4.1 Robust CF steady-state Kalman predictor
The CF system (29) and (35) with known conservative noise statistics Qf, Rf, and Sf are called worst-case conservative system. Under the conditions of Hypotheses 1–5, applying the standard Kalman filtering algorithm [3], for the worst-case conservative system, yields that the steady-state one-step Kalman predictor is given as
$$\hat{x}_{a} (t + 1|t) = \Psi_{ap} \hat{x}_{a} (t|t - 1) + K_{ap} y^{(c)} (t)$$
(43)
$$\varepsilon_{a} (t) = y^{(c)} (t) - H_{a}^{m} \hat{x}_{a} (t|t - 1)$$
(44)
$$\Psi_{ap} = \Phi_{a}^{m} - K_{ap} H_{a}^{m} ,\quad K_{ap} = \left[ {\Phi_{a}^{m} P_{a} ( - 1)H_{a}^{m\text T} + S_{f} } \right]Q_{\varepsilon a}^{ - 1} ,\quad Q_{\varepsilon a}^{{}} = H_{a}^{m} P_{a} ( - 1)H_{a}^{m\text T} + R_{f}$$
(45)
with the initial value \(\hat{x}_{a} (0| - 1) = \left[ {\begin{array}{*{20}c} {\mu_{0}^{\text T} } & {\left( {(0)_{r \times 1} } \right)^{\text T} } & {\left( {(0)_{m \times 1} } \right)^{\text T} } \\ \end{array} } \right]^{\text T}\), and \(\Psi_{ap}\) is stable.
The conservative steady-state prediction error variance Pa(− 1) satisfies the following steady-state Riccati equation
$$P_{a} ( - 1) = \Phi_{a}^{m} P_{a} ( - 1)\Phi_{a}^{m\text T} - \left[ {\Phi_{a}^{m} P_{a} ( - 1)H_{a}^{m\text T} + S_{f}^{{}} } \right]\times\left[ {H_{a}^{m} P_{a} ( - 1)H_{a}^{m\text T} + R_{f} } \right]^{ - 1} \left[ {\Phi_{a}^{m} P_{a} ( - 1)H_{a}^{m\text T} + S_{f}^{{}} } \right]^{\text T} + Q_{f}$$
(46)
Remark 1
The local observations yi(t), produced by the “worst-case” system (1)–(4), are called conservative local observations and are unavailable (unknown). Thus, the conservative CF observations y(c)(t), consisted by conservative local observations yi(t), are also unavailable. The observations yi(t) generated from the actual system (1)–(4) with the actual variances \(\overline{R}_{\eta } ,\overline{R}_{{g_{i} }} ,\overline{\sigma }_{{\xi_{k} }}^{2}\), and \(\overline{P}_{0}\) are called actual observations and are available (known). Furthermore, the actual CF observations y(c)(t), consisted by actual local observations yi(t), are also available. In (43), replacing the conservative CF observations y(c)(t) by the actual CF observations y(c)(t), the actual CF Kalman predictor can be obtained.
