the stability. Thirdly, the SDOF state-space formulation is specialized to deal with discrete time control, where the feedback forces are computed at discrete time points and held constant over time intervals. Stability for discrete time feedback with time delay is examined in detail, and a numerical procedure for determining the time increment corresponding to a stability transition is presented and illustrated with examples. Fourthly, the choice of the optimal magnitudes of the feedback 546 Chapter 8: Dynamic Control Algorithm - Time Invariant Systems parameters is considered. Optimality is related to the magnitude of a quadratic performance index (LQR) which is taken as a time integral involving weighted response and control force terms. This approach is referred to as the “linear quadratic regulator problem” and leads to a time invariant linear relationship between the control forces and state variables. Fifthly, the state-space formulation is extended to MDOF systems. The modal properties for an arbitrary damping scheme are derived, and used to generate the governing equations expressed in terms of the modal coordinates. The last section deals with optimal feedback based on the LQR performance index generalized for MDOF systems. Examples which illustrate the sensitivity of the response and cost parameters to variations in the location and nature of the control forces, and weighting coefficients are presented. 8.2 State-space formulation - time invariant SDOF systems Governing equations Fig. 8.1: SDOF system. The dynamic response of the SDOF linear system shown in Fig. 8.1 is governed by the second order equation, (8.1) where is the applied external loading, F is the active force, and m, k, c are system parameters. Integrating eqn (8.1) in time, and enforcing the initial conditions on and at , one obtains the velocity and displacement as functions of time. These quantities characterize the state of the system in the sense that once and are specified, the acceleration and internal forces can be determined by back substitution. k c m uu g + F u g p mu ˙˙ cu ˙ ku++ ma g – pF++= p u u ˙ t 0= u u ˙ 8.2 State-space Formulation-Time Invariant SDOF System 547 Rather than working with a second order equation, it is more convenient to transform eqn (8.1) to a set of first order equations involving the state variables and . The new form is (8.2) This form is called the state-space representation. The motivation for the state- space representation is mainly the reduced complexity in generating both analytical and numerical solutions. Matrix notation is convenient for expressing the state-space equations in a compact form. Defining as the state vector, (8.3) the matrix equilibrium equation is written as (8.4) where the various coefficient matrices are defined below. (8.5) (8.6) (8.7) The initial conditions at t=0 are denoted by . u u ˙ ud dt u ˙ = u ˙ d dt c m –   u ˙ k m –   u 1–()a g 1 m   p 1 m   F++++= X X u u ˙ X t()== Xd dt X ˙ AX B f F B g a g B p p++ +== A 01 k m – c m – = B f B p 0 1 m == B g 0 1– = X o 548 Chapter 8: Dynamic Control Algorithm - Time Invariant Systems (8.8) With this representation, the problem is reduced to solving a first order equation involving . Free vibration uncontrolled response The free vibration uncontrolled response is governed by a reduced form of eqn (8.4) (8.9) When is constant, the general solution has the form (8.10) where is an unspecified vector of order 2 and is a scalar. Substituting for results in (8.11) where is the identity matrix. According to eqn (8.11), the eigenvalues of define the frequency and damping characteristics of the free vibration response. Expanding , (8.12) leads to the characteristic equation (8.13) and two eigenvalues (8.14) Noting that and , eqn (8.14) is identical to eqn (6.27) which was obtained from the second order equation. Given , eqn (8.11) can be solved for the eigenvectors which define