Structure Preserving Uncertainty Modelling and Robustness Analysis for Spatially Distributed Dissipative Dynamical Systems
Abstract
:1. Introduction
When is discretization of spatially distributed systems good enough for control?
When are reduced order models of spatially discretized and distributed systems good enough for control?
How to model uncertainties for spatially discretized and reduced order spatially distributed dissipative dynamic systems that are suitable for practical robust control?
 (i)
 Systematic modelling of the uncertainty and model order reduction (MOR) at the level of a subsystem gives both modelling freedom and the ability for obtaining less conservative uncertainties on the level of a subsystem.
 (ii)
 For a special class of interconnected dissipative dynamical systems—by employing a newly discovered structurepreserving subsystem partitioning technique—uncertainty at the subsystem level can be reduced, while at the same time preserving the structure and keeping the order of interconnected system low.
2. Materials and Methods
2.1. Preliminaries and Notation
2.2. Creating State Space Subsystems from Spatially Discretized Submodels
2.3. System Interconnection
2.4. Structure Preserving Balanced Truncation Method
Algorithm 1:Generalized square root balanced truncation method. 
For the subsystem defined with the Equation (3) with the transfer function calculated as ${G}_{j}\left(s\right)={C}_{j}{\left(sI{A}_{j}\right)}^{1}{B}_{j}$ compute the reduced order system.

2.5. Additive Uncertainty Model
2.6. Robustness Analysis Using Integral Quadratic Constraints
 1.
 for all $\tau \in [0,1]$ the interconnection defined with Equation (12), with omitted performance channels w and z, is well posed for $\mathsf{\Delta}$ replaced by $\tau \Delta $;
 2.
 for all $\tau \in [0,1]$ and some $\mathsf{\Pi}={\mathsf{\Pi}}^{*}\in {\mathcal{RL}}_{\infty}$, the IQC defined with Equation (14) is satisfied for $\mathsf{\Delta}$ replaced by $\tau \Delta $;
 3.
 the following FDI is satisfied:$${\left(\begin{array}{c}\mathsf{\Gamma}\\ I\end{array}\right)}^{*}\mathsf{\Pi}\left(\begin{array}{c}\mathsf{\Gamma}\\ I\end{array}\right)\prec 0\phantom{\rule{1.em}{0ex}}\forall \omega \in \mathbb{R}\cup \left\{\infty \right\}.$$
 1.
 ${\mathsf{\Gamma}}^{*}P\mathsf{\Gamma}\prec 0$.
 2.
 There exists a matrix $X\in \mathbb{S}$ such that$${\left(\begin{array}{cc}I& 0\\ {A}_{\mathsf{\Gamma}}& {B}_{\mathsf{\Gamma}}\\ {C}_{\mathsf{\Gamma}}& {D}_{\mathsf{\Gamma}}\end{array}\right)}^{T}\left(\begin{array}{ccc}0& X& 0\\ X& 0& 0\\ 0& 0& P\end{array}\right)\left(\begin{array}{cc}I& 0\\ {A}_{\mathsf{\Gamma}}& {B}_{\mathsf{\Gamma}}\\ {C}_{\mathsf{\Gamma}}& {D}_{\mathsf{\Gamma}}\end{array}\right)\prec 0.$$
 nonstrict inequalities, if, in addition, the pair $({A}_{\mathsf{\Gamma}},{B}_{\mathsf{\Gamma}})$ is controllable,
 equalities, if, in addition, ${A}_{\mathsf{\Gamma}}$ is Hurwitz and the pair $({A}_{\mathsf{\Gamma}},{B}_{\mathsf{\Gamma}})$ is controllable.
