A Convex Sum-of-Squares Approach to Analysis, State Feedback and Output Feedback Control of Parabolic PDEs

Abstract

Abstract-We present an optimization-based framework for analysis and control of linear parabolic Partial Differential Equations (PDEs) with spatially varying coefficients without discretization or numerical approximation. For controller synthesis, we consider both full-state feedback and point observation (output feedback). The input occurs at the boundary (point actuation). We use positive definite matrices to parameterize positive Lyapunov functions and polynomials to parameterize controller and observer gains. We use duality and an invertible state variable transformation to convexify the controller synthesis problem. Finally, we combine our synthesis condition with the Luenberger observer framework to express the output feedback controller synthesis problem as a set of LMI/SDP constraints. We perform an extensive set of numerical experiments to demonstrate accuracy of the conditions and to prove necessity of the Lyapunov structures chosen. We provide numerical and analytical comparisons with alternative approaches to control including Sturm Liouville theory and backstepping. Finally we use numerical tests to show that the method retains its accuracy for alternative boundary conditions. Index Terms-Distributed parameter systems, partial differential equations (PDEs), control design, sum of squares.

A Generalized LMI Formulation for Input-Output Analysis of Linear Systems of ODEs Coupled with PDEs

“…Then for allx(t) ∈ X, y ∈ L ny 2 [0, ∞) and w ∈ L ny 2 [0, ∞) which satisfy (7), if (14), i ,B i ,Ĉ i ,D i and I i are as defined in (16), then w, y L2 ≥ 0.…”

“…Furthermore, the effect of the PDE state on the ODE state is not bounded as the input to the ODE state is a single unmeasurable point of a larger distributed state. These limitations preclude obvious approaches such as construction of a Lyapunov functionals which depends on the joint ODE-PDE state (See earlier work in [5], [6], [7], [8]). Our solution to this problem is based on an alternative boundary-condition-free representation of the dynamics using a fundmantal state, x f , as proposed in [9] (See Section VI).…”

“…Other techniques, such as backstepping method [12], [13], [14], [15] and sliding mode control method [16], [17], rely on the use of predetermined Lyapunov functional which is chosen in an ad hoc manner. Other methods, such as [5], [6], [7], [8] use LMIs to search over a given set of Lyapunov functionals. However, these methods are largely restricted to scalar or vector-valued PDE systems with a specific choice of boundary conditions and are not applicable to coupled ODE-PDE systems with a general form of boundary conditions.…”

On the domain of attraction and local stabilization of nonlinear parameter‐varying systems

“…Thus far, SOS-based conditions for the analysis and synthesis of polynomial systems have attracted extensive interests of scholars. [14][15][16][17][18][19][20][21][22] Among those works, a novel type of system, the nonlinear parameter-varying (NPV) system, is put forward in the works of Fu et al 15,16 as . x = A(x, (t))x + B(x, (t))u, in which the system matrices are dependent on both the system state and the time-varying parameter.…”

A New State-Space Representation for Coupled PDEs and Scalable Lyapunov Stability Analysis in the SOS Framework

“…This Recently, Sum-of-Squares (SOS) optimization methods have been applied to the problem of finding Lyapunov functions which prove stability of vector-valued PDEs. Examples of this work from our lab can be found in [13], [14], [15], [16] and work from our colleagues can be found in [13], [17], [18], [19]. While these previous works have proven remarkably effective, they suffered from high computational complexity and the lack of a unifying framework -deficiencies which limit the practical impact and scalability of these results.…”