479 Chapter 7 Quasi-static Control Algorithms 7.1 Introduction to control algorithms Referring back to Fig 6.1, an active structural control system has 3 main components: i) a data acquisition system that collects observations on the excitation and response, ii) a controller that identifies the state of the structure and decides on a course of action and iii) a set of actuators that apply the actions specified by the controller. The decision process utilizes both information about how the structure responds to different inputs and optimization techniques to arrive at an “optimal” course of action. When this decision process is based on a specific procedure involving a set of prespecified operations, the process is said to be algorithmic, and the procedure is called a “control algorithm”. A non-adaptive control algorithm is time invariant, i.e., the procedure is not changed over the time period during which the structure is being controlled. Adaptive control algorithms have the ability to modify their decision making process over the time period, and can deal more effectively with unanticipated loadings. They also can upgrade their capabilities by incorporating a learning mechanism. This text is concerned primarily with time invariant control algorithms which are well established in the control literature. Adaptive control is an on-going research area which holds considerable promise but is not well defined at this time. A brief discussion is included here to provide an introduction to the topic. The topic addressed in this chapter is quasi-static control, i.e., where the 480 Chapter 7: Quasi-static Control Algorithms structural response to applied loading can be approximated as static response. Since time dependent effects are neglected, stiffness is the only quantity available for passive control. Active control combines stiffness with a set of pseudo-static control forces. The quasi-static case is useful for introducing fundamental concepts such as observability, controllability, and optimal control. Both continuous and discrete physical systems are treated. The next chapter considers time- invariant dynamic feedback control of multi-degree-of freedom structural systems. A combination of stiffness, damping, and time dependent forces is used for motion control of dynamic systems. The state-space formulations of the governing equations for SDOF and MDOF systems are used to discuss stability, controllability, and observability aspects of dynamically controlled systems. Continuous and discrete forms of the linear quadratic regulator (LQR) control algorithm are derived, and examples illustrating their application to a set of shear beam type buildings are presented. The effect of time delay in the stability of LQR control, and several other linear control algorithms are also discussed. 7.2 Active prestressing of a simply supported beam Passive prestressing The concept of introducing an initial stress in a structure to offset the stress produced by the design loading is known as prestressing. This strategy has been used for over 60 years to improve the performance of concrete structures, particularly beams. The approach is actually a form of quasi-static control, where the variables being controlled are the stresses. Figure 7.1 illustrates prestressing of a single span beam with a single cable. When the cable shape is parabolic, the tension introduced in the cable creates an “upward” uniform loading, w o , that is related to the tension by (7.1) The initial moment distribution is parabolic, and the moment is negative according to the conventional notation. w o L 2 8d T= 7.2 Active Prestressing of a Simple Supported Beam 481 Fig. 7.1: Passive prestressing scheme Suppose the design loading is a concentrated force that can act at any point on the span. The maximum positive moment due to the force occurs when the force acts at mid-span, and the resultant positive moment at mid-span is given by (7.2) The initial mid-span moment is negative and equal to (7.3) If the prestress level is selected such that (7.4) which requires (7.5) L d w o (-) M+ w o L 2 /8 (+) PL/4 M+ P cable d = distance between top and bottom location of the cable M