^{1}

^{*}

^{2}

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.

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,

x ˙ ( t ) = A x ( t ) + B u ( t ) , x ( 0 ) = x 0 , t ∈ [ 0 , t 1 ] , y ( t ) = C x ( t ) , (1)

with coefficient matrices A ∈ ℝ n × n , B ∈ ℝ n × m , C ∈ ℝ r × n , state vector x ( t ) ∈ ℝ n , control vector u ( t ) ∈ ℝ m and output vector y ( t ) ∈ ℝ r . 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 [

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 [

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.

In order to guarantee the existence and uniqueness of the state and output vectors, respectively in the CLTI (1), we assume that the condition det ( r I n − A ) ≠ 0 holds, for some r. The linear-quadratic regulator (LQR) problem for finite time horizon seeks the optimal control u ( t ) to minimize the cost function:

J ( u , Q 1 , t 1 ) ≡ ∫ 0 t 1 [ y Τ ( t ) y ( t ) + u Τ ( t ) R u ( t ) ] d t + x Τ ( t 1 ) Q 1 x ( t 1 ) ,

for some R > 0 and Q 1 ≥ 0 . The optimal control is given by

u ( t ) = − R − 1 B Τ X ( t ) x ( t ) , t ∈ [ 0, t 1 ] , (2)

with X ( t ) being the solution to the RDE:

X ˙ ( t ) = − A Τ X ( t ) − X ( t ) A − H + X ( t ) G X ( t ) , X ( t 1 ) = Q 1 , (3)

where G ≡ B R − 1 B Τ and H ≡ C Τ C .

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 [

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).

The RDEs (3) study nonlinear matrix differential equations arising in optimal control, optimal filtering, H ∞ -control of linear-time varying systems, differential games, etc.; see, e.g. [

First, we transform from the RDEs (3) with terminal condition into initial value condition. Let P ( t ) = X ( t 1 − t ) , then for 0 ≤ t ≤ t 1

P ˙ ( t ) = H + A Τ P ( t ) + P ( t ) A − P ( t ) G P ( t ) , P ( 0 ) = Q 1 . (4)

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 [

P ˜ ˙ ( t ) = H + A ˜ Τ P ˜ ( t ) + P ˜ ( t ) A ˜ − P ˜ ( t ) G P ˜ ( t ) , P ˜ ( 0 ) = Q 1 + Δ Q 1 , (5)

where A ˜ = A + Δ A is the perturbed coefficient matrix.

Dropping the second and high-order terms in (5) yields

Δ P ˙ ( t ) = A g ( t ) Δ P ( t ) + Δ P ( t ) A g Τ ( t ) + Δ A Τ P ( t ) + P ( t ) Δ A , (6)

Δ P ( 0 ) = Δ Q 1 ,0 ≤ t ≤ t 1 , A g ( t ) ≡ A Τ − P ( t ) G . (7)

Let Φ g satisfy

Φ ˙ g ( t ) = A g ( t ) Φ g ( t ) , Φ g ( 0 ) = I , 0 ≤ t ≤ t 1 . (8)

Define

Ω g − 1 ( Z ) = ∫ 0 t Φ g ( t ) Φ g − 1 ( s ) Z ( s ) Φ g − Τ ( s ) Φ g Τ ( t ) d s (9)

for any continuous matrix function Z = Z ( s ) , s ∈ [ 0, t ] . By variation method, Δ P ( t ) in (6) can be solved

Δ P ( t ) = Φ g ( t ) Δ Q 1 Φ g Τ ( t ) + Θ g ( Δ A ) , (10)

where

Θ g ( Z ) ≡ Ω g − 1 ( Z Τ P ( t ) + P ( t ) Z ) . (11)

Since we only perturb the coefficient matrix A, we modify the condition theory of Rice [

C ϵ A ( t ) = s u p { ‖ Δ X ( t ) ‖ ϵ ‖ X ( t ) ‖ | ‖ Δ A ‖ ≤ ϵ ‖ A ‖ , ‖ Δ Q 1 ‖ ≤ ϵ ‖ Q 1 ‖ } .

Taking the limit as ϵ goes to zero, the condition number is defined:

K R D E A = l i m ϵ → 0 C ϵ A ( t ) .

