Optimal Observer of a Stochastic Decentralized Singularly Perturbed Unified System

For the stochastic decentralized discrete-time system [12], consider the following system model acquired in [5].

where x_i(k)∈Rⁿ is the state vector of the system, u_i(k)∈R^p and y_i(k)∈R^q are the system input and output vectors, respectively. The matrices A_q0i, B_q0i, C_q0i, and D_q0i are constant matrices of appropriate demensions. Independent vectors of Gaussian noise are w_i(k)∈R and v_i(k)∈R with zero mean and intensity V_1q and V_2q, respectively.

In this case of a discrete-time system, original system noise and noise for state monitoring are presumed to be white Gaussian noise having zero mean Σ_w > 0 and intensity Σ_v > 0. The correlation function is described as

In accordance with stochastic control method for observers [14], a state evaluator is able to be formulated for individual subsystem. Because the control input u_i(k) is deterministic, the state evaluator can be described as

Then we have the goal here to discover the optimal observer gain L_q0i(k) that eliminate the noise involved in the system output signals. Here, we adapt the orthogonality priciple [14]:

A necessary condition for a linear estimate to be optimal is that the estimation error be orthogonal(in the probabilistic sense) to al of the data:

Utilizing the orthogonality principle indicates that

where y_i(k) = C_q0ix_i(k) + S_q0iw_i(k) + F_q3iv_i(k).

Using y_i(k) into equation (5) gives

Considering $$ {\widehat{x}}_i\left(k\right)$$ is optimal, $$ E\left[\left\{x_i\left(k+1\right)-{\widehat{x}}_i\left(k+1\right)\right\}{y}_i^T\left(k\right)\right]=0$$ for n<k. As original system noise and state monitoring noise are uncorrelated with past state monitoring, $$ E\left[w_i\left(k\right)y_i^T\left(n\right)\right]=E\left[{\upsilon }_i\left(k\right)y_i^T\left(n\right)\right]=0$$ So, the orthogonality principle is validated to all data such that n < k. For n = k,

Associating these outcome generates

Then, the optimal observer gain is given as

where

Because of the evaluation of the state, the error is described as $$ e_i\left(k\right)=x_i\left(k\right)-{\widehat{x}}_i\left(k\right)$$. Then, the following equation is generated as

For a discrete-time system, the following evaluation covariance matrix for error is demanded to stipulate the Kalman gain. The error model is able to determine this covariance matrix. White noise is the Input to the error model here. Using this outcome to the error model generates

With deploying equation (11), the equation for the error covariance matrix e_i(k) becomes

Now we have a conclusion that the Σ_ei(optimal numerical figure) is achieved by converting the presented Lyapunov equation to the Riccati equation. Then the optimal value of Σ_ei is able to be completed by choosing the optimal Kalman gain, $$ L_{q0i}=\left(A_{q0i}{\sum }_{ei}{C}_{q0i}^T+F_{q0i}{\sum }_w{S}_{q0i}^T\right)V_{q0i}^{-1}$$. From equation (12), the optimal covariance matrix of the Σ_ei is able to be acquired by

where

Ultimately, we can achieve the optimal stochastic observer equation as follows

where L_q0i is the optimal stochastic observer parameter, and $$ L_{q0i}=\left(A_{q0i}{\sum }_{ei}{C}_{q0i}^T+F_{q0i}{\sum }_w{S}_{q0i}^T\right)V_{0i}^{-1}$$, Σ_ei = K_ei.

In equation (14), Σ_ei denotes the solution of the algebraic discrete Riccati equation.

In [5], for devising the steadying optimal controllers, the discrete-time Riccati equation was applied. In this part, by using the Riccati equation method, the steadying observer gain is able to settle the observers when the state parameters are not attainable for evaluation. If the discrete-time system is

Now we have the closed-loop system as the form of

and the LQ performance index is given by

The optimal control for subsystem one is

where $$ G_{0i}={\left(R_{q0i}+B_{0i}^T{K}_i{B}_{q0i}\right)}^{-1}\left(M_{q0i}^T+B_{q0i}^T{K}_i{A}_{q0i}\right).$$

In equation (19), K_i denotes the solution of the fundamental Riccati equation. Then we have

The control feedback gain in equation (19) is the gain that is able to steady the system equation (16) too.

