Randomized LTI Systems

LEARNING AND CONTROL IN LINEAR TIME-VARYING SYSTEMS

4.3 Stability and Identification of Random Asynchronous LTI Systems In this section, we introduce the random asynchronous LTI systems, a new form ofIn this section, we introduce the random asynchronous LTI systems, a new form of

4.3.3 Randomized LTI Systems

Since𝑞 = 1, randomized LTI systems do not have any delays or asynchrony in the system, i.e., if the state element getting updated it has access to the most recent information on all states. The random delay probabilities reduces toP[𝑘_𝑗 =0] =1 for all 𝑗. Thus, we get the following model:

(𝑥_𝑡+1)𝑖 =











(A𝑥_𝑡+B𝑢_𝑡+𝑤_𝑡)𝑖, w.p. 𝑝, (𝑥_𝑡+𝑤_𝑡)𝑖, w.p. 1− 𝑝 .

(4.29)

This model corresponds to the setting first introduced in Teke and Vaidyanathan [261]. In the following, we consider the properties of randomized LTI systems.

Markov Parameters

Markov parameters of an LDS is the unique matrix impulse response of the system.

For a synchronous LTI system, the Markov parameters of the system (A,B) are given asH^𝑘 =A^𝑘−1Bfor 𝑘 ≥ 1. From the input-output viewpoint, the randomized updates on the system with (A,B) can be represented in an average sense as a synchronous LTI system with parameters (A,B) where the average state-transition matrixAand the average input matrixBare given as follows:

A= 𝑝A+ (1− 𝑝)I^𝑑𝑥, B= 𝑝B. (4.30) As a result, Markov parameters of the average system can be obtained as H^𝑘 = A

𝑘−1

B, for 𝑘 ≥ 1. Notice that Markov parameters of the underlying system and the randomized system can be directly obtained from each other. The 𝑘^{𝑡 ℎ} Markov parameter of the randomized system,H^𝑘, can be written as a linear combination of the first𝑘 Markov parameters of the synchronous system:

H^𝑘= 𝑝A+ (1− 𝑝)I^𝑑𝑥𝑘−1

𝑝B=∑︁𝑘 𝑖=1

𝑘−1 𝑖−1

𝑝^𝑖(1− 𝑝)^𝑘−𝑖H^𝑖.

More generally, define the first𝐾Markov parameters matricesG= [H1H2 · · · H^𝐾] andG= [H1 H2 · · · H^𝐾]. We haveG=G(T⊗I^𝑑𝑢), whereT∈R^𝐾×𝐾is an upper triangular matrix with T^{𝑖, 𝑗} = ^𝑗_𝑖−⁻¹

1

𝑝^𝑖(1−𝑝)^𝑗−𝑖 for 𝑗 ≥ 𝑖 ≥ 1. Notice that T does not depend on the system parameters (A,B), and it is determined solely by the probability 𝑝. Moreover, T has diagonal entries (thus eigenvalues) ofT^𝑖,𝑖 = 𝑝^𝑖−1. Thus, T is always invertible since we trivially assumed that 𝑝 > 0. This shows that once the average system behavior and the rate of updates are known, one can identify the underlying system parameters exactly. When the update probability is 𝑝 =1 (synchronous), we getT=I^𝐾 so thatG=G. Note that the properties above could be trivially extended to measurement feedback systems, i.e., randomized partially observed LTI systems.

Mean-Squared Stability

In this section, we precisely characterize the mean-square stability of randomized LTI systems which corresponds to the rightmost vertical axes of plots in Figure 4.4.

As discussed above, the dynamics of the randomized LTI system is determined by

the matrix A in an average sense. The stability of the matrix A is necessary, but not sufficient for the stability of the system. In order to analyze the mean-square stability, we look for the condition that ensures E[𝑥

𝑡𝑥^⊤

𝑡 ] stays finite as 𝑡 → ∞. In fact, the steady-state covariance matrix, i.e., lim𝑡→∞E[𝑥

𝑡𝑥^⊤

𝑡 ] =𝚪, can be found as the solution of the followingextended Lyapunov equation introduced in Teke and Vaidyanathan [262]:

