Điều khiển kết cấu - Chương 6 pdf

421 Part II: Active Control Chapter 6 Introduction to active structural motion control 6.1 The nature of active structural control Active versus passive control The design methodologies presented in the previous chapters provide systematic procedures for distributing passive motion control resources which, by definition, have fixed properties and do not require an external source of energy. Once installed, a passive system cannot be modified instantaneously, and therefore one needs a reliable estimate of the design loading and an accurate numerical model of the physical system for any passive control scheme to be effective. The inability to change a passive control system dynamically to compensate for an unexpected loading tends to result in an over-conservative design. When self-weight is an important design constraint, one cannot afford to be too conservative. Also, simulation studies on the example building structures show that passive control is not very effective in fine tuning the response in a local region. Considering these limitations, the potential for improving the performance by dynamically modifying the loading and system properties exists. An active structural control system is one which has the ability to determine the present state of the structure, decide on a set of actions that will change this state to a more desirable one, and carry out these actions in a controlled manner and in a short period of time. Such control systems can theoretically accommodate unpredictable environmental 422 Chapter 6: Introduction to Active Structural Motion Control changes, meet exacting performance requirements over a wide range of operating conditions, and compensate for the failure of a limited number of structural components. In addition, they may be able to offer more efficient solutions for a wide range of applications, from both technical and financial points of view. Active motion control is obtained by integrating within the structure a control system consisting of three main components: a) monitor, a data acquisition system, b) controller, a cognitive module which decides on a course of action in an intelligent manner, and c) actuator, a set of physical devices which execute the instructions from the controller. Fig 6.1 shows the interaction and function of these components; the information processing elements for active control are illustrated in Fig 6.2. This control strategy is now possible due to significant recent advances in materials that react to external stimuli in a non- conventional manner, sensor and actuator technologies, real-time information processing, and intelligent decision systems. Fig. 6.1: Components of an active control system STRUCTURE RESPONSE EXCITATION SENSORS SENSORS ACTUATOR AGENTS TO CARRY OUT INSTRUCTIONS IDENTIFY THE STATE OF SYSTEM DECIDE ON COURSE OF ACTION DEVELOP THE ACTION PLAN (SET OF INSTRUCTIONS TO BE COMMUNICATED TO THE ACTUATORS) CONTROLLER MONITOR to measure external loading MONITOR to measure response 6.1 The Nature of Active Structural Control 423 Fig. 6.2: Information Processing Elements for an Active Control System Fig. 6.3: Passive and active feedback diagrams Physical System Sensor Sensor Sensor Data Processing Transmission channel Processing Modeling & Analysis Decision Making Action Visualization Archival and Access Fusion hp() pu (a) Passive h' p ∆p e ∆p f ++() u (b) Active observe u u ε u + decide on ∆p f + ∆p f p observe p p ε p + decide on ∆p e + ∆p e decide on changing htoh' 424 Chapter 6: Introduction to Active Structural Motion Control The simple system shown in Fig. 6.3 is useful for comparing active and passive control. Figure 6.3(a) corresponds to passive control. The input, , is transformed to an output, , by the operation (6.1) One can interpret this system as a structure with denoting the loading, the displacement, and the flexibility of the structure. The strategy for passive motion control is to determine such that the estimated output due to the expected loading is contained within the design limits, and then design the structure for this specific flexibility. Active control involves monitoring the input and output, and adjusting the input and possibly also the system itself, to bring the response closer to the desired response. Figure 6.3(b) illustrates the full range of possible actions. Assuming the input corrections and system modifications are introduced instantaneously, the input-output relation for the actively controlled system is given by (6.2) Monitoring the input, and adjusting the loading is referred to as open-loop control. Observing the response, and using the information to apply a correction to the loading is called feedback control. The terminology closed-loop control is synonymous with feedback control. In addition to applying a correction to the input, the control system may also adjust certain properties of the actual system represented by the transformation . For example, one can envision changing the geometry, the connectivity, and the properties of structural elements in real time. One can also envision modifying the decision system. A system that can adjust its properties and cognitive processes is said to be “adaptive”. The distinguishing characteristic of an adaptive system is the self-adjustment feature. Non-adaptive active structural control involves monitoring and applying external forces using an invariant decision system. The make up of the structure is not changed. Adaptive control is the highest level of active control. The role of feedback Feedback is a key element of the active control process. The importance of feedback can be easily demonstrated by considering a linear static system and p uhp() uhp()= pu h hp() uh' p ∆p e ∆p f ++()= hp() 6.1 The Nature of Active Structural Control 425 taking the input correction to be a linear function of the output. For this case, (6.3) (6.4) where and are constants. Substituting in eqn (6.2) specialized for h’=h and solving for results in (6.5) When is positive, the sensitivity of the system to loading is increased by feedback, i.e. the response is amplified. Taking negative has the opposite effect on the response. Specializing eqn (6.5) for negative feedback ( ), the response becomes (6.6) Increasing decreases the effect of external loading. However, the influence of , the noise in the response observation, increases with and, for sufficiently large , is essentially independent of the feedback parameter. This result indicates that the accuracy of the monitoring system employed to observe the response is an important design issue for a control system. Computational requirements and models for active control The monitor component identified in Figs 6.1 and 6.2 employs sensors to measure a combination of variables relevant to motion such as strain, acceleration, velocity, displacement, and other physical quantities such as pressure, temperature, and ground motion. This data is usually in the form of analog signals which are converted to discrete time sequences, fused with other data, and transmitted to the controller module. Data compression is an important issue for large scale remote sensing systems. Wavelet based data compression (Amaratunga, 1997) is a promising approach for solving the data processing problem. The functional requirements of the controller are to compare the observed response with the desired response, establish the control action such as the level of feedback force, and communicate the appropriate commands to the actuator uhp= ∆p f k f u ε u +[]= hk f u u h 1 hk f – p ∆p e +[] hk f 1 hk f – ε u += k f k f k f 0< u h 1 hk f + p ∆p e +[] hk f 1 hk f + ε u += k f ε u k f k f 426 Chapter 6: Introduction to Active Structural Motion Control which then carries out the actual control actions such as apply force or modify a structural property. The controller unit is composed of a digital computer and software designed to evaluate the input and generate the instructions for the actuators. There are 2 information processing tasks: state identification and decision making. Given a limited amount of data on the response, one needs to generate a more complete description of the state of the system. Some form of model characterizing the spatial distributions of the response and data analysis are required. Once the state has been identified, the corrective actions which bring the present state closer to the desired state can be established. In this phase, a model which defines the input - output relationship for the structure is used together with an optimization method to decide upon an appropriate set of actions. For algorithmic non-adaptive systems, the decision process is based on a numerical procedure that is invariant during the period when the structure is being controlled. Time invariant linear feedback is a typical non-adaptive control algorithm. An adaptive controller may have, in addition to a numerical control algorithm, other symbolic computational models in the form of rule-based systems and neural networks which provide the capability of modifying the structure and control algorithm in an intelligent manner when there is a change in the environmental conditions. Examples illustrating time invariant linear feedback control algorithms are presented in the following sections; a detailed treatment of the algorithms is contained in Chapters 7 and 8. 