The steady-state prediction error is \(\tilde{x}_{a} (t + 1|t) = x_{a} (t + 1) - \hat{x}_{a} (t + 1|t)\), subtracting (43) from (29) yields
$$\tilde{x}_{a} (t + 1|t) = \Psi_{ap} \tilde{x}_{a} (t|t - 1) + w_{f} (t) - K_{ap} v_{f} (t) = \Psi_{ap} \tilde{x}_{a} (t|t - 1) + \left[ {I_{n + r + m} , - K_{ap} } \right]\lambda_{f} (t)$$
(47)
where
$$\lambda_{f} (t) = \left[ {\begin{array}{*{20}c} {w_{f}^{\text T} (t)} & {v_{f}^{\text T} (t)} \\ \end{array} } \right]^{\text T}$$
(48)
the actual and conservative steady-state variances of augmented noises \(\lambda_{f} (t)\) are, respectively, calculated by
$$\overline{\Lambda }_{f} = \left[ {\begin{array}{*{20}c} {\overline{Q}_{f} } & {\overline{S}_{f} } \\ {\overline{S}_{f}^{\text T} } & {\overline{R}_{f} } \\ \end{array} } \right],\quad \Lambda_{f} = \left[ {\begin{array}{*{20}c} {Q_{f} } & {S_{f} } \\ {S_{f}^{\text T} } & {R_{f} } \\ \end{array} } \right]$$
(49)
Furthermore, the actual and conservative CF steady-state prediction error variances satisfy the following Lyapunov equations, respectively,
$$\overline{P}_{a} ( - 1) = \Psi_{ap} \overline{P}_{a} ( - 1)\Psi_{ap}^{\text T} + \left[ {I_{n + r + m} , - K_{ap} } \right]\overline{\Lambda }_{f} \left[ {I_{n + r + m} , - K_{ap} } \right]^{\text T}$$
(50)
$$P_{a} ( - 1) = \Psi_{ap} P_{a} ( - 1)\Psi_{ap}^{\text T} + \left[ {I_{n + r + m} ,\; - K_{ap} } \right]\Lambda_{f} \left[ {I_{n + r + m} ,\; - K_{ap} } \right]^{\text T}$$
(51)
Lemma 4
[10] If \(\Theta \in R^{m \times m}\) is the positive semi-definite matrix, i.e., \(\Theta \ge 0\), and \(\Theta_{\delta } = \left( {\Theta_{ij} } \right) \in R^{mL \times mL} ,\Theta_{ij} = \Theta ,i,j = 1,2, \ldots ,L\), then \(\Theta_{\delta } \ge 0\).
Lemma 5
Under the conditions of Hypothesis 4, we have that.
$$\overline{\Lambda }_{f} \le \Lambda_{f}$$
(52)
Proof
Define \(\Delta \Lambda_{f} = \Lambda_{f} - \overline{\Lambda }_{f}\), utilizing (32), (38), and (42), we have that
$$\begin{aligned} \Delta \Lambda_{f} & = \left[ {\begin{array}{*{20}c} {\Delta Q_{f} } & {\Delta S_{f} } \\ {\Delta S_{f}^{T} } & {\Delta R_{f} } \\ \end{array} } \right] \\ & = \left[ {\begin{array}{*{20}c} \begin{gathered} \sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} \Phi_{a}^{\xi k} \Delta X_{a} \Phi_{a}^{\xi kT} } + \sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} \Phi_{a}^{\xi k} X_{a} \Phi_{a}^{\xi kT} } \hfill \\ + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \Phi_{a}^{\zeta i} \Delta X_{a} \Phi_{a}^{\zeta iT} + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} } \Phi_{a}^{ki} \Delta X_{a} \Phi_{a}^{kiT} \hfill \\ + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} } \Phi_{a}^{ki} X_{a} \Phi_{a}^{kiT} \hfill \\ + \Gamma_{a}^{m} \Delta Q_{a} \Gamma_{a}^{mT} + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \Gamma_{a}^{\zeta i} \Delta Q_{a} \Gamma_{a}^{\zeta iT} \hfill \\ \end{gathered} &\vline & \begin{gathered} \sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} \Phi_{a}^{\xi k} \Delta X_{a} H_{a}^{\xi kT} } + \sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} \Phi_{a}^{\xi k} X_{a} H_{a}^{\xi kT} } \hfill \\ + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} \Phi_{a}^{\zeta i} \Delta X_{a} H_{a}^{\zeta iT} } + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} \Phi_{a}^{ki} \Delta X_{a} H_{a}^{kiT} } \hfill \\ + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} \Phi_{a}^{ki} X_{a} H_{a}^{kiT} } \hfill \\ + \Gamma_{a}^{m} \Delta Q_{a} C_{a}^{mT} + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \Gamma_{a}^{\zeta i} \Delta Q_{a} C_{a}^{\zeta iT} \hfill \\ \end{gathered} \\ \hline \begin{gathered} \sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} H_{a}^{\xi k} \Delta