More Like this

1. Robust Analysis of Linear Systems with Uncertain Delays using PIEs

   https://doi.org/10.1016/j.ifacol.2021.11.133  

   Wu, S.; Peet, M.; Sun, F.; Hua, C. (January 2021, IFACPapersOnLine)

   Appeared in the proceedings of the 2021 IFAC Workshop on Time-Delay Systems This paper establishes a PIE (Partial Integral Equation)-based technique for the robust stability and H∞ performance analysis of linear systems with interval delays. The delays considered are time-invariant but uncertain, residing within a bounded interval excluding zero. We first propose a structured class of PIE systems with parametric uncertainty, then propose a Linear PI Inequality (LPI) for robust stability and H∞ performance of PIEs with polytopic uncertainty. Next, we consider the problem of robust stability and H∞ performance of multidelay systems with interval uncertainty in the delay parameters and show this problem is equivalent to robust stability and performance of a given PIE with parametric uncertainty. The robust stability and H∞ performance of the uncertain time-delay system are then solved using the LPI solver in the MATLAB PIETOOLS toolbox. Numerical examples are given to prove the effectiveness and accuracy of the method. This paper adds to the expanding field of PIE approach and can be extended to linear partial differential equations. 

   more » « less
2. H _∞ ‐optimal observer design for systems with multiple delays in states, inputs and outputs: A PIE approach

   https://doi.org/10.1002/rnc.6626  

   Wu, Shuangshuang; Peet, Matthew M.; Sun, Fuchun; Hua, Changchun (May 2023, International Journal of Robust and Nonlinear Control)

   Abstract This article investigates the ‐optimal estimation problem of a class of linear system with delays in states, disturbance input, and outputs. The estimator uses an extended Luenberger estimator format which estimates both the present and history states. The estimator is designed using an equivalent Partial Integral Equation (PIE) representation of the coupled nominal system. The advantage of the resulting PIE representation is compact and delay free—obviating the need for commonly used bounding technique such as integral inequalities which typically introduces conservatism into the resulting optimization problem. The ‐optimal estimator synthesis problem is then reformulated as a Linear Partial Inequality (LPI)—a form of convex optimization using operator variables and inequlities. Such LPI‐based optimization problems can be solved using semidefinite programming via the PIETOOLS toolbox in Matlab. Compared with previous work, the proposed method simplifies the analysis and computation process and resulting in observers which are non‐conservtism to 4 decimal places when compared with Pad‐based ODE observer design methodologies. Numerical examples and simulation results are given to illustrate the effectiveness and scalability of the proposed approach. 

   more » « less
3. Computing Optimal Upper Bounds on the H2-norm of ODE-PDE Systems using Linear Partial Inequalities

   https://doi.org/10.1016/j.ifacol.2023.10.538  

   Braghini, Danilo; Peet, Matthew M. (January 2023, IFAC-PapersOnLine)

   Recently, a broad class of linear delayed and ODE-PDEs systems was shown to have an equivalent representation using Partial Integral Equations (PIEs). In this paper, we use this PIE representation, combined with algorithms for convex optimization of Partial Integral (PI) operators to bound the H2-norm for input-output systems of this class. Specifically, the methods proposed here apply to delayed and ODE-PDE systems (including delayed PDE systems) in one or two spatial variables where the disturbance does not enter through the boundary. For such systems, we define a notion of H2-norm using an initial state-to-output framework and show that this notion reduces to more traditional concepts under the assumption of existence of a strongly continuous semigroup. Next, we consider input-output systems for which there exists a PIE representation and for such systems show that computing a minimal upper bound on the H2-norm of delayed and PDE systems can be equivalently formulated as a convex optimization problem subject to linear PI operator inequalities (LPIs). We convert, then, these optimization problems to Semi-Definite Programming (SDP) problems using the PIETOOLS toolbox. Finally, we apply the results to several numerical examples – focusing on time-delay systems (TDS) for which comparable H2 approximation results are available in the literature. The numerical results demonstrate the accuracy of the computed upper bound on the H2-norm. 

   more » « less
4. A PIE Representation of Scalar Quadratic PDEs and Global Stability Analysis Using SDP

   Jagt, Declan; Seiler, Peter; Peet, Matthew (December 2023, Proceedings of the IEEE Conference on Decision Control)

   It has recently been shown that the evolution of a linear Partial Differential Equation (PDE) can be more conveniently represented in terms of the evolution of a higher spatial derivative of the state. This higher spatial derivative (termed the `fundamental state') lies in $$L_2$$ - requiring no auxiliary boundary conditions or continuity constraints. Such a representation (termed a Partial Integral Equation or PIE) is then defined in terms of an algebra of bounded integral operators (termed Partial Integral (PI) operators) and is constructed by identifying a unitary map from the fundamental state to the state of the original PDE. Unfortunately, when the PDE is nonlinear, the dynamics of the associated fundamental state are no longer parameterized in terms of PI operators. However, in this paper we show that such dynamics can be compactly represented using a new tensor algebra of partial integral operators acting on the tensor product of the fundamental state. We further show that this tensor product of the fundamental state forms a natural distributed equivalent of the monomial basis used in representation of polynomials on a finite-dimensional space. This new representation is then used to provide a simple SDP-based Lyapunov test of stability of quadratic PDEs. The test is applied to three illustrative examples of quadratic PDEs. 

   more » « less
5. Representation of linear PDEs with spatial integral terms as Partial Integral Equations

   https://doi.org/10.23919/ACC55779.2023.10156465  

   Shivakumar, Sachin; Das, Amritam; Peet, Matthew M. (May 2023, Proceedings of the American Control Conference)

   In this paper, we present the Partial Integral Equation (PIE) representation of linear Partial Differential Equations (PDEs) in one spatial dimension, where the PDE has spatial integral terms appearing in the dynamics and the boundary conditions. The PIE representation is obtained by performing a change of variable where every PDE state is replaced by its highest, well-defined derivative using the Fundamental Theorem of Calculus to obtain a new equation (a PIE). We show that this conversion from PDE representation to PIE representation can be written in terms of explicit maps from the PDE parameters to PIE parameters. Lastly, we present numerical examples to demonstrate the application of the PIE representation by performing stability analysis of PDEs via convex optimization methods. 

   more » « less