Perturbation Analysis of Continuous-Time Linear Time-Invariant Systems

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.


Introduction
Many mathematical models of physical, biological and social systems involve partial differential equations (PDEs). In order to understand these systems, we consider problems of control and optimization, leading to PDE boundary control, optimization constrained by stochastic PDEs, model order reduction and some related applications.
Consider the continuous-time linear time-invariant system (CLTI) by discretizing the PDE, ( ) n x t ∈  , control vector ( ) m u t ∈  and output vector ( ) r y t ∈  . We can apply the optimal control u to influence the state vector x for output vector y.
From control theory, we seek to find the optimal control via solving the Riccati differential equation (RDE) in the finite time. For infinite time, we solve the continuous-time algebraic Riccati equation (CARE).
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. For solving RDEs, Leipnik [1] used the canonical form of the self-adjoint RDEs to obtain a convenient explicit solution. Rusnak [2] proposed an almost analytic representation for the solution of the nonhomogeneous and homogeneous, time invariant, and time variant RDEs to discuss the behavior of the optimal estimator on a finite time interval. For solving CAREs, it has been an extremely active area of research in various years. Laub [3] proposed the Schur method.
Byers [4] suggested a stable symplectic orthogonal method as well as the matrix sign function method [5]. Guo and Lancaster [6] applied the Newton's method.
Benner and Byers [7] adopted a modified Newton's method for solving CAREs that used exact line search to improve the convergence behavior of Newton's method. Furthermore, Chu et al. [8] used the SDA.
Perturbation analysis considers the sensitivity of the solution to the small perturbations in the input data of a problem. A condition number, which is a measurement of the sensitivity, is important in the numerical computation.
Furthermore, perturbation bounds are usually discussed. Kenney and Hewer studied the sensitivity of the RDEs developed by Byers [9] in [10]. Konstantinov and Pelova presented linear and nonlinear methods for estimating the sensitivity of the solution to RDEs in [11]. Konstantinov et al. [12] [13] proposed new methods to improve the sensitivity estimate of RDEs in 2-norm. For the sensitivity analysis of the linear differential system, we refer papers [14] [15] [16] and their references therein. For the perturbation analysis and perturbation bounds of CAREs, please see [9] [10] [17] [18] [19] [20] [21]. In this paper, it is the first to consider the perturbation analysis of CLTI via RDEs and CAREs.
The paper is organized as follows. We introduce the CLTI, solve the perturbed CLTI with only one perturbed coefficient matrix via RDEs, discuss the sensitivity of the RDEs, compute condition numbers and perturbation bounds of the CLTI via RDEs and apply backward differentiation formula (BDF) to solve differential Lyapunov matrix equation (DLE) in Section 2. Section 3 discusses the CLTI via CAREs, the sensitivity of the CAREs, condition numbers and perturbation bounds of the CLTI via CAREs. The illustrative numerical examples are presented in Section 4. Section 5 concludes the paper.

Solving Continuous-Time Linear Time-Invariant System Via Riccati Differential Equation
In order to guarantee the existence and uniqueness of the state and output The optimal control is given by with ( ) X t being the solution to the RDE: In this paper, the Bernoulli substitution technique is applied to solve RDEs (3), then 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 . Furthermore, the output vector ( ) y t can be also obtained. Please refer to Weng and Phoa [22] about the details of solving the CLTI (1) via RDEs (3).

Sensitivity of the Riccati Differential Equation
As we solve the CLTI (1) by applying RDEs (3), then the sensitivity of RDEs (3) is studied. We first derive two kinds of condition numbers and perturbation bounds before we present the sensitivity of CLTI (1).
First, we transform from the RDEs (3) with terminal condition into initial value condition. Let ( ) ( ) Suppose we add some small perturbations only to coefficient matrix A in the RDEs (3) due to some applications like electric circuit simulation and multibody dynamics [30]. Other two coefficient matrices B and C in the CLTI (1) are treated similarly step by step. The solution to perturbed Riccati differential is the perturbed coefficient matrix.
Dropping the second and high-order terms in (5) yields for any continuous matrix function ( ) . By variation method, Since we only perturb the coefficient matrix A, we modify the condition theory of Rice [31] into Taking the limit as  goes to zero, the condition number is defined: The following theorem describes the condition numbers of RDEs using 2-and ∞ -norm. Theorem 2.1. Using the notations given above, we can derive the explicit expressions and perturbation bounds for two kinds of condition numbers of the RDEs according to only perturbed matrix A ( ) where ( ) Proof. According to the above definition about the condition number of RDEs (12), we take 2-norm in (10) and substitute t into 1 t t − , then obtain Analogously, we take ∞ -norm in (10) and change t into 1 t t − , then obtain In order to compute two kinds of condition numbers and perturbation bounds of the RDEs efficiently, we let (7) and ( ) P P t = is the solution of RDEs (4). We assume that is a c-stable matrix and therefore (15) has a unique symmetric solution The following theorem is the connection between DLE (15) and partial condition numbers (13) and (14).
g Θ , and l F as in (9), (11), and (15), respectively, , we let u and v be unit left and right singular vectors of ( ) Using (9) and (11), we can obtain Applying the Cauchy-Schwarz inequality [21], we get We can express the solution l F to (15) explicitly using (8) Advances in Pure Mathematics where u is a unit vector. Moreover, where v is a unit vector. Combining (17) and (18), we have ( )