X_{a} \Phi_{a}^{\xi kT} } + \sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} H_{a}^{\xi k} X_{a} \Phi_{a}^{\xi kT} } \hfill \\ + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} H_{a}^{\zeta i} \Delta X_{a} \Phi_{a}^{\zeta iT} } + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} H_{a}^{ki} \Delta X_{a} \Phi_{a}^{kiT} } \hfill \\ + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} H_{a}^{ki} X_{a} \Phi_{a}^{kiT} } \hfill \\ + C_{a}^{m} \Delta Q_{a} \Gamma_{a}^{mT} + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } C_{a}^{\zeta i} \Delta Q_{a} \Gamma_{a}^{\zeta iT} \hfill \\ \end{gathered} &\vline & \begin{gathered} \sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} H_{a}^{\xi k} \Delta X_{a} H_{a}^{\xi kT} } + \sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} H_{a}^{\xi k} X_{a} H_{a}^{\xi kT} } \hfill \\ + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} H_{a}^{\zeta i} \Delta X_{a} H_{a}^{\zeta iT} } + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} H_{k}^{ki} \Delta X_{a} H_{a}^{kiT} } \hfill \\ + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} H_{k}^{ki} X_{a} H_{a}^{kiT} } \hfill \\ + C_{a}^{m} \Delta Q_{a} C_{a}^{mT} + \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } C_{a}^{\zeta i} \Delta Q_{a} C_{a}^{\zeta iT} \hfill \\ \end{gathered} \\ \end{array} } \right] \\ \end{aligned}$$
Partition \(\Delta \Lambda_{f}\) into \(\Delta \Lambda_{f} = \Delta \Lambda_{f}^{(1)} + \Delta \Lambda_{f}^{(2)} + \cdots + \Delta \Lambda_{f}^{(7)}\), with the definitions
$$\begin{aligned} \Delta \Lambda_{f}^{(1)} & = \left[ {\begin{array}{*{20}c} {\sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} \Phi_{a}^{\xi k} \Delta X_{a} \Phi_{a}^{\xi kT} } } &\vline & {\sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} \Phi_{a}^{\xi k} \Delta X_{a} H_{a}^{\xi kT} } } \\ \hline {\sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} H_{a}^{\xi k} \Delta X_{a} \Phi_{a}^{\xi kT} } } &\vline & {\sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} H_{a}^{\xi k} \Delta X_{a} H_{a}^{\xi kT} } } \\ \end{array} } \right],\quad \Delta \Lambda_{f}^{(2)} = \left[ {\begin{array}{*{20}c} {\sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} \Phi_{a}^{\xi k} X_{a} \Phi_{a}^{\xi kT} } } &\vline & {\sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} \Phi_{a}^{\xi k} X_{a} H_{a}^{\xi kT} } } \\ \hline {\sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} H_{a}^{\xi k} X_{a} \Phi_{a}^{\xi kT} } } &\vline & {\sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} H_{a}^{\xi k} X_{a} H_{a}^{\xi kT} } } \\ \end{array} } \right], \\ \Delta \Lambda_{f}^{(3)} & = \left[ {\begin{array}{*{20}c} {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \Phi_{a}^{\zeta i} \Delta X_{a} \Phi_{a}^{\zeta iT} } &\vline & {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} \Phi_{a}^{\zeta i} \Delta X_{a} H_{a}^{\zeta iT} } } \\ \hline {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} H_{a}^{\zeta i} \Delta X_{a} \Phi_{a}^{\zeta iT} } } &\vline & {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} H_{a}^{\zeta i} \Delta X_{a} H_{a}^{\zeta iT} } } \\ \end{array} } \right],\quad \Delta \Lambda_{f}^{(4)} = \left[ {\begin{array}{*{20}c} {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} } \Phi_{a}^{ki} \Delta X_{a} \Phi_{a}^{kiT} } &\vline & {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} \Phi_{a}^{ki} \Delta X_{a} H_{a}^{kiT} } } \\ \hline {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} H_{a}^{ki} \Delta X_{a} \Phi_{a}^{kiT} } } &\vline & {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} H_{a}^{ki} \Delta X_{a} H_{a}^{kiT} } } \\ \end{array} } \right] \\ \Delta \Lambda_{f}^{(5)} & = \left[ {\begin{array}{*{20}c} {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} } \Phi_{a}^{ki} X_{a} \Phi_{a}^{kiT} } &\vline & {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} \Phi_{a}^{ki} X_{a} H_{a}^{kiT} } } \\ \hline {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} H_{a}^{ki} X_{a} \Phi_{a}^{kiT} } } &\vline & {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} H_{a}^{ki} X_{a} H_{a}^{kiT} } } \\ \end{array} } \right],\quad \Delta \Lambda_{f}^{(6)} = \left[ {\begin{array}{*{20}c} {\Gamma_{a}^{m} \Delta Q_{a} \Gamma_{a}^{mT} } &\vline & {\Gamma_{a}^{m} \Delta Q_{a} C_{a}^{mT} } \\ \hline {C_{a}^{m} \Delta Q_{a} \Gamma_{a}^{mT} } &\vline & {C_{a}^{m} \Delta Q_{a} C_{a}^{mT} } \\ \end{array} } \right] \\ \Delta \Lambda_{f}^{(7)} & = \left[ {\begin{array}{*{20}c} {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \Gamma_{a}^{\zeta i} \Delta Q_{a} \Gamma_{a}^{\zeta iT} } &\vline & {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \Gamma_{a}^{\zeta i} \Delta Q_{a} C_{a}^{\zeta iT} } \\ \hline {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } C_{a}^{\zeta i} \Delta Q_{a} \Gamma_{a}^{\zeta iT} } &\vline & {\sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } C_{a}^{\zeta i} \Delta Q_{a} C_{a}^{\zeta iT} } \\ \end{array} } \right] \\ \end{aligned}$$
Noting that \(\Delta \Lambda_{f}^{(1)} ,\Delta \Lambda_{f}^{(2)} ,\Delta \Lambda_{f}^{(3)} ,\Delta \Lambda_{f}^{(4)}\), and \(\Delta \Lambda_{f}^{(5)}\) can equivalently be expressed in the following forms
$$\begin{aligned} \Delta \Lambda_{f}^{(1)} & = \sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} B_{a}^{\xi k} \Delta X_{f} B_{a}^{\xi k\text T} } ,\quad \Delta \Lambda_{f}^{(2)} = \sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} B_{a}^{\xi k} X_{f} B_{a}^{\xi k\text T} } ,\quad \Delta \Lambda_{f}^{(3)} = \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} B_{a}^{\zeta i} \Delta X_{f} B_{a}^{\zeta i\text T} } \\ \Delta \Lambda_{f}^{(4)} & = \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\overline{\sigma }_{{\xi_{k} }}^{2} B_{a}^{ki} \Delta X_{f} B_{a}^{ki\text T} } ,\quad \Delta \Lambda_{f}^{(5)} = \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } \sum\limits_{k = 1}^{q} {\Delta \sigma_{{\xi_{k} }}^{2} } B_{a}^{ki} X_{f} B_{a}^{ki\text T} \\ \end{aligned}$$
where
$$\begin{aligned} B_{a}^{\xi k} & = \left[ {\begin{array}{*{20}c} {\Phi_{a}^{\xi k} } &\vline & {\left( 0 \right)_{(n + r + m) \times (n + r + m)} } \\ \hline {\left( 0 \right)_{m \times (n + r + m)} } &\vline & {H_{a}^{\xi k} } \\ \end{array} } \right],\quad B_{a}^{\zeta i} = \left[ {\begin{array}{*{20}c} {\Phi_{a}^{\zeta i} } &\vline & {\left( 0 \right)_{(n + r + m) \times (n + r + m)} } \\ \hline {\left( 0 \right)_{m \times (n + r + m)} } &\vline & {H_{a}^{\zeta i} } \\ \end{array} } \right] \\ B_{a}^{ki} & = \left[ {\begin{array}{*{20}c} {\Phi_{a}^{ki} } &\vline & {\left( 0 \right)_{(n + r + m) \times (n + r + m)} } \\ \hline {\left( 0 \right)_{m \times (n + r + m)} } &\vline & {H_{a}^{ki} } \\ \end{array} } \right],\quad \Delta X_{f} = \left[ {\begin{array}{*{20}c} {\Delta X_{a} } & {\Delta X_{a} } \\ {\Delta X_{a} } & {\Delta X_{a} } \\ \end{array} } \right],\quad X_{f} = \left[ {\begin{array}{*{20}c} {X_{a} } & {X_{a} } \\ {X_{a} } & {X_{a} } \\ \end{array} } \right] \\ \end{aligned}$$
the application of (27) and Lemma 4 yields \(\Delta X_{f} \ge 0\), which yields \(\Delta \Lambda_{f}^{(1)} \ge 0\), \(\Delta \Lambda_{f}^{(3)} \ge 0\), \(\Delta \Lambda_{f}^{(4)} \ge 0\). According to the positive semi-definiteness of variance matrix, and applying Lemma 4 yields \(X_{f} \ge 0\), which yields \(\Delta \Lambda_{f}^{(2)} \ge 0\) and \(\Delta \Lambda_{f}^{(5)} \ge 0\).