 1.
 for all $\Delta \in \mathsf{\Delta}$ the interconnection defined by Equation (12) is well posed;
 2.
 for all $\Delta \in \mathsf{\Delta}$ and for all $P\in \mathit{P}$ the IQC defined by Equation (14) is satisfied with $\mathsf{\Pi}={\mathsf{\Psi}}^{*}P\mathsf{\Psi}.$ Then the interconnection defined with the Equation (12) is robustly stable and robust performance on the channel $w\to z$ is guaranteed, if there exist $X\in \mathbb{S}$, $P\in \mathit{P}$ and ${P}_{p}\in {\mathit{P}}_{p}$ such that$${\left(\begin{array}{ccc}I& 0& 0\\ {A}_{R}& {B}_{R1}& {B}_{R2}\\ {C}_{R1}& {D}_{R11}& {D}_{R12}\\ {C}_{R2}& {D}_{R21}& {D}_{R22}\end{array}\right)}^{T}\left(\begin{array}{cccc}0& X& 0& 0\\ X& 0& 0& 0\\ 0& 0& P& 0\\ 0& 0& 0& {P}_{p}\end{array}\right)\left(\begin{array}{ccc}I& 0& 0\\ {A}_{R}& {B}_{R1}& {B}_{R2}\\ {C}_{R1}& {D}_{R11}& {D}_{R12}\\ {C}_{R2}& {D}_{R21}& {D}_{R22}\end{array}\right)\left(\begin{array}{cc}I& 0\\ {A}_{\mathsf{\Gamma}}& {B}_{\mathsf{\Gamma}}\\ {C}_{\mathsf{\Gamma}}& {D}_{\mathsf{\Gamma}}\end{array}\right)\prec 0.$$
3. Results
3.1. Design Procedure
3.2. Numerical Example: Series of Simply Supported Euler Beams Mutually Interconnected by Springs and Dampers
4. Discussion
4.1. On the Choice of Weight Design
4.2. Defining the Unique Paths of Energy Transfer throughout the System
4.3. Replacing the Surroundings of a Subsystem with InputOutput Transfer Functions
On the Calculation of the IOTFs and its Practical Applicability
4.4. Refining the Additive Uncertainty Model
5. Conclusions and Future Work
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
BTM  balanced truncation method 
FEM  finite element method 
IOTF(s)  inputoutput transfer function(s) 
IQC(s)  integral quadratic constraint(s) 
LMI(s)  linear matrix inequality(ies) 
LFT  linear fractional transformation 
LTI  linear timeinvariant 
MOR  model order reduction 
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Case  l  d  $\mathit{\rho}$  E  ${\mathit{c}}_{1}$  ${\mathit{k}}_{1}$  ${\mathit{c}}_{2}$  ${\mathit{k}}_{2}$  $\mathit{\zeta}$ 

#  $\mathbf{m}$  ${\mathbf{m}}^{\mathbf{2}}$  $\mathbf{kg}/{\mathbf{m}}^{\mathbf{3}}$  $\mathbf{GPa}$  $\mathbf{N}\xb7\mathbf{s}/\mathbf{m}$  $\mathbf{N}\xb7\mathbf{s}/\mathbf{m}$  $\mathbf{N}/\mathbf{m}$  $\mathbf{N}/\mathbf{m}$   
1  2  $0.018$  7800  210 × ${10}^{9}$  ${10}^{4}$  0  ${10}^{6}$  0  $0.08$ 
2  2  $0.01$  7800  $210\times {10}^{9}$  0  $7\times {10}^{0}$  0  $3\times {10}^{1}$  $0.08$ 
3  1  $0.01$  7800  $210\times {10}^{9}$  ${10}^{2}$  $2\times {10}^{2}$  ${10}^{1}$  ${10}^{1}$  $0.05$ 
Case # 1  Case # 2  Case # 3  

discretization number ${n}_{j}$ per beam ^{2}  ${[6,6,5,4,4,3,3,3,2,1]}_{{n}_{b}=10}$  ${[10,9,8,7,6,4,3]}_{{n}_{b}=7}$  ${[10,10,10,10,10]}_{{n}_{b}=5}$^{1} 
