Convergence of Discrete Algebraic Riccati Equation and Uniform Estimation of Matrix Exponential Approximation

 

Author: Suhang Zhang

                Luoyang

 

 

 

Abstract

 

This paper systematically studies two core problems in sampled-data optimal control and numerical approximation of linear systems. First, under the standard stabilisability and detectability assumptions, we rigorously prove that the unique positive semi-definite stabilising solution P_h of the discrete-time algebraic Riccati equation converges to the solution P_c of the continuous-time algebraic Riccati equation as the sampling period h\to 0^+, with an optimal convergence rate \|P_h-P_c\|=O(h). Second, we establish a uniform convergence estimate for discrete approximations of the matrix exponential evolution operator, proving that the iterated discrete propagators converge to the continuous-time matrix exponential uniformly over the finite-time interval [0,T] with order O(h). The entire analysis follows a rigorous mathematical framework, including asymptotic expansions, uniform boundedness, Fréchet linearisation, Lyapunov operator inversion, and telescoping error decomposition.

 

Keywords: Discrete algebraic Riccati equation; Matrix exponential; Sampled-data system; Convergence rate; Uniform error estimate; Optimal control

 

 

 

1 Introduction

 

In digital control and sampled-data systems, continuous-time optimal control problems are routinely approximated by discrete-time formulations via zero-order hold sampling. A fundamental question is the consistency of the solution to the discrete algebraic Riccati equation (DARE) with respect to the sampling period h: as h\to 0, does the solution of the DARE converge to that of the continuous-time algebraic Riccati equation (CARE)? This convergence guarantees that high-rate digital control designs approximate the continuous-time optimal controller.

 

Meanwhile, the convergence analysis relies crucially on uniform error bounds for the discretised state-transition matrix, i.e., the approximation of the matrix exponential e^{\mathcal{A}t} by iterated discrete propagators L_h^{\lfloor t/h\rfloor}. A sharp uniform O(h) estimate over a finite-time horizon is required to close the error bound for the Riccati solution.

 

This paper consists of two logically connected parts. Part 1 proves the convergence of the DARE solution to the CARE solution with explicit first-order rate. Part 2 establishes a general uniform convergence theorem for matrix exponential approximations, covering exact sampling, forward Euler, and consistent one-step methods, providing the foundational estimate used in Part 1.

 

 

 

Part 1 Convergence of the Discrete Riccati Equation Solution

 

2 Problem Statement and Main Result

 

Consider the continuous-time linear time-invariant system

 

\dot{x}(t) = A x(t) + B u(t), \quad x(0)=x_0,

 

with the infinite-horizon quadratic cost

 

J_c = \int_0^\infty \big( x(t)^T Q x(t) + u(t)^T R u(t) \big) dt,

 

where Q\succeq 0, R\succ 0. We impose the standard assumptions:

 

1. The pair (A,B) is stabilisable;

2. The pair (Q^{1/2},A) is detectable.

 

The continuous-time algebraic Riccati equation (CARE) is

 

A^T P + P A - P B R^{-1} B^T P + Q = 0, \tag{2.1}

 

which admits a unique positive semi-definite stabilising solution P_c, such that A-BR^{-1}B^TP_c is Hurwitz.

 

For sampling period h>0, the system is discretised as

 

x_{k+1} = A_d x_k + B_d u_k,\quad A_d = e^{Ah},\quad B_d = \int_0^h e^{As}B\,ds,

 

with cost

 

J_d = \sum_{k=0}^\infty \big( x_k^T Q_d x_k + u_k^T R u_k \big),\quad Q_d = \int_0^h e^{A^Ts}Q e^{As}ds.

 

The discrete-time algebraic Riccati equation (DARE) reads

 

A_d^T P_h A_d - P_h - A_d^T P_h B_d (R+B_d^T P_h B_d)^{-1} B_d^T P_h A_d + Q_d = 0. \tag{2.2}

 

For all sufficiently small h>0, (2.2) has a unique positive semi-definite stabilising solution P_h.