L 2   PL 4 w o L 2 8 –= M L 2   w o L 2 8 –= w o L 2 8 1 2 PL 4   M * == T M * d T * ≡= 482 Chapter 7: Quasi-static Control Algorithms then the maximum positive and negative moments are equal. The cross section can now be proportioned for , which is 1/2 the design moment corresponding to the case of no prestress. This reduction is the optimal value; taking will increase the initial moment beyond and result in the cross-section being controlled by the initial prestress. The limitation of this approach is the need to apply the total prestress loading prior to the application of the actual loading. Since the tension is not adjusted while the loading is being applied, the scheme can be viewed as a form of passive control. The best result that can be obtained with prestressing for this example is equal design moment values for the unloaded and loaded states. Active prestressing Suppose the cable tension can be adjusted at any time. The equivalent uniform upward loading due to the cable action can now be considered to be an active loading. Deforming as the equivalent active loading and noting eqn (7.1), the loading is related to the “active” tension force, T(t ), by (7.6) The time history of T(t ) can be established using simple static equilibrium relations. Given the spatial distribution and time history of the loading, T(t )is determined such that the maximum moment at any time is less than the design moment for the cross-section. When the applied loading is uniformly distributed, the moment distributions are similar in form. The expression for the net positive moment has the form (7.7) Enforcing the constraint on the maximum moment, which occurs at mid-span, (7.8) results in the following control algorithm, (7.9) M * TT * > M * w a w a 8d L 2   Tt()= M * Mxt,() wt() w a t()–() 1 2 Lx x 2 –()= wt() w a t()–() L 2 8 M * ≤ 1. w a t() 0 for wt() 8 L 2 M* w s ≡≤= 2. w a t() wt() w s for wt() w s >–= 7.2 Active Prestressing of a Simple Supported Beam 483 No action needs to be taken until reaches , since the maximum moment is less than . Above this load level, the active loading counteracts the difference between and . With active prestressing, the constraint imposed on the initial prestressing is eliminated. Theoretically, the total applied load can be carried by the active system for this example. This result is due to the fact that the moment distributions for the actual and active loadings have the same form. When these distributions are different, the effectiveness of active prestressing depends on the difference between the distributions. The following discussion addresses this point. Consider the case where the loading is a concentrated force that can act at any point on the span, and the prestressing action is provided by a single cable. The moment diagrams for the individual loadings are shown in Fig 7.2. When these distributions are combined, there is a local positive maximum at point B, the point of application of the load, and possibly also at another point, say C. Whether the second local negative maximum occurs depends on the level of prestressing. As is increased, the positive moment at B decreases, and the negative moment at C increases. For a given position of the loading, the control problem involves establishing whether can be selected such that the magnitudes of both local moment maxima are less than the prescribed target design value, , indicated in Fig 7.2. With passive prestressing, the optimal prestressing scheme produced a 50% reduction in the required design moment, i.e., it resulted in =0.5(PL/4). Whether an additional reduction can be achieved with active prestressing remains to be determined. ww s M * wt() w s w a w a M * M * 484 Chapter 7: Quasi-static Control Algorithms Fig. 7.2: Active prestressing scheme for a concentrated load The net moment is given by Region A-B (7.10) (+) P P ab L aLa– w a L 2 8 – (-) M+ M+ parabola low prestress high prestress M+ M net M * + M * – AB C D x c x Mxt,() Px L a–() L w a Lx x 2 – 2   –= 7.2 Active Prestressing of a Simple Supported Beam 485 Region B-C-D (7.11) Specializing eqn (7.10) for leads to (7.12) The location of the second maxima is established by differentiating