6.2 An introductory example of quasi-static feedback control Consider the cantilever beam shown in Fig 6.4. Suppose the beam acts like a bending beam, and the design objective is to control the deflected shape such that it has constant curvature. The target displacement distribution corresponding to this constraint has the form (6.7) where is the desired curvature. One option is to select the bending rigidity according to (6.8) u * x() 1 2 χ * x 2 = χ * D B x() Mx() χ * = 6.2 An Introductory Example of Quasi-static Feedback Conrol 427 where M(x) is the moment at location x due to the design loading. This strategy is a stiffness based passive control approach. A second option is to select a representative bending rigidity distribution, and apply a set of control forces which produce a displacement distribution that, when combined with the displacement due to the design loading, results is a displacement profile that is close to the desired distribution. In what follows, the latter option is discussed. Fig. 6.4: Cantilever beam with control force Suppose the control force system consists of a single force applied at x=L. Assuming linear elastic behavior, and using the linear technical theory of beams as the model for the structure, the displacement due to F is estimated as (6.9) where is the bending rigidity, considered constant in this example. The displacement due to the design loading is also determined with the technical beam theory. This term is denoted as , and expressed as (6.10) Combining the 2 displacement patterns results in the total displacement, u(x). (6.11) The expanded form of eqn (6.11) corresponding to the particular choice of control force location for this example is (6.12) The difference between and is defined as and interpreted L u(x) x F u c x() F 2D B Lx 2 x 3 3 –   F D B hx()== D B u o x() u o x() 1 D B gx()= ux() u o x() u c x()+= ux() 1 D B gx() Fh x()+[]= ux() u * x() ex() 428 Chapter 6: Introduction to Active Structural Motion Control as the displacement error. (6.13) For this example, is considered to be fixed, and therefore is a function only of the single control force magnitude F. (6.14) Ideally, one wants for . This goal cannot be achieved, and it is necessary to work with a relaxed condition. The simplest choice is collocation, which involves setting equal to zero at a set of prescribed locations. For example, setting at x=L leads to (6.15) A more demanding condition is a least square requirement, which involves first forming the sum of evaluated at a set of prescribed points, and then selecting F such that the sum is a minimum. The continuous least square sum is given by the following integral (6.16) Taking J(F) as the measure of the square error sum, F is determined with the stationary condition (6.17) Differentiating the integral expression for J, (6.18) and using eqn (6.14), which defines for this particular example, results in (6.19) The value of F defined by eqn (6.19) produces the absolute minimum value of J. A proof of this statement is presented in Section 7.2 which treats in more detail the ex() u o u c u * –+= D B ex() ex() 1 D B gx() Fh x()+[]u * x()–= e 0= 0 xL≤≤ e e 0= F D B u * L() gL()– hL() = e 2 J 1 2 e 2 xd 0 L ∫ JF()== F∂ ∂J 0= F∂ ∂J e F∂ ∂e xd 0 L ∫ 0== ex() F hx()D B u * x() gx()–[]xd 0 L ∫ hx()() 2 xd 0 L ∫ = 6.2 An Introductory Example of Quasi-static Feedback Conrol 429 least square procedure for quasi-static loading. Example 6.1: Shape control for uniform loading This example illustrates the application of the approach described above to the case where the design loading is a uniform distributed load extending over the entire length of the beam. The corresponding deflected shape is (1) Applying collocation at x=L leads to (2) The least square solution is (3) Both solutions are approximate since they do not satisfy . One can improve the performance by taking additional control forces. Selecting the spatial distribution of the control forces is a key decision for the design of a control system. The above discussion assumes that there is some initial loading, and one can determine the corresponding displacement field with the physical model of the structural system. This control strategy is similar to the concept of prestressing. A more general scenario is the case where one is observing the response at a set of “observation” points and the loading is being applied gradually so that there is negligible dynamic amplification and sufficient time to u o x() 1 D B gx() 1 D B w 24 x 4 4x 3 L– 6x 2 L 2 +()== F eL() 0= 3 2   D B χ * L   3 2   wL–= F ls 91 66   D B χ * L 2065 5280   wL–= 1.379() D B χ * L 0.391()wL–= ex() 0= 430 Chapter 6: Introduction to Active Structural Motion Control adjust the control forces. Here, one needs to establish using the observed displacement data. Suppose there are observation points located at ( j= 1, 2, , s), and at time t the monitoring system produces the data set . This data can be used together with an interpolation scheme to generate an estimate of for the region adjacent to the observation points. A typical spatial interpolation model has the form (6.20) where are interpolation functions. Given , one forms the displacement error, (6.21) and determines F(t) with either collocation or a least square method. The continuous least square estimate for F(t) is given by (6.22) Example 6.2: Discrete displacement data Suppose the displacement observation points are located at x=L/2 and x=L. Given these 2 values of displacement, one needs to employ an interpolation scheme in order to estimate . Taking a quadratic expansion, (1) u o x() sx j u o x j t,() u o u o xt,() u o x j t,()Ψ j x() j 1= s ∑ = Ψ j x() u o xt,() ext,()u o xt,()u * x()– 1 D B hx()Ft()+= Ft() D B hx()u * x() u o xt,()–()xd 0 L ∫ hx()() 2 xd 0 L ∫ = u o x() u o x() a o a 1 xa 2 x 2 ++= [...]