That is,

K R D E A = lim ϵ → 0 sup { ‖ Δ P ( t 1 − t ) ‖ ϵ ‖ P ( t 1 − t ) ‖ | ‖ Δ A ‖ ≤ ϵ ‖ A ‖ , ‖ Δ Q 1 ‖ ≤ ϵ ‖ Q 1 ‖ } . (12)

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

K 2 _ R D E A = m 1 ‖ P ( t 1 − t ) ‖ , (13)

K ∞ _ R D E A = m 2 ‖ P ( t 1 − t ) ‖ ∞ , 0 ≤ t ≤ t 1 , (14)

where

m 1 = ‖ Φ g ( t 1 − t ) ‖ 2 ‖ Q 1 ‖ + ‖ Θ g ‖ ‖ A ‖ ,

m 2 = ‖ Φ g ( t 1 − t ) ‖ ∞ 2 ‖ Q 1 ‖ ∞ + ‖ Θ g ‖ ∞ ‖ A ‖ ∞ .

Proof. According to the above definition about the condition number of RDEs (12), we take 2-norm in (10) and substitute t into t 1 − t , then obtain

‖ Δ P ( t 1 − t ) ‖ ≤ ‖ Φ g ( t 1 − t ) ‖ 2 ‖ Δ Q 1 ‖ + ‖ Θ g ‖ ‖ Δ A ‖ .

For ϵ sufficiently small, with ‖ Δ A ‖ / ‖ A ‖ , ‖ Δ Q 1 ‖ / ‖ Q 1 ‖ ≤ ϵ , we can get

‖ Δ P ( t 1 − t ) ‖ ≤ ‖ Φ g ( t 1 − t ) ‖ 2 ϵ ‖ Q 1 ‖ + ‖ Θ g ‖ ϵ ‖ A ‖ .

Divide by ϵ ‖ P ( t 1 − t ) ‖ to get

‖ Δ P ( t 1 − t ) ‖ ϵ ‖ P ( t 1 − t ) ‖ ≤ ‖ Φ g ( t 1 − t ) ‖ 2 ‖ Q 1 ‖ + ‖ Θ g ‖ ‖ A ‖ ‖ P ( t 1 − t ) ‖ .

From (12), let ϵ → 0 give

K 2 _ R D E A ≤ m 1 ‖ P ( t 1 − t ) ‖ , 0 ≤ t ≤ t 1 .

Analogously, we take ∞ -norm in (10) and change t into t 1 − t , then obtain

‖ Δ P ( t 1 − t ) ‖ ∞ ϵ ‖ P ( t 1 − t ) ‖ ∞ ≤ ‖ Φ g ( t 1 − t ) ‖ ∞ 2 ‖ Q 1 ‖ ∞ + ‖ Θ g ‖ ∞ ‖ A ‖ ∞ ‖ P ( t 1 − t ) ‖ ∞ .

Let ϵ → 0 , we can get

K ∞ _ R D E A ≤ m 2 ‖ P ( t 1 − t ) ‖ ∞ , 0 ≤ t ≤ t 1 .

In order to compute two kinds of condition numbers and perturbation bounds of the RDEs efficiently, we let

where

Theorem 2.2. For

Proof. For

Using (9) and (11), we can obtain

Applying the Cauchy-Schwarz inequality [

We can express the solution

However,

where u is a unit vector. Moreover,

where v is a unit vector. Combining (17) and (18), we have

Thus,

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

We can replace the perturbed optimal control

and obtain

where

Dropping the second and higher-order terms in (20) yields

where the pRDEs (21) are solved. Let

where

Define

for any continuous matrix function

The above relation discusses a first-order perturbation

Taking the limit as

The following theorem describes the condition numbers of the CLTI (1) via RDEs and perturbation bounds in 2- and

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

where

Proof. We can investigate condition numbers in 2- and

For

Therefore, we can obtain

Thus

Analogously, we take

Let

When we compute condition numbers and perturbation bounds of CLTI efficiently via solving RDEs, we let

where

Theorem 2.4. [

where

Therefore,

There is a large variety of methods to compute the solution of DLEs, see, e.g. [

Consider

Applying the fixed-coefficients BDF method to the DLEs (29), we obtain the matrix valued BDF scheme

where

It leads to solving the following Lyapunov-BDF difference equation

with

for

The Lyapunov Equation (30) can be solved by applying various methods such as the Schur vector method, symplectic SR methods, the matrix sign function, the matrix disk function or the doubling method; see, e.g. [

p | ||||||
---|---|---|---|---|---|---|

1 | 1 | −1 | ||||

2 | ||||||

3 | ||||||

4 | ||||||

5 |

For infinite time horizon,

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 symmetric positive semidefinite solution X to the CAREs (31), then replace the optimal control

