Abstract
In this paper, we consider the perturbation analysis of linear time-invariant systems, which arise from the linear optimal control in continuous-time. We provide a method to compute condition numbers of continuous-time linear time-invariant systems. It solves the perturbed linear time-invariant systems via Riccati differential equations and continuous-time algebraic Riccati equations in finite and infinite time horizons. We derive the explicit expressions of measuring the perturbation bounds of condition numbers with respect to the solution of the linear time-invariant systems. Furthermore, condition numbers and their upper bounds of Riccati differential equations and continuous-time algebraic Riccati equations are also discussed. Numerical simulations show the sharpness of the perturbation bounds computed via the proposed methods.
Highlights
Many mathematical models of physical, biological and social systems involve partial differential equations (PDEs)
We first derive two kinds of condition numbers and perturbation bounds before we present the sensitivity of continuous-time linear time-invariant system (CLTI) (1)
We investigate two kinds of condition numbers according to only perturbed matrix A in the CLTI (1) via continuous-time algebraic Riccati equation (CARE) (31) in the following theorem
Summary
Many mathematical models of physical, biological and social systems involve partial differential equations (PDEs). We seek to find the optimal control via solving the Riccati differential equation (RDE) in the finite time. We solve the (perturbed) CLTI to get the relative errors in the exact solutions via RDEs and CAREs in the finite and infinite time horizons respectively. The Bernoulli substitution technique is applied to solve RDEs (3), we can take the optimal control u (t ) (2) into the CLTI (1) and solve the ordinary differential equation (ODE) to get the state vector x (t ). Please refer to Weng and Phoa [22] about the details of solving the CLTI (1) via RDEs (3)
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