eqn (7.11) with respect to x and setting the resulting expression equal to 0. This operation yields (7.13) The value for M at has the following form: (7.14) When , the maximum negative moment occurs outside the span, and is taken as 0. The control algorithm is established by requiring (7.15) for all . Starting at , no action is required as increases until the moment due to the force P is equal to . The limiting value is denoted as , and determined with (7.16) When is greater than , the maximum positive moment, , is set equal to , (7.17) Solving eqn (7.17) for leads to (7.18) The last step involves checking whether for this value of , the maximum negative moment, , exceeds . Mxt,() Pa L x–() L w a Lx x 2 – 2   –= xa= M + Mat,()≡ aL a–() P L 1 2 w a –   = x c L 2 Pa Lw a += xx c = M - Mx c t,()≡ Lx c –() Pa L w a x c 2 – for x c L<= x c L> M – M + M * ≤ M - M * ≤ aa0= a M * a s P L a s La s –()M * = aa s M + M * aL a–() P L 1 2 w a –   M * = w a w a 2 P L M * aL a–() –   = w a M - M * – 486 Chapter 7: Quasi-static Control Algorithms It is convenient to work with dimensionless variables for x and M. (7.19) where (7.20) The factor, f, can be interpreted as the “reduction” due to prestressing. No prestress corresponds to f=1; passive prestress for this loading and prestressing scheme corresponds to f=0.5. Using this notation, is given by (7.21) The dimensionless form of eqn (7.18) is written as (7.22) Lastly, the dimensionless peak negative moment is expressed as (7.23) where (7.24) The peak negative moment is a function of the position coordinate, , and the design moment reduction factor, f. Since must be less than 1 for all values of between 0 and 0.5, the magnitude of f is constrained to be above a limiting value, f min . Figure 7.3 shows plots of vs. for a range of values f. For this case, the limiting value of f is equal to 0.26. Therefore, with active prestressing, the design moment can be reduced to 50% of the corresponding value for passive prestressing. The influence line for the cable tension required for the “optimal” active prestressing algorithm is plotted in Fig 7.4. Also plotted is the required tension corresponding to f=0.5, the optimal passive value. As expected, lowering the cross-sectional design moment results in an increase in the required cable tension. In order to arrive at an optimal design, the costs associated with the material (cross-section) and prestressing need to be considered. x x L = M M M * = M * f PL 4   = a s a s 1 2 11f–() 12⁄ –[]= w a L 2 M * 8d M *   T≡ 8 f 2 a 1 a–() – ga f,()== M - M * M - 1 x c –() 4a f gx c 2 –== x c 1 2 4a fg += a M - a M - a 7.2 Active Prestressing of a Simple Supported Beam 487 Fig. 7.3: Influence lines for the peak negative moment 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 M – aL⁄ f=0.2 f=0.3 f=0.4 f=0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 M – aL⁄ 0.27 0.26 0.25 488 Chapter 7: Quasi-static Control Algorithms Fig. 7.4: Influence lines for the optimal cable tension Active prestressing with concentrated forces In this section, the use of concentrated forces to generate prestress moment fields is examined. The design loading is assumed to be a single concentrated force that can act anywhere along the span. Example 7.1: A single force actuator Consider the structure shown in Fig (1). The active prestressing is provided by a single force, F, acting at mid-span. This loading produces 2 local moment maxima, M 1 and M 2 . The moment at mid-span may be negative for certain combinations of and F, and therefore it is necessary to check both M 1 and M 2 when selecting a value for the control force. Adopting the strategy discussed earlier, the control algorithm is based on the following requirements (1) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 1 2 3 4 5 6 aL⁄ 8d PL 4⁄   T f=0.26 f=0.5 a M 1 M * ≤ M 2 M * ≤ [...]... 4 5 6 7 8 9 -1 -2 Figure 1: Moment due to design loading 10 11 504 Chapter 7: Quasi-static Control Algorithms * Mc 3 upper bound M* = 1 2 1 1 2 3 4 5 6 7 8 9 10 11 -1 -2 lower bound -3 -4 Figure 2: Upper and lower bounds on the prestress moment field 7. 