Duality and H<sub>∞</sub>-Optimal Control Of Coupled ODE-PDE Systems

Shivakumar, Sachin; Das, Amritam; Weiland, S.; Peet, Matthew M.

doi:10.1109/cdc42340.2020.9303989

Cited by 12 publications

(12 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…11, letting v = D v, (v, z) is a solution to the PIE (21) with input w. Since E 1 = 0, it follows by Thm. 9 that T 1 = 0, and therefore (v, z) is a solution to the PIE (10) with T = T 0 . Finally, by Prop.…”

Section: An Lmi For L 2 -Gain Analysis Of 2d Pdesmentioning

confidence: 98%

“…Parameterizing the cone of positive PI operators by positive matrices, the authors then pose this LPI as an SDP, allowing problems of L 2 -gain analysis of 1D PDEs to be efficiently solved [10]- [12]. However, despite a PIE framework having recently been introduced for 2D PDEs [13], deriving an SDP test for bounding the L 2 -gain of general systems of the form (4) still offers several challenges.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

$L_2$-Gain Analysis of Coupled Linear 2D PDEs using Linear PI Inequalities

Declan¹,

Peet²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we present a new method for estimating the L 2 -gain of systems governed by 2nd order linear Partial Differential Equations (PDEs) in two spatial variables, using semidefinite programming. It has previously been shown that, for any such PDE, an equivalent Partial Integral Equation (PIE) can be derived. These PIEs are expressed in terms of Partial Integral (PI) operators mapping states in L 2 [Ω], and are free of the boundary and continuity constraints appearing in PDEs. In this paper, we extend the 2D PIE representation to include input and output signals in R n , deriving a bijective map between solutions of the PDE and the PIE, along with the necessary formulae to convert between the two representations. Next, using the algebraic properties of PI operators, we prove that an upper bound on the L 2 -gain of PIEs can be verified by testing feasibility of a Linear PI Inequality (LPI), defined by a positivity constraint on a PI operator mapping R n × L 2 [Ω]. Finally, we use positive matrices to parameterize the cone of positive PI operators on R n × L 2 [Ω], allowing feasibility of the L 2 -gain LPI to be tested using semidefinite programming. We implement this test in the MATLAB toolbox PIETOOLS, and demonstrate that this approach allows an upper bound on the L 2 -gain of PDEs to be estimated with little conservatism.

show abstract

Section: An Lmi For L 2 -Gain Analysis Of 2d Pdesmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

$L_2$-Gain Analysis of Coupled Linear 2D PDEs using Linear PI Inequalities

Declan¹,

Peet²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…For given x 0 ∈ X, we say that {x} satisfies the PDE defined by {n, G b , G p } (defined in Eqs. (10) and ( 12)) with initial condition x 0 if 11) is satisfied for almost all t ≥ 0.…”

Section: Termsmentioning

confidence: 99%

“…Consequently, many numerical methods designed for control, analysis, and simulation of ODEs in state-space form can be extended to PIEs. For example, computational tests for analysis and control of PIEs were presented in [3], [9], [10], etc.…”

Section: Introductionmentioning

confidence: 99%

Computational stability analysis of PDEs with integral terms using the PIE framework

Shivakumar¹,

Peet²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

The Partial Integral Equation (PIE) framework was developed to computationally analyze linear Partial Differential Equations (PDEs) where the PDE is first converted to a PIE and then the analysis problem is solved by solving operator-valued optimization problems. Previous works on the PIE framework focused on the analysis of PDEs with spatial derivatives up to 2 nd -order. In this paper, we extend the class of PDEs by including integral terms and performing stability analysis using the PIE framework. More specifically, we show that PDEs with the integral terms where the integration is with respect to the spatial variable and the kernel of the integral operator is matrix-valued polynomials can be converted to PIEs if the boundary conditions satisfy certain criteria. The conversion is performed by using a change of variable where every PDE state is substituted in terms of its highest derivative and boundary values to obtain a new equation (a PIE) in a variable that does not have any continuity requirements. Later, we show that this change of variable can be represented using explicit maps from the parameters of the PDE to the parameters of the PIE and the stability test can be posed as an optimization problem involving these parameters. Lastly, we present numerical examples to demonstrate the simplicity and application of this method.

show abstract

“…This toolbox offers a framework for implementation and manipulation of 1D-PI operators in MATLAB, allowing e.g. Lyapunov stability analysis [8], robust stability analysis [12], and H ∞ -optimal control [13] of systems involving 1D PDEs. For a detailed manual of the PIETOOLS toolbox we refer to [9].…”

Section: Pietools Implementationmentioning

confidence: 99%

A PIE Representation of Coupled Linear 2D PDEs and Stability Analysis using LPIs

Declan¹,

Peet²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We introduce a Partial Integration Equation (PIE) representation of Partial Differential Equations (PDEs) in two spatial variables. PIEs are an algebraic state-space representation of infinite-dimensional systems and have been used to model 1D PDEs and time-delay systems without continuity constraints or boundary conditions -making these PIE representations amenable to stability analysis using convex optimization. To extend the PIE framework to 2D PDEs, we first construct an algebra of Partial Integral (PI) operators on the function space L 2 [x, y], providing formulae for composition, adjoint, and inversion. We then extend this algebra to R n × L 2 [x] × L 2 [y] × L 2 [x, y] and demonstrate that, for any suitable coupled, linear PDE in 2 spatial variables, there exists an associated PIE whose solutions bijectively map to solutions of the original PDE -providing conversion formulae between these representations. Next, we use positive matrices to parameterize the convex cone of 2D PI operators -allowing us to optimize PI operators and solve Linear PI Inequality (LPI) feasibility problems. Finally, we use the 2D LPI framework to provide conditions for stability of 2D linear PDEs. We test these conditions on 2D heat and wave equations and demonstrate that the stability condition has little to no conservatism.

show abstract

Duality and H_∞-Optimal Control Of Coupled ODE-PDE Systems

Cited by 12 publications

References 10 publications

$L_2$-Gain Analysis of Coupled Linear 2D PDEs using Linear PI Inequalities

$L_2$-Gain Analysis of Coupled Linear 2D PDEs using Linear PI Inequalities

Computational stability analysis of PDEs with integral terms using the PIE framework

A PIE Representation of Coupled Linear 2D PDEs and Stability Analysis using LPIs

Contact Info

Product

Resources

About

Duality and H∞-Optimal Control Of Coupled ODE-PDE Systems

Cited by 12 publications

References 10 publications

$L_2$-Gain Analysis of Coupled Linear 2D PDEs using Linear PI Inequalities

$L_2$-Gain Analysis of Coupled Linear 2D PDEs using Linear PI Inequalities

Computational stability analysis of PDEs with integral terms using the PIE framework

A PIE Representation of Coupled Linear 2D PDEs and Stability Analysis using LPIs

Contact Info

Product

Resources

About

Duality and H_∞-Optimal Control Of Coupled ODE-PDE Systems