A Constrained Optimization Approach to Integrated Active Fault Detection and Control

Abstract

In this paper, a novel approach is proposed to formulate and solve the problem of active fault detection and control for multi-model discrete-time systems. The main objectives are to design an optimal fixed-order controller for stabilizing the healthy and faulty models by minimizing a well-defined quadratic performance index, meanwhile synthesizing a test signal for active fault detection in the presence of bounded energy uncertainty. To do this, an optimal solution is proposed to obtain the trade-off between the optimal fixed-order stabilizing controller signal and optimal test signal for active fault detection. These objectives could be achieved by solving a finite-dimensional constrained optimization problem. The dynamic nonlinear optimization problem is solved by two constructive recursive solution algorithms which are finally applied to a numerical example. Simulation results show the effectiveness of the proposed algorithms.

Appendices

Appendices

1.1 Appendix 1: Proof 4.3

Let ${\mathcal{L}} = \mathop \sum \limits_{i = 1}^{2} {\text{Tr}}[W_{i} Z_{i} ] + \mathop \sum \limits_{i = 1}^{2} {\text{Tr}}[ \varGamma_{i} (\bar{A}_{i}^{\text{T}} W_{i} \bar{A}_{i} - W_{i} + S)]$, where the positive definite matrix $_{i}$ is $i{\text{th}}$ Lagrangian multiplayer. The following sufficient conditions for existence of optimal fixed-order DOF controller are given by:

$${\begin{array}{*{20}c} {1.\quad \frac{{\partial {\mathcal{L}}}}{{\partial W_{i} }} = Z_{i} + (\bar{A}_{i} )\varGamma_{i} (\bar{A}_{i}^{{\text{T}}} ) - _{i} = 0,i = 1,2} \\ {2.\quad \frac{{\partial {\mathcal{L}}}}{{\partial _{i} }} = S + (\bar{A}_{i}^{{\text{T}}} )W_{i} (\bar{A}_{i} ) - W_{i} = 0,i = 1,2} \\ {3.\quad \frac{{\partial {\mathcal{L}}}}{{\partial E}} = \sum\limits_{{i = 1}}^{2} {(\tilde{B}_{{\bar{M}_{i}^{{\text{T}}} }} W_{i} \bar{M}_{i} \gamma _{0} \tilde{C}_{{\bar{M}_{i}^{{\text{T}}} }} + \tilde{B}_{{\bar{A}_{i}^{{\text{T}}} }} W_{i} \bar{A}_{i} \Gamma _{i} \tilde{C}_{{\bar{A}_{i}^{{\text{T}}} }} + \hat{R}E^{{\text{T}}} T \varGamma_{i} T^{{\text{T}}} )} = 0} \\ \end{array} }$$

(24)

Now, by solving the set of coupled nonlinear equations (24) simultaneously, the optimal DOF controller can be obtained. According to Lagrange’s theorem, to achieve the minimum point, the unknown matrices $W_{i} \quad {\text{and}}\quad \varGamma_{i}$ should be positive definite. To complete the optimal DOF controller synthesis, the following theorem is discussed.

1.2 Appendix 2: Proof 4.4

By extracting (6.3), we have

$$\left\{ {\begin{array}{*{20}l} {1.\quad \sum\limits_{i = 1}^{2} {\left[ \begin{aligned} B_{i}^{\text{T}} W_{{11_{i} }} M_{i} \gamma_{0} N_{i}^{\text{T}} + B_{i}^{\text{T}} W_{{11_{i} }} A_{i} \varGamma_{{11_{i} }} C_{i}^{\text{T}} + B_{i}^{\text{T}} W_{{11_{i} }} B_{i} G\varGamma_{{21_{i} }} C_{i}^{\text{T}} \hfill \\ + B_{i}^{\text{T}} W_{{12_{i} }} KN_{i} \gamma_{0} N_{i}^{\text{T}} + B_{i}^{\text{T}} W_{{12_{i} }} KC_{i} \varGamma_{{11_{i} }} C_{i}^{\text{T}} + B_{i}^{\text{T}} W_{{12_{i} }} (P + KDG)\varGamma_{{21_{i} }} C_{i}^{\text{T}} \hfill \\ \end{aligned} \right] = 0} } \hfill \\ {2.\quad \sum\limits_{i = 1}^{2} {\left[ \begin{aligned} RG\varGamma_{{22_{i} }} + B_{i}^{\text{T}} W_{{11_{i} }} A_{i} \varGamma_{{12_{i} }} + B_{i}^{\text{T}} W_{{11_{i} }} B_{i} G\varGamma_{{22_{i} }} \hfill \\ + B_{i}^{\text{T}} W_{{12_{i} }} KC_{i} \varGamma_{{12_{i} }} + B_{i}^{\text{T}} W_{{12_{i} }} (P + KDG)\varGamma_{{22_{i} }} \hfill \\ \end{aligned} \right] = 0} } \hfill \\ {3. \quad \mathop \sum \limits_{i = 1}^{2} \left[ {\begin{array}{*{20}c} {W_{{21_{i} }} M_{i} \gamma_{0} N_{i}^{\text{T}} + W_{{21_{i} }} A_{i} \varGamma_{{11_{i} }} C_{i}^{\text{T}} + W_{{21_{i} }} B_{i} G \varGamma_{{21_{i} }} C_{i}^{\text{T}} + W_{{22_{i} }} KN_{i} \gamma_{0} N_{i}^{T} } \\ { + W_{{22_{i} }} KC_{i} \varGamma_{{11_{i} }} C_{i}^{\text{T}} + W_{{22_{i} }} (P + KDG)\varGamma_{{21_{i} }} C_{i}^{\text{T}} } \\ \end{array} } \right] = 0 } \hfill \\ {4. \quad \mathop \sum \limits_{i = 1}^{2} \left[ {\begin{array}{*{20}c} {W_{{21_{i} }} A_{i} \varGamma_{{12_{i} }} +W_{{22_{i} }} (P + KDG)\varGamma_{{21_{i} }} C_{i}^{\text{T}} } \\ \end{array} } \right] = 0 } \hfill \\ \end{array} } \right.$$