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 (4), then we get the perturbed continuous-time algebraic Riccati equation (pCARE):

where

Set

and let

Due to solvable conditions of CAREs (31), it is known that the matrix

Therefore, we can solve the Lyapunov Equation (34)

where

To connect

Taking the limit as

The following theorem derives two kinds of condition numbers of CAREs (31) in 2- and

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

where

Proof. For

Divide by

Take

Analogously, we take

Let

To solve two kinds of condition numbers and perturbation bounds of CAREs (31) efficiently, we let

where

Theorem 3.2. For

Proof. For

By (36) and (38), we get

Applying the Cauchy-Schwarz inequality, we obtain

We can express the solution

But

where u is a unit vector. Moreover,

where v is a unit vector. Therefore, we combine (44) and (45), so

Thus,

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 technique such as variation method to solve

and we obtain

with

where

The above relation (47) states a first-order perturbation

Theorem 3.3. Using the above notations, the explicit expressions and perturbation bounds for two kinds of condition numbers in the CLTI (1) via CAREs (31) according to only perturbed matrix A are

where

Proof. We consider condition numbers of the CLTI (1) via CAREs (31) according to only perturbed matrix A in 2- and

For

According to the above definition of the condition number, we obtain

Let

Analogously, we take

Take

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.

Theorem 3.4. For

Thus,

The numerical simulations are conducted on a desktop with a 3.40 GHz Intel Core 2 Duo processor and 32 GB RAM, with machine accuracy

We have chosen one example for demonstration:

1) The example 1 illustrates condition numbers and perturbation bounds of CLTI via solving RDEs and CAREs with finite and infinite time horizons, respectively to present the effectiveness of the theoretical results.

Example 1 (CLTI)

Consider the CLTI (1) with

satisfying

with the optimal controls

In the example, the perturbed coefficient matrix is constructed such as

Moreover, some parameters are set below:

for perturbation bounds of RDEs, CLTI and CAREs, respectively; the time range is

From

t | |||||||
---|---|---|---|---|---|---|---|

0 | 2.5870e−04 | 7.6082e−04 | 8.0831e−04 | 9.2242e−04 | 9.2242e−04 | 9.2978e−04 | 9.2978e−04 |

0.0667 | 2.5856e−04 | 7.6823e−04 | 8.1539e−04 | 8.6703e−04 | 1.0657e−03 | 8.7716e−04 | 1.1148e−03 |

0.1333 | 2.5803e−04 | 7.7703e−04 | 8.2377e−04 | 8.1119e−04 | 1.2269e−03 | 8.4001e−04 | 1.3251e−03 |

0.2000 | 2.5677e−04 | 7.8723e−04 | 8.3342e−04 | 7.5528e−04 | 1.3740e−03 | 8.0345e−04 | 1.5178e−03 |

0.2667 | 2.5430e−04 | 7.9861e−04 | 8.4408e−04 | 6.9985e−04 | 1.5005e−03 | 7.6736e−04 | 1.6842e−03 |

0.3333 | 2.4983e−04 | 8.1055e−04 | 8.5505e−04 | 6.4563e−04 | 1.6089e−03 | 7.3152e−04 | 1.8274e−03 |

0.4000 | 2.4218e−04 | 8.2175e−04 | 8.6497e−04 | 5.9361e−04 | 1.7044e−03 | 6.9567e−04 | 1.9545e−03 |

0.4667 | 2.2956e−04 | 8.2989e−04 | 8.7140e−04 | 5.4514e−04 | 1.7931e−03 | 6.5941e−04 | 2.0740e−03 |

0.5333 | 2.0947e−04 | 8.3130e−04 | 8.7058e−04 | 5.0210e−04 | 1.8820e−03 | 6.2220e−04 | 2.1966e−03 |

