VIBRATION CONTROL OF CIVIL ENGINEERING STRUCTURES VIA LINEAR PROGRAMMING

4 th World Conference on Structural Control and Monitoring 4WCSCM-65 VIBRATION CONTROL OF CIVIL ENGINEERING STRUCTURES VIA LINEAR PROGRAMMING P. Rentzos, G.D. Halikias and K.S. Virdi School of Engineering and Mathematical Science, City University, London ECV HB, UK p.rentzos@city.ac.uk, g.halikias@city.ac.uk, k.s.virdi@city.ac.uk Abstract This paper presents a novel active-control design approach which minimises the peak response of regulated signals rather than, e.g., r.m.s or energy levels optimised by traditional control techniques. This objective is more relevant for active control of civil-engineering structures, as failure occurs after a maximum displacement is exceeded in a structural member, while control constraints typically arise from hard saturation limits on the actuator signal and its rate. The design method is formulated in discrete-time and involves the parametrisation of all finite settling-time stabilizing controllers. This leads to a linear programming optimisation framework, in which the peak response of the structure is directly minimised, subject to linear constraints on the actuator s peak level signal and its rate. The design method is illustrated via a simulation study based on a simple model corresponding to a benchmark design problem. The simulation results compare favourably to those obtained via LQG active control. Finally, some practical implementation issues related to the method are discussed. Introduction Many modern control design methods are formulated as optimisation problems involving the minimisation of a norm, such as the H 2 or H norm, of the closed-loop transfer function between an input disturbance signal and the regulated output. For example, the H 2 norm measures the expected power of the regulated signal (mean-squared value). Normally, the input disturbance signal in this case is assumed to be a random white-noise process. Weighting factors or filters can be employed to emphasise specific frequency ranges of the input or output spectrum. In this paper a novel approach is presented in minimizing the peak value of the regulated signal, subject to peak magnitude and rate constraints on the control signal. The method is developed in discrete time, using a finite-settling time (dead-beat) parametrisation, leading to a linear-programming optimisation framework. The method is particularly relevant to active vibration control of civil engineering structures: Structural members fail after a maximum displacement is exceeded, and thus direct optimisation of peak output levels is more significant than, say, r.m.s. or output energy levels. In addition, control constraints for systems of this type normally arise in the form of hard saturation limits on actuator signals and their rates. Again, using the proposed method such constraints can be directly addressed. In contrast, in the LQG or H design framework the designer can only penalise control signal energy (possibly frequency weighted). Minimisation of peak responses in the context of active vibration control has been investigated by various researchers, e.g. 6, which uses an adaptive bang-bang control methodology. The proposed method is more straightforward as it relies on a fixed parameter-controller which does not require on-line tuning. 2 The design algorithm The design algorithm is described in a step-by-step procedure. A simple benchmark design problem from the area of active vibration control is also presented alongside the algorithm for illustration purposes. The structure model chosen for the example employs active tendon control, since this is reported in the literature to achieve the best results (disregarding cost considerations). A model structure described in 9 was proposed as a benchmark problem and has been investigated Rentzos, Halikias and Virdi