 

Theorem 2.1

 

Under the stabilisability and detectability assumptions, as h\to 0^+,

 

\lim_{h\to 0^+} P_h = P_c, \qquad \|P_h-P_c\|=O(h).

 

3 Preliminary Asymptotic Expansions

 

For small h>0, the following uniform expansions hold in the operator norm:

 

\begin{aligned}

A_d &= I + Ah + \tfrac12 A^2 h^2 + O(h^3),\\

B_d &= Bh + \tfrac12 AB h^2 + O(h^3),\\

Q_d &= Qh + \tfrac12 (A^TQ+QA)h^2 + O(h^3).

\end{aligned}

 

These follow from term-by-term integration of the matrix exponential power series, with uniform remainders.

 

4 Uniform Boundedness of P_h

 

By stabilisability, there exists a constant matrix K such that A-BK is Hurwitz. The discretised closed-loop matrix satisfies

 

A_d-B_dK = I+(A-BK)h+O(h^2).

 

For small enough h, \rho(A_d-B<1, so the discrete system is exponentially stable. Consider the Lyapunov equation

 

(A_d-B_dK)^T\hat{P}_h(A_d-B_dK)-\hat{P}_h+Q_d+K^TRK=0.

 

Its unique solution \hat{P}_h is uniformly bounded for h\in(0,h_0]. Since P_h is the optimal cost matrix, P_h\preceq\hat{P}_h, so \|P_h\| is uniformly bounded.

 

5 Riccati Residual Operator and Linearisation

 

Define the Riccati residual operator

 

\mathcal{R}_h(P) = A_d^TPA_d - P + Q_d - A_d^TPB_d(R+B_d^TPB_d)^{-1}B_d^TPA_d.

 

Then DARE is equivalent to \mathcal{R}_h(P_h)=0. Substituting P_c and using the CARE to cancel the first-order terms yields

 

\mathcal{R}_h(P_c)=O(h^2).

 

Let E_h=P_h-P_c. The Fréchet linearisation gives

 

0=\mathcal{R}_h(P_c)+\mathcal{L}_h(E_h)+O(\|E_h\|^2),

 

where the linearised operator \mathcal{L}_h takes the form

 

\mathcal{L}_h(\Delta)=A_{c,d}^T\Delta A_{c,d}-\Delta+h\Phi_h(\Delta),

 

with A_{c,d}=A_d-B_dK_c, K_c=R^{-1}B^TP_c, and \Phi_h uniformly bounded.

 

Lemma 5.1

 

The operator \mathcal{L}_0(\Delta)=A_{c,d}^T\Delta A_{c,d}-\Delta satisfies

 

\|\mathcal{L}_0^{-1}\|=O(1/h).

 

Since \mathcal{L}_h=\mathcal{L}_0(I+h\Phi_h\mathcal{L}_0^{-1}), for small h, \mathcal{L}_h is invertible and \|\mathcal{L}_h^{-1}\|=O(1/h).

 

6 Convergence Rate Estimate

 

From the error equation

 

\|E_h\|\le\|\mathcal{L}_h^{-1}\|\big(\|\mathcal{R}_h(P_c)\|+O(\|E_h\|^2)\big),

 

substituting \|\mathcal{R}_h(P_c)\|=O(h^2) and \|\mathcal{L}_h^{-1}\|=O(1/h) gives

 

\|E_h\| \le C h + \frac{C'}{h}\|E_h\|^2.

 

By uniform boundedness and a bootstrap argument, \|E_h\|=O(h). Thus Theorem 2.1 is proved.

 

 

 

Part 2 Uniform Estimation of Matrix Exponential Convergence

 

7 Problem Setup

 

Consider the linear system

 

\dot{x}(t)=\mathcal{A}x(t),\quad x(0)=x_0,

 

with exact solution x(t)=e^{\mathcal{A}t}x_0. For h>0, define the discrete approximation

 

x_h(t)=L_h^{\lfloor t/h\rfloor}x_0,

 

where L_h satisfies the local consistency condition

 

\|L_h-(I+h\mathcal{A})\|\le C_0 h^2<h\le h_0.