Sensitivity of the CLTI via RDEs
In this subsection, we discuss the perturbation analysis of the CLTI (1) using RDEs (3) and derive two kinds of condition numbers. Furthermore, we also present their perturbation bounds.
Suppose we introduce some small perturbation A ∆ only to coefficient matrix A and the state vector to the perturbed system is ( ) ( ) ( ) We can replace the perturbed optimal control is the solution of perturbed Riccati differential equation (pRDE): Dropping the second and higher-order terms in (20) yields for any continuous matrix function ( ) . By variation method, we can solve (22) and get The above relation discusses a first-order perturbation ( ) Taking the limit as  goes to zero, we can get the condition number The following theorem describes the condition numbers of the CLTI (1) via RDEs and perturbation bounds in 2-and ∞ -norm according to only perturbed matrix A. Theorem 2.3. Using the notations given above, we can derive the explicit expressions and perturbation bounds for two kinds of condition numbers of the CLTI (1) via RDEs Proof. We can investigate condition numbers in 2-and ∞ -norm according to only perturbed matrix A  defined by For  sufficiently small, with Analogously, we take ∞ -norm in (26) and apply (27), then obtain  (24) and (25), respectively, the unique solution of the DLEs (28) is defined by is defined in (23). Furthermore, we can obtain and .

Backward Differentiation Formula Method for Solving DLEs
There is a large variety of methods to compute the solution of DLEs, see, e.g.
Applying the fixed-coefficients BDF method to the DLEs (29), we obtain the matrix valued BDF scheme α , β are the determining coefficients of the p-step BDF method as listed in Table 1 (see, e.g. [33]).
It leads to solving the following Lyapunov-BDF difference equation

Solving Continuous-Time Linear Time-Invariant System via Continuous-Time Algebraic Riccati Equation
In this case, the time-invariant solution X leads to the optimal control In this paper, we used the MATLAB function "care" to compute the unique

Sensitivity of the Continuous-Time Algebraic Riccati Equation
Before we discuss the sensitivity of the CLTI (1) via solving CAREs (31), we first consider the sensitivity of the CAREs. Suppose we add some small perturbations only to the coefficient matrix A in the CAREs (31) similar to that in the RDEs

0, A X XA H XGX
where X X X ≡ + ∆  . Dropping the second and high-order terms in (33) yields ( ) To connect X ∆ to only A ∆ , we modify the condition theory of Rice [31] into sup | .
Taking the limit as  goes to zero, we obtain the condition number 0 lim .
The following theorem derives two kinds of condition numbers of CAREs (31) in 2-and ∞ -norm.
Theorem 3.1. Using the notations given above, we can derive the explicit expressions and perturbation bounds for two kinds of condition numbers of CAREs (31) according to only perturbed matrix A ≤ , we take 2-norm in (37) according to the definition of the condition number (39) and get ( ) Analogously, we take ∞ -norm in (37) and divide X ∞  , then we obtain To solve two kinds of condition numbers and perturbation bounds of CAREs (31) efficiently, we let k E be the solution to the Lyapunov equation where v is a unit vector. Therefore, we combine (44) and (45), so ( )

Sensitivity of the CLTI via CAREs
We consider the perturbed CLTI (19) and take the perturbed optimal control ( ) ( ) in (19), then obtain By dropping the second and higher-order terms in (46), we apply the similar Analogously, we take ∞ -norm in (47) by applying (51) and obtain To solve two kinds of condition numbers of CLTI (1) via CAREs (31) efficiently, we apply Theorem 3.2 to compute condition numbers (49) and (50) efficiently.
with the optimal controls ( ) u t chosen through minimizing the cost functional are solutions of pRDEs, pCLTI and pCAREs, respectively, then we obtain relative differences of solutions between original and perturbed equations in 2-and ∞ -norm such as and the corresponding perturbation bounds 2_ Moreover, some parameters are set below: From Table 2, we skip the relative differences of RDEs in ∞ -norm as We can observe sharper perturbation bounds of the relative differences in RDEs and CLTI such as To sum up, perturbation bounds of CLTI are tight around ( ) 3 10 O − according to the weighted coefficient 10 −3 whatever we solve via RDEs or CAREs.

Conclusion
We have proposed, tested and analyzed CLTI for the condition numbers and