Mixed control of vibrational systems

Nakić, Ivica; Tomljanović, Zoran; Truhar, Ninoslav

doi:10.1002/zamm.201800328

Cited by 3 publications

(3 citation statements)

References 53 publications

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“…In the input-output dynamical systems settings, the optimization criteria are usually based on system norms such as H 2 or H ∞ system norm (see, e.g., [15,21]). Moreover, mixed performance measure that was introduced in [51]. This specific choice of optimization criteria strongly depends on the application at hand.…”

Section: A Motivating Example: Damping Optimizationmentioning

confidence: 99%

“…The main reason lies in the fact that we need to optimize damping parameters (viscosities) together with damping positions that is a demanding combinatorial optimization problem. Optimization of damping parameters for the case of criteria based on system norms was studied, e.g., in [15,21,51,61]. In this section we will present algorithms that allow parameter optimization in structured systems and in the section with numerical experiments we will apply these algorithms for efficient optimization of damping parameters.…”

Section: Parameter Optimization For Systems With Low-rank Parameteriz...mentioning

confidence: 99%

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Sampling-free parametric model reduction of systems with structured parameter variation

Beattie

Gugercin

Tomljanović

2018

Preprint

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We consider a parametric linear time invariant dynamical systems represented in state-space form as $$E \dot x(t) = A(p) x(t) + Bu(t), \\ y(t) = Cx(t),$$ where $E, A(p) \in \mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m} $ and $C\in \mathbb{R}^{l\times n}$. Here $x(t)\in \mathbb{R}^{n} $ denotes the state variable, while $u(t)\in \mathbb{R}^{m}$ and $y(t)\in \mathbb{R}^{l}$ represent, respectively, the inputs and outputs of the system. We assume that $A(p)$ depends on $k\ll n$ parameters $p=(p_1, p_2, \ldots, p_k)$ such that we may write $$A(p)=A_0+U\,\diag (p_1, p_2, \ldots, p_k)V^T,$$ where $U, V \in \mathbb{R}^{n\times k}$ are given fixed matrices.We propose an approach for approximating the full-order transfer function $H(s;p)=C(s E -A(p))^{-1}B$ with a reduced-order model that retains the structure of parametric dependence and (typically) offers uniformly high fidelity across the full parameter range. Remarkably, the proposed reduction process removes the need for parameter sampling and thus does not depend on identifying particular parameter values of interest. Our approach is based on the classic Sherman-Morrison-Woodbury formula and allows us to construct a parameterized reduced order model from transfer functions of four subsystems that do not depend on parameters, allowing one to apply well-established model reduction techniques for non-parametric systems. The overall process is well suited for computationally efficient parameter optimization and the study of important system properties. One of the main applications of our approach is for damping optimization: we consider a vibrational system described by $$ \begin{equation}\label{MDK} \begin{array}{rl} M\ddot q(t)+(C_{int} + C_{ext})\dot q(t)+Kq(t)&=E w(t),\\ z(t)&=Hq(t), \end{array} \end{equation} $$ where the mass matrix, $M$, and stiffness matrix, $K$, are real, symmetric positive-definite matrices of order $n$. Here, $q(t)$ is a vector of displacements and rotations, while $ w(t) $ and $z(t) $ represent, respectively, the inputs (typically viewed as potentially disruptive) and outputs of the system. Damping in the structure is modeled as viscous damping determined by $C_{int} + C_{ext}$ where $C_{int}$ and $C_{ext}$ represent contributions from internal and external damping, respectively. Information regarding damper geometry and positioning as well as the corresponding damping viscosities are encoded in $C_{ext}= U\diag{(p_1, p_2, \ldots, p_k)} U^T$ where $U \in \mathbb{R}^{n\times k}$ determines the placement and geometry of the external dampers. The main problem is to determine the best damping matrix that is able to minimize influence of the disturbances, $w$, on the output of the system $z$. We use a minimization criteria based on the $\mathcal{H}_2$ system norm. In realistic settings, damping optimization is a very demanding problem. We find that the parametric model reduction approach described here offers a new tool with significant advantages for the efficient optimization of damping in such problems.

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Section: A Motivating Example: Damping Optimizationmentioning

confidence: 99%

Section: Parameter Optimization For Systems With Low-rank Parameteriz...mentioning

confidence: 99%

Sampling-free parametric model reduction of systems with structured parameter variation

Beattie

Gugercin

Tomljanović

2018

Preprint

Self Cite

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“…More details can be found, e.g., in [3,5,8,9,[13][14][15]. Moreover, this criterion can be extended to the case where we consider Multiple-Input Multiple-Output systems that appear in the control theory in many applications, e.g., in paper [16] authors consider mixed control performance measure that includes also the total average energy into account.…”

Section: Introductionmentioning

confidence: 99%

Fast Computation of Optimal Damping Parameters for Linear Vibrational Systems

2022

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We propose a fast algorithm for computing optimal viscosities of dampers of a linear vibrational system. We are using a standard approach where the vibrational system is first modeled using the second-order structure. This structure yields a quadratic eigenvalue problem which is then linearized. Optimal viscosities are those for which the trace of the solution of the Lyapunov equation with the linearized matrix is minimal. Here, the free term of the Lyapunov equation is a low-rank matrix that depends on the eigenfrequencies that need to be damped. The optimization process in the standard approach requires O(n3) floating-point operations. In our approach, we transform the linearized matrix into an eigenvalue problem of a diagonal-plus-low-rank matrix whose eigenvectors have a Cauchy-like structure. Our algorithm is based on a new fast eigensolver for complex symmetric diagonal-plus-rank-one matrices and fast multiplication of linked Cauchy-like matrices, yielding computation of optimal viscosities for each choice of external dampers in O(kn2) operations, k being the number of dampers. The accuracy of our algorithm is compatible with the accuracy of the standard approach.

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Control of parameter dependent systems

Lazar

Lohéac

2022

Numerical Control: Part A

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Mixed control of vibrational systems

Cited by 3 publications

References 53 publications

Sampling-free parametric model reduction of systems with structured parameter variation

Sampling-free parametric model reduction of systems with structured parameter variation

Fast Computation of Optimal Damping Parameters for Linear Vibrational Systems

Control of parameter dependent systems

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