the state-space modes. Since is complex, the eigenvectors occur as complex conjugates. X 0() u 0() u ˙ 0() X o ≡= X X ˙ AX= A XVe λt = V λ X A λI–()V0= IA A λI– 0= λ– 1 k m – c m – λ– 0= λ 2 c m λ k m ++ 0= λ 12, 1 2 c m – i 4 k m   c m   2 –±λ R iλ I ±== km⁄ω 2 = cm⁄ 2ξω= λ λ 8.2 State-space Formulation-Time Invariant SDOF System 549 (8.15) The total free vibration response is obtained by combining the 2 complex solutions such that the resulting expression is real. Starting with (8.16) and taking (8.17) where and are real scalars, results in (8.18) The constants and are determined by enforcing the initial conditions on at t=0. (8.19) Lastly, the solution for u(t) is given by the first scalar equation in eqn (8.18). (8.20) V 12, 1 λ R i 0 λ I ± V R iV I ± V 1 V ˜ 1 ,=== X A 1 e λ 1 t V 1 A 2 e λ ˜ 1 t V ˜ 1 += A 1 1 2 A R iA I +()= A 2 A ˜ 1 = A R A I X t() e λ R t A R V R A I V I –()λ I tA– R V I A I V R –()λ I sin t+cos{}= A R A I X X 0() A 1 V 1 A ˜ 1 V ˜ 1 + A R V R A I V I –== ⇒ u o u ˙ o A R A R λ R A I λ I – = ⇒ A R u o = A I 1 λ I u ˙ o λ R u o +()–= ut() e λ R t A R λ I tA I λ I tsin–cos()= 550 Chapter 8: Dynamic Control Algorithm - Time Invariant Systems General solution - time invariant systems The general solution for an arbitrary loading can be expressed as a Duhamel integral involving a specialized form of the free vibration response. Considering first a first order scalar equation, (8.21) where is constant, and g is a function of t, the complete solution has the form (8.22) A similar form can be generated for the first order matrix equation, (8.23) The free vibration solution defined by eqn (8.18), can be expressed as (8.24) where is defined by the following series: (8.25) This matrix exponential function has the “same” property as the corresponding scalar function. Using eqn (8.24), the Duhamel integral matrix form of the total solution for eqn (8.4) is (8.26) where (8.27) The corresponding scalar form of the solution for u(t) is y ˙ ay g+= a yt() e at t o –() y o e at τ–() g τ()τd t o t ∫ += X ˙ t() AX G+= X t() e At X o = e At e At IAt 1 2 AAt 2 … 1 n! A n t n …++ ++ += td d e At ()Ae At = X t() e A tt o –() X o e A t τ–() G τ()τd t o t ∫ += G t() B f F B g a g B p p++= 8.2 State-space Formulation-Time Invariant SDOF System 551 (8.28) Equation (8.26) applies for an arbitrary linear time invariant system. It is convenient for establishing a discrete formulation of the governing equations. This topic is addressed in the next section. Example 8.1: Equivalence of equations (8.18) and (8.24) Consider eqn (8.16). The total free vibration response is given by (1) Noting eqn (8.11), the and V terms are related by (2) Expanding the product, , and using eqn (2), leads to (3) It follows that eqn (1) can be written as (4) ut() e λ R t u o λ I t 1 λ I u ˙ o λ R u o +()λ I tsin+cos   += 1 λ I e λ R t τ–() λ I t τ–()a g τ()– p τ() m F τ() m ++   sin τd 0 t ∫ + X t() A 1 e λt V 1 A ˜ 1 e λ ˜ t V ˜ 1 += X 0() A 1 V 1 A ˜ 1 V ˜ 1 += λ AV 1 λV 1 = AV ˜ 1 λ ˜ V ˜ 1 = e λ t V 1 e λt V 1 V 1 λV 1 ()t λλV 1 () t 2 2 …++ += IAt AA t 2 2 …++ +   V 1 = e At V 1 = X t() e At A 1 V 1 A ˜ 1 V ˜ 1 +()e At X o == 552 Chapter 8: Dynamic Control Algorithm - Time Invariant Systems Stability criterion Another advantage of the state-space representation is the ability to relate the stability of the physical system to the eigenvalues of . A system is said to be stable when the motion resulting from some initial disturbance is bounded. Assuming the system state is at time , stability requires (8.29) where defines the bound on the perturbation from . Equation (8.18) defines the general homogeneous solution for a SDOF time invariant system. The terms contained inside the brackets depend on the initial conditions and are bounded since the time dependency is harmonic. Therefore, it follows that the exponential term must be bounded. This requirement is satisfied when the exponent