Additionally, \(\Delta \Lambda_{f}^{(6)}\) and \(\Delta \Lambda_{f}^{(7)}\) can be expressed as
$$\Delta \Lambda_{f}^{(6)} = D_{a}^{m} \Delta Q_{g} D_{a}^{m\text T} ,\quad \Delta \Lambda_{f}^{(7)} = \sum\limits_{i = 1}^{L} {\sigma_{{\zeta_{iz} }}^{2} } D_{a}^{\zeta i} \Delta Q_{g} D_{a}^{\zeta i\text T} .$$
where
$$D_{a}^{m} = \left[ {\begin{array}{*{20}c} {\Gamma_{a}^{m} } &\vline & {\left( 0 \right)_{(n + r + m) \times (r + m)} } \\ \hline {\left( 0 \right)_{m \times (r + m)} } &\vline & {C_{a}^{m} } \\ \end{array} } \right],\quad D_{a}^{\zeta i} = \left[ {\begin{array}{*{20}c} {\Gamma_{a}^{\zeta i} } &\vline & {\left( 0 \right)_{(n + r + m) \times (r + m)} } \\ \hline {\left( 0 \right)_{m \times (r + m)} } &\vline & {C_{a}^{\zeta i} } \\ \end{array} } \right],\quad \Delta Q_{g} = \left[ {\begin{array}{*{20}c} {\Delta Q_{a} } & {\Delta Q_{a} } \\ {\Delta Q_{a} } & {\Delta Q_{a} } \\ \end{array} } \right].$$
the application of (20) and Lemma 4 yields \(\Delta Q_{g} \ge 0\), which yields \(\Delta \Lambda_{f}^{(6)} \ge 0\) and \(\Delta \Lambda_{f}^{(7)} \ge 0\).
In conclusion, we obtain \(\Delta \Lambda_{f} = \Delta \Lambda_{f}^{(1)} + \Delta \Lambda_{f}^{(2)} + \cdots + \Delta \Lambda_{f}^{(7)} \ge 0\), i.e., (52) holds. This completes the proof. □
Lemma 6
[34] Consider the following Lyapunov equation.
$$U = CUC^{{\text{T}}} + V$$
where U, C and V are the n × n matrices, V is a symmetric matrix, C is a stable matrix (i.e., all its eigenvalues are inside the unit circle). If V ≥ 0, then U is symmetric and unique, and U ≥ 0.
Theorem 1
For the time-invariant augmented CF system (29) and (35), on the basis of Hypotheses 1–5, the actual CF steady-state Kalman predictor given by (43) is robust, i.e., for all admissible uncertainties, we have that.
$$\overline{P}_{a} ( - 1) \le P_{a} ( - 1)$$
(53)
, and Pa(− 1) is the minimal upper bound of \(\overline{P}_{a} ( - 1)\).