2 Active Prestressing of a Simple Supported Beam 505 1 1 2 3 4 5 6 7 8 10 9 11 Ψ1 -1 1 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 2 3 4 5 6 7 8 9... 11 Ψ2 -1 1 Ψ3 -1 1 1 -1 (a) r=4 1 1 5 3 m5 m1 2 3 m3 m2 m1 m6 4 m4 7 (b) r=4 m3 m2 m 7 5 7 6 (c) r =7 11 9 m4 8 9 10 11 Figure 3: Prestress moment fields for 4 and 7 actuators Ψ4 506 Chapter 7: Quasi-static Control Algorithms 1 0.5 0 3 −0.5 2 −1 Target distribution 1 4 actuators (1) −1.5 4 actuators (2) o o 7 actuators (3) −2 −2.5 1 2 3 4 5 6 7 8 9 10 11 Figure 4: Prestress moment fields 7. 3 Quasi-static... “target” distribution for Mc ( x ) 3 Md 1 - 0 M* -1 -2 (a) upper bound Mc 1 - 0 M* -1 -2 a b c e d (b) lower bound Permissible region -4 1 * Mc - 0 M * -1 -2 a b c d e (c) Optimum solution Fig 7. 5: Limiting prestress moment fields Figure (7. 5) illustrates the process of establishing the desired distribution for the control moment The curves shown in Fig (7. 5b) correspond to M c = – M d ± M * ; allowable.. .7. 2 Active Prestressing of a Simple Supported Beam 489 for all a where Pa ( L – a ) Fa M 1 = - – L 2 (2) Pa FL M 2 = - – -2 4 (3) and M * is the design moment for the cross-section a P L –a L/2 F a(L – a) P L (+) M+ (-) M+ FL – -4 M1 M2 Figure 1 M+ 490 Chapter 7: Quasi-static Control Algorithms Shifting to dimensionless variables, a a = -L PL M * = f -4 Mi M i... the errors at locations x = 0.5 and x = 1.0 (10) 512 Chapter 7: Quasi-static Control Algorithms x = 0.5 x = 0.5 x = 1.0 x = 1.0 Case f g f g 1 Continuous least square F 1 0.0035 0.0 373 -0 .0053 -0 .0808 2 Discrete least square F 1 and 2 obs points 0.0048 0.0 570 -0 .0015 -0 .0 178 3 Discrete least square F 2 and 2 obs points -0 .0049 0.1292 0.0020 -0 .0521 4 Discrete least square F 1, F 2 and 2 obs points 0...  384  (11) 5  -  - L 3  48  Φ = -   DB  1  - 3 Substituting in eqns (7. 55) and (7. 56) results in (12) 7. 3 Quasi-static Displacement Control of Beams 513 L6 a = a 11 = - ( 0.1220 ) 2 DB (13) wL 7 L4 b = b 1 = ( – 0.0463 ) + - ( 0.3594 ) 2 αD B DB (14) DB F 1 = - ( 2.94 67 ) – wL ( 0. 379 5 ) αL 2 (15) and This solution approaches the continuous solution, eqn (9), as the... a 11 F 1 = b 1 (6) L7 a 11 = - ( 0.0262 ) 2 DB (7) L5 wL 8 b 1 = – ( 0.0102 ) + - ( 0. 072 2 ) 2 αD B DB (8) where Solving for F 1 leads to: DB F 1 = - ( 2 .75 76 ) – wL ( 0.3911 ) L2α (9) The error function associated with the solution is: wL 4 L e ( x ) = f ( x ) + g ( x ) DB α wL 4 = { 0.0545x 2 – 0.1015x 3 + 0.417x 4 } + DB L + { 0. 378 9x 2 – 0.4596x 3 } α Table 1 contains... point 0.0043 0. 070 3 0 0 Table 1 Comparison of displacement errors To illustrate the discrete formulation, observation points at x = 0.5 and x = 1.0 are selected For this case, n = 2 and r = 1 Using eqns (1), (3), and (5), the terms defining e are: 17   - Uo – U* 1  L   =   –  4  DB  1  α   - 1   8  wL 4  384  (11) 5  -  - L 3  48  Φ = -   DB  1  - 3 Substituting... moment By definition, M ( x ) = Md ( x ) + Mc ( x ) (7. 25) The design objective is to limit the magnitude of M ( x ) to be less than M * , the design moment for the cross-section 500 Chapter 7: Quasi-static Control Algorithms Md + Mc ≤ M * 0≤x≤L (7. 26) Equation (7. 26) imposes a constraint on the magnitude of M c – Md – M * ≤ Mc ≤ M * – Md 0≤x≤L (7. 27) These limits establish the lower and upper bounds... following form: L3 wL 4 L e ( x = 1 ) = – + 3D B 8D B α 5 L 3  F1  - -   48 D B  F  2 (20) e = U o – U c + ΦF Expanding a and b , 1 L3 2   a = 5 1  3D  B 16 1 L3 5   - =  3D  5 16 B 16 5 16 5 5 -  - - 16  16 1 L3 wL 4 L - b =   5  – +    3D  -  8D B B α 16 (21) (22) shows that a is singular; the second row is 5/16 times . acts at mid-span, and the resultant positive moment at mid-span is given by (7. 2) The initial mid-span moment is negative and equal to (7. 3) If the prestress level is selected such that (7. 4) which. Chapter 7: Quasi-static Control Algorithms Fig. 7. 2: Active prestressing scheme for a concentrated load The net moment is given by Region A-B (7. 10) (+) P P ab L aLa– w a L 2 8 – (-) M+ M+ parabola low. prestress M+ M net M * + M * – AB C D x c x Mxt,() Px L a–() L w a Lx x 2 – 2   –= 7. 2 Active Prestressing of a Simple Supported Beam 485 Region B-C-D (7. 11) Specializing eqn (7. 10) for leads to (7. 12) The location of the second maxima