4 th World Conference on Structural Control and Monitoring 4WCSCM-65 Table : Stuctural parameters Floor m i (Kg) c i (Ns/m) k i (N/m) Base (i = ) 5 6 First (i = ).72.78 26 Actuator k f = 2N/A k e = 2V s/m R =.5Ω by a number of researchers. The model represents a simple and regular 3-storey structure. A schematic of the structure is shown in Figure below and its parameters summarised in Table. For simplicity only the ground and first floors have been considered in this example. The tendons are connected between the ground and first floor and produce a pair of equal and opposite forces. The structure is a scaled-down version of a real building with small masses and dimensions, suitable for experimental work. A high value is assumed for the base stiffness to account for the interaction between the base of the building and the surrounding ground. The main objective of the controller is to minimise first-floor acceleration when subjected to an impulsive force at the base. The structure is idealised as a mass-spring-damper system shown in figure. In this diagram, u s is the actuator force and ν is the external-disturbance acceleration signal (representing an earthquake) assumed to act on its base. The main design objective is to minimise the peak value of the the first regulated signal, chosen to represent first-floor acceleration. This is equivalent to minimising the l norm of the impulse response of the system, corresponding to the transfer function between the external disturbance and first-floor acceleration. Constraints on the amplitude and rate on the actuation signal will be subsequently imposed. The design algorithm follows the following steps:. Define the generalised plant: The two regulated signals are chosen to be first-floor acceleration and the control signal u representing actuator input voltage. It is required that the controller stabilises the system and min max ẍ, subject to: u(t) u max for all t () K S t In addition, to avoid highly discontinuous or high-rate signals we may impose constraints on the derivative of the control, i.e., u(t) u max for all t (2) Choosing as state-variables the displacements and velocities x, x, ẋ and ẋ, a state-space description of the model is given as ẋ = Ax + B ν + B 2 u where ν denotes the disturbance input and u is the input actuator voltage. The state-space matrices defining the model are given as: A = k +k k c m m m + k f k e m R c m + k f k e m R k m k m c m + k f k e m R c m + k f k e m R, B =, B 2 = k f m R k f m R Choosing as the only measurement the first-floor acceleration, defines the output equation of the system as y = C 2 x + D 22 u, where C 2 = k +k k m m c +c m + k f k e c m R m + k f k e m and, D R 22 = k f (4) m R Rentzos, Halikias and Virdi 2 (3)

4 th World Conference on Structural Control and Monitoring 4WCSCM-65 m x z z 2 P ν k u s c y u k c m x ν K Figure : One-storey structure Figure 2: Generalised plant Next we define the generalised plant. Choosing the vector of regulated signals as z = (ẍ u) where ẍ is the first-floor acceleration and u is the actuator voltage, the generalised plant (see figure 2) has a state-space description: ẋ = Ax + B ν + B 2 u (5) z z 2 y = C 2 C 2 x + ν + D 22 D 22 u (6) Note that there is no direct feed-through term from the disturbance ν to z or y. Rows and 2 of the output matrix define the two regulated outputs, in this case the first-floor acceleration and the control input effort u. The last row defines the measured output, in this case also first-floor acceleration. 2. System discretisation: The solution to the optimisation problem will be obtained in discretetime. Thus we first need to discretise the system using an appropriate sampling interval. The zero-order hold discretisation was employed using the standard procedure for transforming between continuous and discrete-time state-space models 4. The sampling period was chosen as T =.s. The corresponding Nyquist frequency f N = f s /2 = 5 Hz is significantly higher than the frequencies of all system modes. We will still use the same notation for the discrete-time state-space realisation of the generalised plant, by appropriately redefining the matrices A, B i, C i and D ij (i, j {, 2}) given above. 3. Youla parametrisation of all stabilising controllers: All stabilizing controllers and the corresponding closed-loop systems can be defined in terms of two matrices F and H (stabilizing state-feedback and output injection matrices, respectively). Matrices F and H can be any two matrices such that A B 2 F and A HC 2 are asymptotically stable (all eigenvalues inside unit circle). The parametrisation proceeds by first expressing the discrete plant G(z) as the ratio of two stable, relatively prime transfer functions. Note that the procedure is identical in the continuous and discrete domains, with the exception that stability needs to be defined appropriately in each domain. In addition, note that we have complete freedom in the choice of state-feedback and output injection matrices, as long as A B 2 F and A HC 2 are asymptotically stable; here F and H will be chosen so that all eigenvalues of A B 2 F and A HC 2 are placed at the origin; this is always possible under appropriate controllability and observability assumptions, which are satisfied in this case. The set of stabilising controllers is parametrised in bilinear (linear-fractional) form, while the set Rentzos, Halikias and Virdi 3

