A Mathematical Framework for Structural Control Integration

In this paper, some control strategies to design decentralized controllers are developed and discussed. These strategies are based on the Inclusion Principle, a very useful mathematical framework to obtain decentralized controllers, mainly when the systems are composed by overlapped subsystems sharing common parts. A five-story building model serves as example to show the advantages provided by this approach. Numerical simulations are conducted to assess the performance of the proposed control laws with positive results. Introduction Vibration control of flexible structures, in particular tall buildings, under strong winds and/or seismic excitations has attracted considerable attention in recent years. In order to avoid the undesirable effects of vibrations produced by external stimuli, a large variety of control strategies has been developed, including passive, active, hybrid or semiactive damping methods. Passive devices, as the base isolation systems, offer well-known mechanisms to be implemented in the foundation of the buildings. Mass dampers placed on the top or in the middle of the buildings, or the use of viscoelastic dampers, are other passive control possibilities. Unfortunately, these kinds of devices are not able to react properly to different structural change of conditions [1, 2]. To overcome this difficulty, active, hybrid and semiactive control systems are frequently designed to attenuate building vibrations, achieving better performance than passive control methods [3, 4, 5, 6, 7, 8]. Tall buildings can be considered as complex systems and, in this case, they can be decomposed into disjoint subsystems. For this class of systems a set of local controllers may be independently obtained to design decentralized controllers. Some advantages in designing and using local controllers are the following: (1) lower-dimension computation is required; (2) minimization of the information exchange; (3) increment of the global robustness; and (4) reduction of the effect of perturbations and failures on communications. In order to design decentralized controllers, a mathematical framework called Inclusion Principle will be used in the paper. The inclusion principle deals with systems composed by overlapped subsystems which, by means of appropriate linear transformations, can be treated as disjoint. Then, local controllers are designed to be transformed and implemented into the initial systems to control them [9, 10, 11]. This useful approach has been applied in a large variety of complex control problems appearing in different areas, such as electric power generation, automated highway traffic management, civil structural engineering or aerospace structural engineering. As a good example of structural control integration, we consider a complex control configuration consisting in several semiactive dampers and sensors installed in different floors of the building together with an appropriate communications system and a suitable feedback control strategy. To improve the robustness of the communications, we suppose that the controllers operate using only local information supplied by neighboring sensors. Consequently, a decentralized control is required for a realistic treatment of wireless networked control systems [12, 13, 14] and an overlapping approach may be specially convenient [15, 16]. In order to present the main ideas, a five-story building model excited by a seismic disturbance has been selected. For this building, three kinds of LQR controllers are designed and compared: (1) a centralized controller, which serves as reference; (2) a semi-decentralized two-overlapping controller; and (3) a semi-decentralized multi-overlapping controller. In all the cases, the El Centro North-South 1940 seismic record has been used as excitation. The Inclusion Principle We summarize some basic definitions and results related to the Inclusion Principle in order to design overlapping controllers. A more detailed treatment can be found in [9, 10, 11]. Consider a pair of linear systems S : { ẋ(t)=Ax(t) + B u(t) y(t)=Cy x(t) S̃ : { ̇̃ x(t)= à x̃(t) + B̃ ũ(t) ỹ(t)= C̃y x̃(t) (1) where x(t)∈Rn , u(t)∈Rm , y(t)∈Rl are the state, the input, and the output of S at time t>0; x̃(t)∈Rñ , ũ(t)∈Rm̃ , ỹ(t)∈Rl̃ are the state, the input, and