𝚪=𝜙(𝚪) +B U B

⊤+1 𝑝

−1 B U B

⊤

⊙I+𝜎²

𝑤I, (4.31) where the function𝜙(·)is defined as follows:

𝜙(𝑥)=A𝑥A

⊤+ (𝑝−𝑝²) (A−I)𝑥(A^⊤−I)

⊙I=A𝑥A

⊤+ 1

𝑝

−1

(A−I)𝑥(A

⊤−I)

⊙I. The function 𝜙(·) is a positive linear map that controls the evolution of the state covariance matrix in the extended Lyapunov equation. It can be vectorized as vec(𝜙(𝑥)) =S vec(𝑥)where

S=A⊗A+ (𝑝−𝑝²)J(A−I) ⊗ (A−I), (4.32) forJ =Í^𝑑𝑥

𝑖=1 𝑒_𝑖𝑒^⊤

𝑖

⊗ 𝑒_𝑖𝑒^⊤

𝑖

. S ∈R^𝑑

2 𝑥×𝑑²_𝑥

is the matrix representation of the linear map 𝜙(·) and corresponds to S^ℎ introduced in Section 4.3.2 at ℎ = 0. Note that to extend this to complex valued systems, (4.32) needs element-wise conjugate operations on the left-side of Kronecker products.2 Recall that in Section 4.3.2, the mean-square stability of the random asynchronous LTI systems is demonstrated numerically. However, with the closed-form expression of S in (4.32), we can analytically characterize the stability of the randomized LTI system.

Lemma 4.4. [260, 262] The randomized LTI systems given in (4.29) are mean- square stable if and only if𝜌(S) < 1.

This result is due to the fact that one can recursively represent the covariance matrix of the state variables at time 𝑡+1 as a function of the covariance matrix at time 𝑡. Since S represents this mapping, stability of S is a necessary and sufficient condition for the convergence of the covariance matrix, which implies mean-square stability for the state variables. The key observation in Lemma 4.4 is that the

2Element-wise conjugation ensures thatSalways has a real nonnegative eigenvalue that is equal to its spectral radius, and the corresponding eigenvector is the vectorized version of a positive semidefine matrix. This follows from the extensions of the Perron-Frobenius theorem to positive maps in more general settings, Theorem 5 of Karlin [139].

mean-square stability of the randomized LTI system and the stability of Ado not imply each other, i.e.,𝜌(S)<1 and𝜌(A)<1 are not equivalent in general. Note that Lemma 4.4 provides the precise characterization of the rightmost axes of the plots in Figure 4.4.

In order to visualize the convergence behavior of the randomized updates, we consider a numerical test example of size 𝑑_𝑥 =2 with a constant input (i.e., fixed- point iteration), and initialize𝑥₀with independent Gaussian random variables (left- most block in Figure 4.5). Then, the distribution of the state vector 𝑥_𝑡 (at time 𝑡) follows a Gaussian mixture model (GMM) due to the randomized nature of the updates (See Figure 4.5). Furthermore, the stability of the matrix S ensures that the mean of𝑥_𝑡 converges to the fixed-point of the system while the variance of 𝑥_𝑡 converges to zero.

The key insight to the convergence behavior in Figure 4.5 is as follows: When rep- resented as a switching system, the randomized LTI model (4.29) switches between 2^𝑑𝑥 systems randomly, and it can be shown that all these 2^𝑑𝑥 systems (including the original system) have the same fixed-point. It should also be noted that not all 2^𝑑𝑥 systems are stable by themselves, and an arbitrary switching does not necessarily ensure the convergence. Nevertheless, with a careful selection of the probability, the randomized model can obtain convergence even when the synchronous system is unstable.

When the system is mean-square stable, the steady-state covariance matrix, 𝚪, is given as

vec(𝚪) =(I−S)⁻¹ 𝑝²I+ (𝑝− 𝑝²)J

vec(B U B^⊤) +𝜎²

𝑤vec(I)

. (4.33) When𝑝 =1, we have𝜙(𝑥) =A𝑥A^⊤, which implies that𝚪=A𝚪A^⊤+B U B^⊤+𝜎²

𝑤I and 𝜌(S) =𝜌²(A). So, we have 𝜌(S) < 1 if and only if𝜌(A) < 1 for synchronous LTI systems, which recovers the well-known stability result in the classical systems theory.