4 th World Conference on Structural Control and Monitoring 4WCSCM-65 of corresponding closed-loop systems is given in linear (more precisely affine) form, i.e. T (z ) = T (z ) T 2 (z )Q(z )T 3 (z ) (7) where Q(z ) is a free stable parameter. Concrete state-space realisations of T i (z ) can be found in 3. 4. Formulation of optimisation problem in terms of linear constraints: First partition the closedloop equations as: y(z ) t u(z = (z ) t ) t 2 2 (z ) (z ) t 2 q(z )t 2 (z ) 3 (z ) (8) where y(z ) and u(z ) are the regulated output responses to a discrete pulse δ(z ) =. Hence the transfer function between ν(z ) y(z ) can be written as: T (z ) = y(z ) ν(z ) = b(z ) + c(z )q(z ) a(z ) (9) Note that under the assumptions made earlier (all eigenvalues of A B 2 F and A HC 2 placed at the origin), we have that a(z ) =. The degree of both b(z ) and c(z ) is r, where r denotes the number of state variables (in this example r = 4). Parametrise q(z ) as a finite-impulse-response filter of degree p, i.e. q(z ) = q + q z + q 2 z 2 +... + q p z p () Also write: Then: b(z ) = b + b z + b 2 z 2 +... + b r z r () c(z ) = c + c z + c 2 z 2 +... + c r z r (2) y(z ) = y + y z + y 2 z 2 +... + y N z N (3) y(z ) = b +... + b r z r + (c +... + c r z r )(q +... + q p z p ) (4) so that degy(z ) = N = r + p. The equations can be written in matrix form as: y b c y b c c........ y r = b r + c y r+ r c r c c r c....... y N c r Note that the response is forced to be dead-beat, i.e. y r+p is the last non-zero sample of the regulated output. This is due to the restriction on q(z ) which is taken to be an FIR filter and may lead to a conservative solution unless r is taken to be large. Ideally r should be selected to make N T, a reasonable transient before the structure is fully stabilised. It is expected (but needs to be established formally) that by increasing N the deviation from optimality can be made arbitrarily small. The equations can be written compactly in matrix form as y = b + Cq where vector q contains the coefficients of the polynomial q(z ) and where C is a Toeplitz matrix. q q. q p (5) Rentzos, Halikias and Virdi 4

4 th World Conference on Structural Control and Monitoring 4WCSCM-65 5. Formulation into a linear programming problem: Since all constraints are linear, the minimisation of the peak response of the regulated signal can be formulated as a linear programming problem of the form: min c x subject to Ax b. Let δ be the maximum output of the regulated signal (first-floor acceleration) that we wish to minimise. Then: δ y k δ for all k N (6) Now y(k) = c k x+ˆb k, where c k denotes the k-th row of the C-matrix, and ˆb k = b k for k r and ˆbk = for k > r. Thus, separating the two equations we can write: c k q δ ˆb k and c k q δ ˆb k. for all k N, which can be written in matrix form as: C C δ q ˆb ˆb where represents a column vector of ones. Setting x = (δ q), the problem is now in the standard linear programming form: min δ = x subject to (7). The solution to the problem will result in the optimal peak-value of the regulated signal and the coefficients of the optimal q(z ), from which the optimal controller can be recovered via the Youla parametrisation in bilinear form. 6. Introducing constraints to the problem: In the above formulation, the peak value of the regulated output is minimised for an impulsive loading without any constraints on the size or rate of the control input. This is unrealistic and may result in highly discontinuous control signals that would be difficult to implement or could cause stability problems, especially in the presence of model uncertainty, due to the potentially excessive bandwidth of the closed-loop. The first constraint limits the magnitude of the control signal and corresponds to the actuator s saturation limits. Hence we require that: u(k) u max for all k (8) Now using Youla parametrisation and the fact that the control effort has been chosen as the second regulated output, equations 9-4 are applied to u respectively, to obtain the linear programming minimisation problem in matrix form as: T2 T 2 δ q umax + ˆt u max ˆt and solved via a standard linear programme to minimise δ. In order to make the response smoother an extra constraint needs to be added limiting the rate of actuator signal, u (slewrate constraint). Now, and we require u = u(k + ) u(k) = (ˆt (k + ) ˆt (k)) + (t 2 (k + ) t 2 (k))q This may be written as two pairs of linear inequalities: for all k, or, in matrix form as: (T 2 T 2 ) (7) (9) u(k) ( u) max for all k (2) (t 2 (k + ) t 2 (k))q ( u) max (ˆt (k + ) ˆt(k)) (2) (t 2 (k + ) t 2 (k))q ( u) max + (ˆt (k + ) ˆt(k)) (22) T 2 T 2 δ q ( u)max (t t ) ( u) max (t + t ) (23) Rentzos, Halikias and Virdi 5