the output corresponding to S̃; A, B, Cy and Ã, B̃, C̃y are n×n, n×m, l×n and ñ×ñ, ñ×m̃, l̃×ñ dimensional matrices, respectively. Let us consider the following linear transformations: V : Rn−→ Rñ , R : Rm−→ Rm̃ , T : Rl −→ Rl̃ , U : Rñ−→ Rn , Q : Rm̃−→ Rm , S : Rl̃−→ Rl , (2) where V , R, T are called expansion matrices with rank(V )=n, rank(R)=m, rank(T )=l, and U , Q, S are contraction matrices obtained by computing U=(V V )−1V T , Q=(RTR)−1RT , S=(T T )−1T T , which satisfy UV =In, QR=Im, ST=Il, where In, Im, Il denote the identity matrices of indicated dimensions. Definition 1. (Inclusion Principle) A system S̃ includes the system S if there exists a quadruplet of matrices (U, V,R, S) such that, for any initial state x0 and any fixed input u(t) of S, the choice of x̃0=V x0 , ũ(t)=Ru(t) for all t>0 as initial state x̃0 and input ũ(t) for the system S̃, implies x(t;x0 , u)=Ux̃(t; x̃0 , ũ), y[x(t)]=ỹ[x̃(t)], for all t>0. An expanded system S̃ can be defined in the form Ã=V AU+M A , B̃=V BQ+N B , C̃y=TCyU+ L C , where M A , N B , L C are complementary matrices of appropriate dimensions. In order to assure that the system S and the expanded system S̃ satisfy the Inclusion Principle, the complementary matrices have to satisfy the following theorem. Theorem 2. S̃ includes the system S if and only if UM i A V=0, UM i−1 A N B R=0, SL C M i−1 A V=0 and SL C M i−1 A N B R=0 for all i=1, 2, ..., ñ. A special kind of expansion-contraction scheme, called restriction, is particularly simple and suitable for the design of overlapping controllers. Definition 3. (Restriction) Let S̃ be an expansion of the system S defined by the expanded system matrices Ã, B̃, C̃y. The system S is said to be a restriction of S̃ if and only if MAV =0, N B R=0 and L C V =0. Expansions of overlapping systems If a system S can be split into three subsystems S1, S2, S3 in such a way that no direct interaction between S1 and S3 occurs, then it admits an overlapping decomposition. From the three subsystems Si, two overlapping subsystems S (1) =[S1, S2], S (2) =[S2, S3] can be considered. More precisely, we assume that A, B and Cy present a block tridiagonal structure A=  A11 A12 ppp 0 −−− p A21 A22 A23 −−− p p −−− 0 p A32 A33  , B=  B11 B12 ppp 0 −−− p B21 B22 B23 −−− p p −−− 0 p B32 B33  , Cy = (Cy)11 (Cy)12 p p p 0 −−− p −−− (Cy)21 (Cy)22 (Cy)23 −−− p p p −−− 0 p (Cy)32 (Cy)33  , (3) where Aii, Bij, (Cy)ij, for i, j=1, 2, 3, are ni×ni, ni×mj, li×nj dimensional matrices, respectively. The partition of the state x=(x1 , x T 2 , x T 3 ) T has components of respective dimensions n1, n2, n3, satisfying n1+n2+n3=n; the partition of u=(u T 1 , u T 2 , u T 3 ) T has components of dimensions m1, m2, m3, such that m1+m2+m3=m; and y=(y T 1 , y T 2 , y T 3 ) T has components of respective dimensions l1, l2, l3, satisfying l1+l2+l3=l. Given a linear system S, a usual choice of the expansion matrices is V= [ In1 0 0 0 In2 0 0 In2 0 0 0 In3 ] , R= [ Im1 0 0 0 Im2 0 0 Im2 0 0 0 Im3 ] , T=  Il1 0 0 0 Il2 0 0 Il2 0 0 0 Il3  , (4) which provides the corresponding pseudoinverse contractions U , Q and S. A first set of expanded matrices is computed in the form Ā=V AU , B̄=V BQ, C̄y=TCyU . Then, we form an expanded system S̃ by adding adequate complementary matrices. If the complementary matrices are chosen in the form M A =  0 1 2 A12 − 12A12 0 0 1 2 A22 − 12A22 0 0 − 1 2 A22 1 2 A22 0 0 − 1 2 A32 1 2 A32 0  , NB =  0 1 2 B12 − 12B12 0 0 1 2 B22 − 12B22 0 0 − 1 2 B22 1 2 B22 0 0 − 1 2 B32 1 2 B32 0  , LC =  0 1 2 (Cy)12 − 1 2 (Cy)12 0 0 1 2 (Cy)22 − 1 2 (Cy)22 0 0 − 1 2 (Cy)22 1 2 (Cy)22 0 0 − 1 2 (Cy)32 1 2 (Cy)32 0  , (5) then, the system S is a restriction of S̃, and the expanded system S̃ presents an almost-decoupled structure. More specifically, the system matrices of S̃ are given by Ã= Ā + M A = [ Ã11 Ã12 Ã21 Ã22 ] =  A11 A12 p p 0 0 A21 A22 p p 0 A23 −−− −−− − −−− −−− A21 0 p p A22 A23 0 0 p A32 A33  (6) and similar structures have the matrices B̃ and C̃y. The state, input and output vectors of the expanded system S̃ : { ̇̃ x(t) = à x̃(t) + B̃ ũ(t) ỹ(t) = C̃y x̃(t) (7) can be written in the form x̃=(x1 , x T 2 , x T 2 , x T 3 ), ũ =(u1 , u T 2 , u T 2 , u T 3 ) and ỹ = (y 1 , y T 2 , y T 2 , y T 3 ). Using the block notation given in (6), and removing the interconnection blocks, two decoupled expanded subsystems result