 

We aim to prove uniform convergence over t\in[0,T] with explicit O(h) error bound.

 

8 Main Theorem

Theorem 8.1

Let \mathcal{A}\in\mathbb{R}^{n\times n} with logarithmic norm \mu(\mathcal{A})\le\omega, so that \|e^{\mathcal{A}t}\|\le e^{\omega t}. Suppose:

1. \|L_h-(I+h\mathcal{A})\|\le C_0 h^2\<h\le h_0;
​
2. \|L_h\|\le 1+\omega h+M h^2 uniformly in h.

Then there exists a constant C=C(\omega,T,\|\mathcal{A}\|,C_0,M) such that for all 0<h\le h_0 and t\in[0,T],

\big\|L_h^{\lfloor t/h\rfloor}-e^{\mathcal{A}t}\big\|\le C h.

9 Proof of the Uniform Estimate

9.1 Matrix Logarithm Expansion

Define the effective generator

\mathcal{A}_h=\frac{1}{h}\log L_h.

By consistency, L_h=I+h\mathcal{A}+O(h^2), so

\mathcal{A}_h=\mathcal{A}+O(h).

9.2 Telescoping Error Decomposition

For k=\lfloor t/h\rfloor and t_k=kh,

L_h^k-e^{\mathcal{A}t_k}=\sum_{j=0}^{k-1}L_h^{k-1-j}\big(L_h-e^{\mathcal{A}h}\big)e^{\mathcal{A}jh}.

Let E_h=L_h-e^{\mathcal{A}h}. From the expansion

e^{\mathcal{A}h}=I+h\mathcal{A}+\frac{h^2}{2}\mathcal{A}^2+O(h^3),

we obtain

\|E_h\|\le C_E h^2.

9.3 Growth Bound and Summation

Lemma 9.1

There exists c_1>0 such that

\|L_h^k\|\le e^{\omega t_k+c_1 k h^2}.

The error sum satisfies

\|L_h^k-e^{\mathcal{A}t_k}\|\le\|E_h\|\sum_{j=0}^{k-1}\|L_h\|^{k-1-j}\|e^{\mathcal{A}h}\|^j
\le C_E h^2 \cdot \frac{T}{h}\cdot e^{\omega T+O(h)}=O(h).

9.4 Interpolation Error Between Sampling Instants

For t\in[kh,(k+1)h), write \tau=t-kh\in[0,h). Then

L_h^{\lfloor t/h\rfloor}-e^{\mathcal{A}t}=\big(L_h^k-e^{\mathcal{A}t_k}\big)e^{\mathcal{A}\tau}+L_h^k\big(I-e^{\mathcal{A}\tau}\big).

Since \|I-e^{\mathcal{A}\tau}\|\le\|\mathcal{A}\|h e^{\omega h}, both terms are O(h). The uniform bound over [0,T] follows.

10 Corollaries

Corollary 10.1 (Forward Euler Method)

For L_h=I+h\mathcal{A}, the local error is O(h^2), so

\big\|(I+h\mathcal{A})^{\lfloor t/h\rfloor}-e^{\mathcal{A}t}\big\|=O(h).

Corollary 10.2 (Exact Sampling)

For L_h=e^{\mathcal{A}h}, the sampling-point error vanishes, and the uniform error is dominated by the interpolation term, still O(h).

 

11 Conclusion

This paper establishes two foundational results for sampled-data systems and numerical linear approximations. The solution of the discrete-time algebraic Riccati equation converges to the continuous-time solution with first-order rate as the sampling period tends to zero. A general uniform O(h) convergence theorem for discrete approximations of the matrix exponential is proved, covering common one-step methods. The results provide a rigorous consistency guarantee for digital optimal control design and numerical analysis of linear evolution equations.

 

References

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