4 th World Conference on Structural Control and Monitoring 4WCSCM-65.25 st floor acceleration 3 u.2 2.5..5 x m/s 2 volts.5..5 2.2.25.5.5 2 2.5 time sec 3.5.5 2 2.5 time sec Figure 3: Unconstrained LP Acceleration Figure 4: Unconstrained LP voltage Table 2: Comparison between LP and LQR methods LQR LP Constrained LP ẍ max m/s 2 3.2 5.5 u max volts 6 24 5 where T 2 and t denote the matrix T 2 and vector ˆt with the first row eliminated, while ˆT 2 and t denote the matrix T 2 and vector ˆt with the last row eliminated. The inequalities can now be augmented to the previous set of linear inequalities (eq. 7, 9) and solved in a linear programme to impose additional rate constraints on the control signal. 3 Application results and Discussion The LP design method was first applied to the structure without any control constraints with a filter length of r = 2 samples, corresponding to a deadbeat response of approximately 2 seconds. The two regulated signals (st floor acceleration and actuator voltage) are shown in Figures 3 and 4 respectively. An LQR design was also carried out for comparison purposes, as this is one of the most widely accepted design methods in the area of active vibration control. The design involves a quadratic cost-function consisting of two penalty terms, acceleration and control effort. Both weighting factors were set to, penalising equally the two terms. The design was carried out both in continuous and discrete-time (with a sampling rate of Hz), producing almost identical results. Subsequently, the LP design was again carried out, this time with control constraints on the peak control signal and its rate. The peak-magnitude control constraint was set at 5 Volts, slightly less than the peak control signal obtained from the LQR simulation (around 6 Volts) and a maximum rate constraint of 4 Volts/s. The two regulated signals resulting from the two designs (LQR and constrained LP) are shown in figures 5 and 6. The main results of all simulations are also summarised in Table 2. The unconstrained LP method yields excellent results in terms of optimising the peak signal level. The maximum acceleration is about 6 times smaller than the peak acceleration resulting from the Rentzos, Halikias and Virdi 6

4 th World Conference on Structural Control and Monitoring 4WCSCM-65 st floor acceleration u LP LQR 5 LP LQR 5 5 x m/s 2 volts 5 5..2.3.4.5.6 time sec 5..2.3.4.5.6 time sec Figure 5: Acceleration (Constrained LP and LQR) Figure 6: and LQR) Voltage (Constrained LP LQR design, the peak voltage control level increasing by a factor of.5. However, the resulting acceleration profile (Figure 3) clearly indicates that the response is unrealistic for practical implementation. The acceleration reaches its peak positive value of.2 m/s 2 extremely fast and swings to to its minimum negative value.2 m/s 2 almost ms later, requiring a huge slew-rate from the actuator. Subsequently, the acceleration fluctuates between the two extreme values for a few cycles of progressively increasing frequency before decaying to zero after about.5 seconds (.5 seconds earlier than the set deadbeat horizon) exhibiting highly-damped oscillations. Thus the method works theoretically in the sense that it succeeds to minimise peak acceleration, as indicated by the flat regions of the acceleration signal at positive and negative peaks of the same magnitude. However, the response is clearly unrealistic and thus the controller cannot be implemented in practice. By setting an acceptable limit in the rate of change of the control signal ( u max = 4 Volts/s which is about ten times less than the fast rates of the early response observed in Figure 4) the response of the system to the impulsive loading becomes acceptable. The maximum acceleration for the constrained LP design is almost 2.5 times less than the peak value obtained by LQR, while the controller peak signal (5 volts) is slightly less than the peak value obtained from the LQR design (6 Volts). This improvement is made despite the fact that the LQR controller is based on statefeedback (all four states assumed measurable), whereas the LP controller uses output feedback only (first-floor acceleration being the only measurement). 4 Conclusions The paper presents a LP-based algorithm aiming to minimise the peak value of the regulated signal, an objective which is especially relevant for the design of active vibration control of civil engineering structures. Linear constraints are introduced to limit the magnitude of the control signal and its rate, resulting in smooth control signals and low-bandwidth control schemes which can be implemented in practice. The design algorithm was developed in parallel to a simple example involving a scaled-down scalar benchmark model of a one-storey building, although extensions to the multivariable case and multiple regulated signals are straightforward. It was demonstrated via simulations that the design method is capable to reduce significantly the peak acceleration response of the model compared to the LQR design, even after the introduction of constraints on the control-signal. Other advantages of the method include the ability to formulate realistic constraints Rentzos, Halikias and Virdi 7