Then, the observer can be described as

The poles of the matrix (A_q0i-L_q0iC_q0i) are the observer eigenvalues. And (A_q0i-L_q0iC_q0i) and (A_q0i-L_q0iC_q0i)^T have the same eigenvalues. So

The following equation is derived by means of applying the similarities between equations (17) and (21).

Now we can attain the steadying observer gain for sub-system one can be attained from

where

So, the $$ L_{q0i}^T$$ can place the poles of the matrix (A_q0i-L_q0iC_q0i)^T inward the circle of the z-plane where the radius of the circle is unit. Because the sign of transpose does not change the pole placement of a matrix, the observer gain L_q0i is able to steady the matrix (A_q0i-L_q0iC_q0i).

By applying delta operator approach [15], a stochastic decentralized unified system can be given by

where $$ A_{\rho 0i}=\frac{A_{q0i}-I}{∆}, B_{\rho 0i}=\frac{B_{q0i}}{∆}, F_{\rho 0i}=\frac{F_{q0i}}{∆}, C_{\rho 0i}=C_{q0i},D_{\rho 0i}=D_{q0i},S_{\rho 0i}=S_{q0i},F_{\rho 3i}=F_{q3i}. $$ And x_i(τ)∈Rⁿ is the state vector of the system, u_i(τ)∈R^p and y_i(τ)∈R^q are the system input and output vectors, respectively. The matrices A_ρ0i, B_ρ0i, C_ρ0i, and D_ρ0i are constant matrices of appropriate demensions. Independent vectors of Gaussian noise are w_i(τ)∈R and v_i(τ)∈R with zero mean and intensity V_1ρ and V_2ρ, respectively.

It can be presumed that the noise to represent white Gaussian noise having zero mean Σ_w > 0 and intensity Σ_v > 0 where the sampling period is Δ. Such noises are presumed to hold the subsequent correlation functions.

In accordance with stochastic control method for observers [14], a state evaluator is able to be formulated for individual sub-system. Because the control input u_i(τ) is deterministic, the state evaluator can be described as

Then we have the goal here is to discover the optimal observer gain L_ρ0i that remove the noise involved in the system output signals. With applying the optimal observer gain in equation (14), the optimal observer gain can be acquired as

where

From equation (28), the Σ_ρei(covariance matrix of the error) can be rewritten as subsequent equation

Ultimately, we can represent the optimal stochastic observer equations as follows

where

Fig. 1 illustrates that the reduced-order optimal observer in unified system for the ith sub-system.

Fig. 1.

Optimal observer of a decentralized singularly perturbed unified system with ith reduced-order

Assuming that the unified system is given by

then the closed-loop unified system is written as

and the LQ performance index is given by

The optimal control for sub-system one is

where G_ρ0i =

In equation (34), K_i denotes the solution of the unified Riccati equation. Then we have

The control feedback gain in equation (34) is the gain that can steady the system equation (31) too.

Then, the observer can be described as

Now we can recognize that the eigenvalues of (A_ρ0i-L_ρ0iC_ρ0i) and (A_ρ0i-L_ρ0iC_ρ0i)^T are the same, so

The following equation is derived by means of applying the similarities between equations (32) and (36).

The steadying gain of the observer for sub-system one can be achieved from

where

So, the $$ L_{\rho 0i}^T$$ can place the poles of the matrix (A_ρ0i-L_ρ0iC_ρ0i)^T within the circle of the plane with the stable region of radius $$ \frac{1}{∆}$$. Since the sign of transpose does not change the pole placement of a matrix. It means that the observer gain L_ρ0i can steady the matrix (A_ρ0i-L_ρ0iC_ρ0i).

The diagrams illustrating stability region [16] for different operators are given in Fig. 2. The variables s, z, and γ denote transformed variables in discrete-time and unified systems.

Fig. 2.

Stability region and diagrams for discrete-time and unified systems