Proof
Letting \(\Delta P_{a}^{{}} ( - 1) = P_{a}^{{}} ( - 1) - \overline{P}_{a}^{{}} ( - 1)\), from (50) and (51) one has.
$$\Delta P_{a} ( - 1) = \Psi_{ap} \Delta P_{a} ( - 1)\Psi_{ap}^{\text T} + \Delta_{f} ,\quad \Delta_{f} = \left[ {I_{n + r + m} , - K_{ap} } \right]\Delta \Lambda_{f} \left[ {I_{n + r + m} , - K_{ap} } \right]^{\text T} ,$$
Using (52) yields \(\Delta_{f} \ge 0\). Noting that \(\Psi_{ap}\) is stable, accordingly, using Lemma 6 yields \(\Delta P_{a} ( - 1) \ge 0\), i.e., (53) holds. Taking \(\overline{R}_{\eta } = R_{\eta } ,\overline{R}_{{g_{i} }} = R_{{g_{i} }} ,\overline{\sigma }_{{\xi_{k} }}^{2} = \sigma_{{\xi_{k} }}^{2}\), and \(\overline{P}_{0} = P_{0}\), then the Hypothesis 4 still holds. From \(\overline{R}_{{g_{i} }} = R_{{g_{i} }}\) , one has \(\overline{R}_{g}^{(c)} = R_{g}^{(c)}\), further, we have that \(\overline{Q}_{a} = Q_{a}\). From \(\overline{X}(0) = \overline{P}_{0} + \mu_{0} \mu_{0}^{\text T}\) and \(X(0) = P_{0} + \mu_{0} \mu_{0}^{\text T}\) , we get that \(\overline{X}(0) = X(0)\), furthermore, we have that \(\overline{X}_{a} (0) = X_{a} (0)\). By way of recurrence, it is easy to prove that \(X_{a} (t) = \overline{X}_{a} (t)\). From Lemma 3, we have that \(X_{a} = \overline{X}_{a}\). Comparing (30) and (31) yields \(Q_{f} = \overline{Q}_{f}\), comparing (36) and (37) yields \(R_{f} = \overline{R}_{f}\), comparing (40) and (41) yields \(S_{f} = \overline{S}_{f}\). Accordingly, from (49) we obtain that \(\Lambda_{f} = \overline{\Lambda }_{f}\), further, we have that \(\Delta_{f} = 0\). Applying Lemma 6 yields \(\Delta P_{a} ( - 1) = 0\), that is \(\overline{P}_{a}^{{}} ( - 1) = P_{a}^{{}} ( - 1)\). If \(P_{a}^{*}\) is an arbitrary other upper bound of \(\overline{P}_{a}^{{}} ( - 1)\), then \(P_{a}^{{}} ( - 1) = \overline{P}_{a}^{{}} ( - 1) \le P_{a}^{*}\), this means that Pa(− 1) is the minimal upper bound of \(\overline{P}_{a}^{{}} ( - 1)\). The proof is completed. □
The actual CF steady-state Kalman predictor given by (43) is called robust CF steady-state Kalman predictor. The relation given by (53) is called its robustness.