is negative, (8.30) In words, the real part of the eigenvalues of must be equal to or less than zero. When , the response is pure harmonic oscillation. A negative produces a damped harmonic response. Plotting in the complex plane provides a geometric interpretation of the stability. For the SDOF case, there are two eigenvalues, (8.31) Figure 8.2 shows the corresponding points in the complex plane. These points are referred to as poles. Undamped motion has poles on the imaginary axis. Holding stiffness constant and increasing causes the poles to move along the circle of radius toward the critical damping point, . With further increase in damping, the curves bifurcate with one branch heading in the negative (real axis) A X o t 0= X t() X o – ε≤ for all t ε X o λ R 0≤ A λ R 0= λ R λ λλ R iλ I ±= λ R c 2m – ξω–== λ I k m c 2m   2 – ω 1 ξ 2 –== c ωξ1= 8.2 State-space Formulation-Time Invariant SDOF System 553 direction, and the other toward the origin. Increasing the stiffness with held constant moves the poles in the imaginary direction. With this terminology, the stability criterion requires all the poles corresponding to the eigenvalues of A to be on or to the left of the imaginary axis, as shown in Fig. 8.3. The uncontrolled SDOF system is, according to this definition, always stable since . Fig. 8.2: Poles for SDOF system. Fig. 8.3: Stability condition for SDOF system. Linear negative feedback The response of a SDOF time invariant system with negative linear feedback is governed by eqn (8.4) with F taken as a linear function of the state variables c ξ 0≤ ω λ R λ I θ c increases k increases ξ 1= ξ 0= ξ 0= ω λ R λ I stable unstable 554 Chapter 8: Dynamic Control Algorithm - Time Invariant Systems (8.32) Substituting for , the governing equation is transformed to (8.33) where (8.34) The general form of the free vibration solution of eqn (8.33) is (8.35) where and are the eigenvalues and eigenvectors of , the modified coefficient matrix. They are related by (8.36) The eigenvalues of are written as (8.37) Since is positive for negative feedback (note that the minus sign is incorporated in the definition equation, eqn (8.32)), the system is stable for arbitrary . Displacement feedback moves the poles in the imaginary direction, and consequently has no effect on the stability. It follows that increasing the negative F K f X– k d k v u u ˙ –== F X ˙ A c XB g a g B p p++= A c AB f K f – 01 k m – k d m – c m – k v m – == XVe λt = λ V A c AB f K f – λI–[]VO= A c λλ R i±λ I = λ R ck ν + 2m – ξ eq ω eq –== λ I ω eq 1 ξ eq 2 –= ω eq ω 1 k d ω 2 m + 12⁄ = ξ eq ξξ a += ξ a k ν 2mω eq ξ 1 ω ω eq –   –= k v k v [...]... -2 td Ω 1 + tan 2 (8. 63) 562 Chapter 8: Dynamic Control Algorithm - Time Invariant Systems 2 td Ω 1 – tan 2 cos ( t d Ω ) = -2 td Ω 1 + tan 2 (8. 64) eqn (8. 59) can be expressed as td Ω 2 td Ω ( g v Ω – 2ξωΩ )tan - + 2g d tan - – ( g v Ω + 2ξωΩ ) = 0 2 2 (8. 65) The two roots of eqn (8. 65) are td Ω – g d − g d + g v Ω – 4ξ ω Ω + tan - = ... ( τ ) = – kx j tj ≤ τ ≤ tj + 1 (8. 94) is given by x j + 1 = cx j (8. 95) bk c = e a∆t + - ( 1 – e a∆t ) a (8. 96) where Taking j=0, 1, 2, leads to x j = ( c ) j xo (8. 97) 572 Chapter 8: Dynamic Control Algorithm - Time Invariant Systems For x j to be bounded, the absolute magnitude of c must be less than 1 c . …++= ξ eq ω eq 8. 2 State-space Formulation-Time Invariant SDOF System 557 (8. 46) The modified equivalent frequency and damping are related to the time delay by (8. 47) (8. 48) Equation (8. 46) is convenient. ω 2 g d e it d Ω– ig v Ωe it d Ω– +++ + 0= 8. 2 State-space Formulation-Time Invariant SDOF System 561 Replacing the exponential term by (8. 57) yields (8. 58) Equation (8. 58) is satisfied when the real and. tan 2 t d Ω 2 + = 562 Chapter 8: Dynamic Control Algorithm - Time Invariant Systems (8. 64) eqn (8. 59) can be expressed as (8. 65) The two roots of eqn (8. 65) are (8. 66) Finally, the maximum time