... + p (6. 23) and introducing the definitions for frequency and damping ratio leads to the 432 Chapter 6: Introduction to Active Structural Motion Control k m F p c ug u + ug Fig 6. 5: Single-degree-of-freedom system standardized form of the governing equation 2 F p ˙˙ ˙ u + 2ξωu + ω u = – a g + - + m m (6. 24) The free vibration response of the uncontrolled system has the general form u = Ae λt (6. 25)... eqn (6. 31) • applying F with an actuator Mathematically, one incorporates feedback by substituting for F in eqn (6. 24) The result is kv 2 kd p ˙˙ u +  2ξω +  u +  ω + - u = – a g + - ˙     m m m (6. 32) Equation (6. 32) can be transformed to the standardized form by defining equivalent damping and frequency parameters as follows: 2 2 kd ω eq = ω + m (6. 33) kv 2ξ eq ω eq = 2ξω + -m (6. 34)... this force scheme 4 36 Chapter 6: Introduction to Active Structural Motion Control 0. 16 Maximum displacement - m System 1 T1=0.99s El Centro Taft 0.14 0.12 0.1 0.08 0. 06 0.04 0.05 0.1 0.15 ξa 0.2 0.25 0.3 0.35 System 2 T1=3.14s El Centro Maximum displacement - m 0.3 Taft 0.25 0.2 0.15 0.1 0.05 0.1 0.15 ξa 0.2 0.25 0.3 Fig 6. 6: Variation of maximum displacement with active damping 6. 3 An Introductory... 2.2 Maximum force - N El Centro 2 Taft 1.8 1 .6 1.4 1.2 1 0.8 0 .6 0.4 0.05 0.1 0.15 ξa 0.2 0.25 0.3 7000 T1=3.14s System 2 60 00 El Centro Maximum force - N Taft 5000 4000 3000 2000 1000 0.05 0.1 0.15 ξa 0.2 0.25 0.3 Fig 6. 7: Variation of maximum control force level with active damping 438 Chapter 6: Introduction to Active Structural Motion Control 160 00 System 1 T1=0.99s Maximum power - N.m/s 14000 El... (6. 29) It follows that the effect of linear feedback is to change the fundamental frequency and damping 434 Chapter 6: Introduction to Active Structural Motion Control ratio Solving eqns (6. 33) and (6. 34) for ω eq and ξ eq , results in kd ω eq = ω 1 + k (6. 35) ξ eq = ξ + ξ a (6. 36) where ξ a is the increment in damping ratio due to active control kv kd 1 ⁄ 2   1 – 1 ξ a = -. .. ratio due to active control kv kd 1 ⁄ 2   1 – 1 ξ a = - – ξ  1 + -k d 1 ⁄ 2 2ωm k   1 + -k (6. 37) Critical damping corresponds to ξ eq = 1 kv = 2mω ξ = 1 eq kd 1 + - – ξ k (6. 38) Equation (6. 35) shows that negative displacement feedback increases the frequency According to eqn (6. 37), the damping ratio is increased by velocity feedback and decreased by displacement... and eqn (2) Evaluating the integrals leads to DB χ * DB  91 - – -  98 u + 133 u  -F≈  66  L L 3  33 1 66 2 (3) as an estimate for F 6. 3 An introductory example of dynamic feedback control To gain further insight on the nature of feedback control, the simple SDOF system shown in Fig 6. 5 is considered The system is assumed to be subjected to both an external force and ground motion,... eqn (6. 24), one obtains two possible solutions u = A1 e λ1 t + A2 e λ2 t (6. 26) 2 λ 1, 2 = – ξω ± iω 1 – ξ = – ξω ± iω′ (6. 27) Considering A 1 and A 2 to be complex conjugates, 1 A 1, 2 = [ A R ± iA I ] 2 (6. 28) where A R and A I are real numbers representing the real and imaginary parts of A , the solution takes the form u = e – ξωt 2 2 A R cos  ωt 1 – ξ  + A I sin  ωt 1 – ξ      (6. 29) 6. 3... a number of these actuator-rod configurations, one can generate a piecewise linear bending moment distribution 6. 4 Actuator Technologies 443 F d F a) A Fd B Fd b) B A (-) Fd c) Fig 6. 12: Constant moment field L F/2 F/2 F F a) b) C A B (-) c) FL/4 Fig 6. 13: Triangular moment field 444 Chapter 6: Introduction to Active Structural Motion Control Linear actuators generate control force systems composed of... Chapter 6: Introduction to Active Structural Motion Control stress, i.e., the phase remains austenite for arbitrary applied stress As Volume fraction martensite Mf 1 0.5 0 Ms Af Md Temperature, T Fig 6. 23: Martensitic transformation on cooling and heating The stress-strain behavior is strongly dependent on temperature Figure 6. 24 shows the limiting stress-strain curves for Nitinol, a nickel-titanium .   2 ++≈ F 91 66   D B χ * L D B L 3 98 33 u 1 133 66 u 2 +   –≈ F mu ˙˙ cu ˙ ku++ ma g – Fp++= 432 Chapter 6: Introduction to Active Structural Motion Control Fig. 6. 5: Single-degree-of-freedom. Chapter 6: Introduction to Active Structural Motion Control ratio. Solving eqns (6. 33) and (6. 34) for and , results in (6. 35) (6. 36) where is the increment in damping ratio due to active control (6. 37) Critical. governing equation (6. 24) The free vibration response of the uncontrolled system has the general form (6. 25) Substituting for in eqn (6. 24), one obtains two possible solutions (6. 26) (6. 27) Considering