4.2 Robust CF steady-state Kalman filter and smoother
For the worst-case time-invariant CF system (29) and (35) with conservative noise statistics Qf, Rf, and Sf, based on the actual CF steady-state Kalman one-step predictor \(\hat{x}_{a}^{{}} (t|t - 1)\), the actual CF steady-state Kalman filter (N = 0) and smoother (N > 0) \(\hat{x}_{a}^{{}} (t|t + N)\) are given as [35]
$$\hat{x}_{a}^{{}} (t|t + N) = \hat{x}_{a}^{{}} (t|t - 1) + \sum\limits_{k = 0}^{N} {K_{ap} (k)\varepsilon_{a}^{{}} (t + k)} ,\quad N \ge 0$$
(54)
$${\rm K}_{ap} (k) = P_{a} ( - 1)\Psi_{ap}^{{{\rm T}k}} H_{a}^{{m{\rm T}}} Q_{\varepsilon a}^{ - 1} ,\quad k \ge 0$$
(55)
Similar to the derivation in [35], the steady-state filtering and smoothing errors \(\tilde{x}_{a}^{{}} (t|t + N) = x_{a} (t) - \hat{x}_{a}^{{}} (t|t + N)\) are given as
$$\tilde{x}_{a}^{{}} (t|t + N) = \Psi_{aN}^{{}} \tilde{x}_{a}^{{}} (t|t - 1) + \sum\limits_{\rho = 0}^{N} {\left[ {K_{a\rho }^{Nw} ,K_{a\rho }^{Nv} } \right]} \lambda_{f} (t + \rho )$$
(56)
where \(\lambda_{f} \left( {t + p} \right)\) is defined by (48), and
$$\begin{aligned} \Psi_{aN}^{{}} & = I_{(n + r + m) \times (n + r + m)} - \sum\limits_{k = 0}^{N} {K_{ap} (k)H_{a}^{m} \Psi_{ap}^{k} } , \\ K_{a\rho }^{Nw} & = - \sum\limits_{k = \rho + 1}^{N} {K_{ap} (k)H_{a}^{m} \Psi_{ap}^{k - \rho - 1} } ,\quad N > 0,\quad \rho = 0, \ldots ,N - 1,\quad K_{aN}^{Nw} = 0,\quad N \ge 0,\quad \rho = N, \\ K_{a\rho }^{Nv} & = \sum\limits_{k = \rho + 1}^{N} {K_{ap} (k)H_{a}^{m} \Psi_{ap}^{k - \rho - 1} K_{ap} - K_{ap} } (\rho ),\quad N > 0,\quad \rho = 0, \ldots ,N - 1,\quad K_{aN}^{Nv} = - K_{ap} (N),\quad N \ge 0,\quad \rho = N \\ \end{aligned}$$
Utilizing (56) yields that the actual and conservative steady-state estimation error variances are, respectively, computed by
$$\overline{P}_{a}^{{}} (N) = \Psi_{aN}^{{}} \overline{P}_{a}^{{}} ( - 1)\Psi_{aN}^{\text T} + \sum\limits_{\rho = 0}^{N} {\left[ {K_{a\rho }^{Nw} ,K_{a\rho }^{Nv} } \right]\overline{\Lambda }_{f} \left[ {K_{a\rho }^{Nw} ,K_{a\rho }^{Nv} } \right]^{\text T} } ,\quad N \ge 0$$
(57)
$$P_{a}^{{}} (N) = \Psi_{aN}^{{}} P_{a}^{{}} ( - 1)\Psi_{aN}^{\text T} + \sum\limits_{\rho = 0}^{N} {\left[ {K_{a\rho }^{Nw} ,K_{a\rho }^{Nv} } \right]\Lambda_{f} \left[ {K_{a\rho }^{Nw} ,K_{a\rho }^{Nv} } \right]^{\text T} } ,\quad N \ge 0$$
(58)
Theorem 2
Under the conditions of Hypotheses 1–5, the actual CF steady-state Kalman filter and smoother given by (54) are robust, i.e.,
$$\overline{P}_{a}^{{}} (N) \le P_{a}^{{}} (N),\quad N \ge 0$$
(59)
, and Pa(N) is the minimal upper bound of \(\overline{P}_{a}^{{}} (N)\).
Proof
Letting \(\Delta P_{a}^{{}} (N) = P_{a}^{{}} (N) - \overline{P}_{a}^{{}} (N)\), from (57) and (58) one has.
$$\Delta P_{a}^{{}} (N) = \Psi_{aN}^{{}} \Delta P_{a}^{{}} ( - 1)\Psi_{aN}^{\text T} + \sum\limits_{\rho = 0}^{N} {\left[ {K_{a\rho }^{Nw} ,K_{a\rho }^{Nv} } \right]\Delta \Lambda_{f} \left[ {K_{a\rho }^{Nw} ,K_{a\rho }^{Nv} } \right]^{\text T} }$$
utilizing (52) and (53) yields \(\Delta P_{a}^{{}} (N) \ge 0\), i.e., (59) holds. In a similar way to the proof of Theorem 1, we can prove that Pa(N) is the minimal upper bound of \(\overline{P}_{a}^{{}} (N)\), the details are omitted. The proof is completed. □
Corollary 1
From the definition \(x_{a} (t) = \left[ {\begin{array}{*{20}c} {x^{\text T} (t)} & {w^{\text T} (t)} & {\delta^{(c)\text T} (t - 1)} \\ \end{array} } \right]^{\text T}\), the robust CF steady-state Kalman estimator of the original system (1)–(4) can be obtained as \(\hat{x}_{c} (t|t + N) = \left[ {\begin{array}{*{20}c} {I_{n} } & {(0)_{n \times r} } & {(0)_{n \times m} } \\ \end{array} } \right]\hat{x}_{a}^{{}} (t|t + N),N = - 1,N \ge 0\), and their actual and conservative CF steady-state estimation error variances are, respectively, given as.