4 th World Conference on Structural Control and Monitoring 4WCSCM-65 involving the magnitude and rate of regulated signals (rather than rms or energy content) and to provide indirect control of the overall damping by specifying the settling-time horizon. Although the disturbance signal was assumed to be an impulse, more general disturbance models can be accommodated by introducing dynamic-weights/filters absorbed in the generalised plant. Some issues related to the design require further investigation. These include a full robustness analysis and the possibility of incorporating the method within a larger multi-objective optimisation framework (e.g. using multiple regulated signals and a mixture of linear and quadratic constraints). Another important issue is related to controller complexity. The design method tends to produce high-order controllers, in the form of a bilinear transformation of a high-order FIR filter. This can be approximated by a low-order IIR filter resulting in an overall low-order controller using a recently derived Hankel-norm model-reduction algorithm for discrete-time descriptor systems and FIR filters, 5. Alternative control design aiming to minimise the peak response of the regulated signal have recently been reported both in the area of active vibration control 6 and also in general control literature, 2, 2. Reference 6 is based on an adaptive bang-bang methodology, which clearly offers advantages in the case of uncertainty about the disturbance-signal, but is also difficult to apply in practice. A systematic general approach is l optimal control, which attempts to minimise the peak amplification gain between disturbance input and regulated output 2, 7. Interestingly, the method also results in a Linear Programming optimisation framework. Potential merits of this method relative to our own approach (which only takes impulsive loading into account) needs further investigation. C C C 2 C 2 C 2 (n + ) c 2 (n) (C 2 (n + ) c 2 (n)) δ q b b u max + b 2 u max b 2 u max + b 2 (n + ) b 2 (n) u max (b 2 (n + ) b 2 (n)) (24) References M.M. Al-Hussari, I.M. Jaimoukha and D.J.N. Limebeer, A descriptor approach for the solution of one-block distance problems, Proc. IFAC World Congress, Sydney, Australia, 993. 2 M.A. Dahleh and J.J. Diaz Bobillo, Control of uncertain systems: A linear programming approach, Prentice-Hall, Englewood-Cliffs, New Jersey, 995. 3 B.A. Francis, A course in H optimal control theory, Springer Verlag, Lecture Notes in Control and Information Sciences, New York, 987. 4 Franklin G.F., Powell J.D. and Workman M.L., Digital Control of Dynamical Systems, Addison-Wesley, Reading, Massachusetts, 99. 5 G.D. Halikias, I.M. Jaimoukha and D.A. Wilson, A numerical solution to the matrix H 2 /H optimal control problem, Int. Journal of Robust and Nonlinear Control, Vol. 7, No 7, pp. 7-726, July 987. 6 C. W. Lim, T. Y. Chung, S. J. Moon, Adaptive bang-bang control for the vibration control of structures under earthquakes, Journal of earthquake engineering and structural dynamics, pp.977-994, July 23. 7 J.S. MacDonald and J.P. Pearson, l Optimal Control of multivariable systems with output norm constraints, Automatica, Vol. 27, pp. 37-329, 99. 8 J.M. Maciejowski, Multivariable Feedback Design, Addison Wesley Publishing Company, 989. Rentzos, Halikias and Virdi 8

4 th World Conference on Structural Control and Monitoring 4WCSCM-65 9 H. Nishimura and Akihito Kojima. Seismic Isolation Control, IEEE control Systems, 99. T.T. Soong, Active structural control:theory and Practise, Longman Scientific and Technical, 99. M. Sznaier, T. Amishama, and T. Inanc, H 2 control with domain constraints: Theory and applications, IEEE Transactions on Automatic Control, Vol. 48, No 3, March 23. 2 M. Sznaier, A mixed l /H optimization approach to robust controller design, SIAM Control and Optimization, Vol. 33, No 4, pp. 86-, July 995. Rentzos, Halikias and Virdi 9