orders of reduced order models ${m}_{j}$ per beam  $[4,3,3,2,2,2,1,1,1,1]$  $[6,3,3,2,2,2,2]$  $[6,3,3,2,2]$ 
orders ${o}_{j}$ of initial weights ${W}_{j,1}$ per beam  $[3,3,2,2,2,2,1,1,1,1]$  $[4,3,2,2,2,2,2]$  $[4,3,2,2,2]$ 
orders ${o}_{j,e}$ of refined weights ${W}_{j,1e}$ per beam  $[3,3,2,2,2,2,2,2,2,1]$  $[3,3,2,2,2,1,1]$  $[4,3,2,2,2]$ 
number of: inputs × outputs, states of [reference] and (reduced order) system, and decision variables to the IQC COP ^{3}  $22\times 22$$\left[433\right]$ RO: $\left(92\right)$${4280}_{v=0}$ RW: $\left(92\right)$${4280}_{v=0}$ SW: $\left(32\right)$${530}_{v=0}$  $16\times 16$$\left[387\right]$ RO: $\left(88\right)$${6791}_{v=1}$ RW: $\left(88\right)$${6791}_{v=1}$ SW: $\left(36\right)$${668}_{v=0}$  $12\times 24$$\left[310\right]$ RO: (120), n/a RW: (120), n/a SW: $\left(48\right)$${5896}_{v=2}$ 
induced ${\mathcal{L}}_{2}$gains of the [nominal system] ^{4}, the best achievable $\gamma $ (worst case gain using $\mu $tools) for  RO: $\left[0.0643\right]$ $0.06462$ $\left(0.0657\right)$ RW: $\left[0.0643\right]$ $0.06593$ $\left(0.0657\right)$ SW: $\left[0.0643\right]$ $0.06594$ $\left(0.0657\right)$  RO: $\left[0.0670\right]$ $0.06969$ $\left(0.0671\right)$ RW: $\left[0.0670\right]$ $0.06695$ $\left(0.0670\right)$ SW:$\left[0.0670\right]$ $0.06696$ $(0.0670$)  RO: $\left[4.6115\right]$ n/a $\left(4.6324\right)$ RW: $\left[4.6115\right]$ n/a $\left(4.6124\right)$ SW: $\left[4.6115\right]$ $4.622$ $\left(4.6138\right)$ 
obtained [robust stability margins using $\mu $tools] ^{5} and ($\nu $gaps) ^{6} for  RO: $\left[5.5192\right]$ $0.0118$ RW: $\left[992.4105\right]$ $0.0118$ SW: $\left[992.4107\right]$ $0.0118$  RO: $\left[3.7588\right]$ $3.1686\times {10}^{5}$ RW: $\left[159.7053\right]$ $3.1686\times {10}^{5}$ SW: $\left[91.2962\right]$ $3.1690\times {10}^{5}$  RO: $\left[2.7540\right]$ 0.0263 RW: $\left[348.1343\right]$ 0.0263 SW: $\left[80.8150\right]$ 0.0265 
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Dogančić, B.; Jokić, M.; Alujević, N.; Wolf, H. Structure Preserving Uncertainty Modelling and Robustness Analysis for Spatially Distributed Dissipative Dynamical Systems. Mathematics 2022, 10, 2125. https://doi.org/10.3390/math10122125
Dogančić B, Jokić M, Alujević N, Wolf H. Structure Preserving Uncertainty Modelling and Robustness Analysis for Spatially Distributed Dissipative Dynamical Systems. Mathematics. 2022; 10(12):2125. https://doi.org/10.3390/math10122125
Chicago/Turabian StyleDogančić, Bruno, Marko Jokić, Neven Alujević, and Hinko Wolf. 2022. "Structure Preserving Uncertainty Modelling and Robustness Analysis for Spatially Distributed Dissipative Dynamical Systems" Mathematics 10, no. 12: 2125. https://doi.org/10.3390/math10122125