$$\overline{P}_{c} (N) = \left[ {\begin{array}{*{20}c} {I_{n} } & {(0)_{n \times r} } & {(0)_{n \times m} } \\ \end{array} } \right]\overline{P}_{a}^{{}} (N)\left[ {\begin{array}{*{20}c} {I_{n} } & {(0)_{n \times r} } & {(0)_{n \times m} } \\ \end{array} } \right]^{\text T}$$
(60)
$$P_{c} (N) = \left[ {\begin{array}{*{20}c} {I_{n} } & {(0)_{n \times r} } & {(0)_{n \times m} } \\ \end{array} } \right]P_{a}^{{}} (N)\left[ {\begin{array}{*{20}c} {I_{n} } & {(0)_{n \times r} } & {(0)_{n \times m} } \\ \end{array} } \right]^{\text T}$$
(61)
the robust CF steady-state Kalman estimators \(\hat{x}_{c} (t|t + N)\) are robust, i.e.,
$$\overline{P}_{c} (N) \le P_{c} (N),\quad N = - 1,\quad N \ge 0$$
(62)
, and Pc(N) is the minimal upper bound of \(\overline{P}_{c} (N)\).
Corollary 2
It is completely similar to the derivation of (7)–(62), we easily obtain the robust local steady-state Kalman estimators \(\hat{x}_{i} (t|t + N),N = - 1,N \ge 0,i = 1, \ldots ,L\), of the original system (1)–(4), and their actual estimation error variances \(\overline{P}_{i} (N)\) have the corresponding minimal upper bounds Pi(N), i.e.,
$$\overline{P}_{i} (N) \le P_{i} (N),\quad N = - 1,\quad N \ge 0,\quad i = 1, \ldots ,L$$
(63)
Remark 2
Applying the projection theory, it can be proved that
$$P_{c} (N) \le P_{i} (N),\quad N = - 1,\quad N \ge 0,\quad i = 1, \ldots ,L$$
(64)
$$P_{c} (N) < P_{c} (N - 1) < \cdots < P_{c} (1) < P_{c} (0) < P_{c} ( - 1),\quad N \ge 1$$
(65)
Taking the trace operations to (62)–(65) yields the accuracy relations with the matrix trace inequalities as
$${\text{tr}}\overline{P}_{c} (N) \le {\text{tr}}P_{c} (N),\quad {\text{tr}}\overline{P}_{i} (N) \le {\text{tr}}P_{i} (N),\quad {\text{tr}}P_{c} (N) \le {\text{tr}}P_{i} (N),\quad N = - 1,\quad N \ge 0,\quad i = 1, \ldots ,L$$
(66)
$${\text{tr}}P_{c} (N) < {\text{tr}}P_{c} (N - 1) < \cdots < {\text{tr}}P_{c} (1) < {\text{tr}}P_{c} (0) < {\text{tr}}P_{c} ( - 1),\quad N \ge 1$$
(67)
Remark 3
In Remark 2, \({\text{tr}}\overline{P}_{c} (N)\) and \({\text{tr}}\overline{P}_{i} (N)\) are defined as the actual accuracies of the corresponding robust Kalman estimators, while trPc(N) and trPi(N) are defined as their robust accuracies (or global accuracies). The smaller trace means the higher accuracy. The robust accuracy of CF is higher than that of each local estimator.