0.6000 | 1.7867e−04 | 8.2098e−04 | 8.5741e−04 | 4.6699e−04 | 1.9797e−03 | 5.8328e−04 | 2.3359e−03 |

0.6667 | 1.3375e−04 | 7.9373e−04 | 8.2674e−04 | 4.4295e−04 | 2.0980e−03 | 5.4158e−04 | 2.5112e−03 |

0.7333 | 7.2418e−05 | 7.4724e−04 | 7.7649e−04 | 4.3366e−04 | 2.2528e−03 | 4.9563e−04 | 2.7494e−03 |

0.8000 | 4.5173e−06 | 6.8682e−04 | 7.1256e−04 | 4.4279e−04 | 2.4637e−03 | 4.4333e−04 | 3.0829e−03 |

0.8667 | 9.2024e−05 | 6.3045e−04 | 6.5392e−04 | 4.7364e−04 | 2.7408e−03 | 3.9502e−04 | 3.5256e−03 |

0.9333 | 1.8188e−04 | 6.1786e−04 | 6.4218e−04 | 5.2947e−04 | 3.0334e−03 | 5.1072e−04 | 3.9783e−03 |

1.0000 | 2.6472e−04 | 7.9124e−04 | 8.2675e−04 | 6.1487e−04 | 3.0669e−03 | 6.3318e−04 | 3.9355e−03 |

To sum up, perturbation bounds of CLTI are tight around ^{−3} whatever we solve via RDEs or CAREs.

We have proposed, tested and analyzed CLTI for the condition numbers and perturbation bounds according to only one perturbed coefficient matrix via solving RDEs and CAREs. Numerical simulations show that condition numbers provide tight perturbation bounds of the solutions to CLTI under some small

t | ||||
---|---|---|---|---|

0 | 9.2242e−04 | 1.7511e−03 | 9.2978e−04 | 2.0709e−03 |

0.0667 | 8.6124e−04 | 1.7768e−03 | 8.8827e−04 | 2.1070e−03 |

0.1333 | 8.0087e−04 | 1.8030e−03 | 8.6181e−04 | 2.1434e−03 |

0.2000 | 7.4210e−04 | 1.8297e−03 | 8.3565e−04 | 2.1801e−03 |

0.2667 | 6.8599e−04 | 1.8570e−03 | 8.0980e−04 | 2.2171e−03 |

0.3333 | 6.3401e−04 | 1.8848e−03 | 7.8425e−04 | 2.2543e−03 |

0.4000 | 5.8810e−04 | 1.9132e−03 | 7.5903e−04 | 2.2917e−03 |

0.4667 | 5.5074e−04 | 1.9421e−03 | 7.3413e−04 | 2.3294e−03 |

0.5333 | 5.2480e−04 | 1.9716e−03 | 7.0956e−04 | 2.3673e−03 |

0.6000 | 5.1314e−04 | 2.0016e−03 | 6.8534e−04 | 2.4054e−03 |

0.6667 | 5.1787e−04 | 2.0322e−03 | 6.6146e−04 | 2.4437e−03 |

0.7333 | 5.3970e−04 | 2.0633e−03 | 6.3793e−04 | 2.4822e−03 |

0.8000 | 5.7779e−04 | 2.0948e−03 | 6.1476e−04 | 2.5208e−03 |

0.8667 | 6.3025e−04 | 2.1269e−03 | 5.9196e−04 | 2.5596e−03 |

0.9333 | 6.9480e−04 | 2.1595e−03 | 7.0861e−04 | 2.5984e−03 |

1.0000 | 7.6931e−04 | 2.1926e−03 | 8.3791e−04 | 2.6374e−03 |

change in the only one coefficient matrix. In summary, we introduce some efficient measurement tools for the sensitivity analysis of CLTI via solving RDEs and CAREs respectively.

This work was supported by Academia Sinica (Taiwan) grant number 103-CDA-M04, and Ministry of Science and Technology (Taiwan) grant numbers 104-2118-M-001-016-MY2 and 105-2118-M-001-007-MY2.

The authors declare no conflicts of interest regarding the publication of this paper.

Weng, P.C.-Y. and Phoa, F.K.H. (2020) Perturbation Analysis of Continuous-Time Linear Time-Invariant Systems. Advances in Pure Mathematics, 10, 155-173. https://doi.org/10.4236/apm.2020.104010