(25)

With regard to (25.1)–(25.4) for $i = 1,2 ,$ the new variables are rewritten as follow:

$$\left\{ {\begin{array}{*{20}l} {1. \check{S}_{1} + \mathop \sum \limits_{i = 1}^{2} \check{a}_{i} G\check{b}_{i} + \check{c}_{i} K\check{d}_{i} + \check{c}_{i} (P + KDG)\check{b}_{i} = 0} \hfill \\ {2. \check{S}_{3} + \mathop \sum \limits_{i = 1}^{2} \check{g}_{i} G\check{m}_{i} + \check{c}_{i} K\check{h}_{i} + \check{c}_{i} (P + KDG)\check{m}_{i} = 0} \hfill \\ {3. \check{S}_{2} + \mathop \sum \limits_{i = 1}^{2} \check{e}_{i} G\check{b}_{i} + \check{f}_{i} K\check{d}_{i} + \check{f}_{i} (P + KDG)\check{b}_{i} = 0 } \hfill \\ {4. \check{S}_{4} + \mathop \sum \limits_{i = 1}^{2} \check{e}_{i} G\check{m}_{i} + \check{f}_{i} K\check{h}_{i} + \check{f}_{i} (P + KDG)\check{m}_{i} = 0} \hfill \\ \end{array} } \right.$$

(26)

where

$$\begin{aligned} \check{S}_{1} & = - \mathop \sum \limits_{i = 1}^{2} [B_{i}^{\text{T}} W_{{11}_i} M_{i} \gamma_{0} N_{i}^{\text{T}} + B_{i}^{\text{T}} W_{{11}_i} A_{i} \varGamma_{{11}_i} C_{i}^{\text{T}} ], \\ \check{S}_{2} & = - \mathop \sum \limits_{i = 1}^{2} [W_{{21}_i} M_{i} \gamma_{0} N_{i}^{\text{T}} + W_{{21}_i} A_{i} \varGamma_{{11}_i} C_{i}^{\text{T}} ], \\ \check{S}_{3} & = - \mathop \sum \limits_{i = 1}^{2} [B_{i}^{\text{T}} W_{{11}_i} A_{i} \varGamma_{{12}_i} ],\quad \check{S}_{4} = - \mathop \sum \limits_{i = 1}^{2} [W_{{21}_i} A_{i} \varGamma_{{12}_i} ]. \\ [\check{a}_{i} ] & = [B_{i}^{\text{T}} W_{{11}_i} B_{i} ],\quad [\check{b}_{i} ] = [\varGamma_{{21}_i} C_{i}^{\text{T}} ],\quad [\check{c}_{i} ] = [B_{i}^{\text{T}} W_{{12}_i} ],\quad [\check{d}_{i} ] = [N_{i} \gamma_{0} N_{i}^{\text{T}} + C_{i} \varGamma_{{11}_i} C_{i}^{\text{T}} ] \\ [\check{e}_{i} ] & = [W_{{21}_i} B_{i} ], \quad [\check{f}_{i} ] = [W_{{22}_i} ],\quad [\check{g}_{i} ] = [R + B_{i}^{\text{T}} W_{{11}_i} B_{i} ],\quad [\check{m}_{i} ] = [\varGamma_{{22}_i} ], \quad [\check{h}_{i} ] = [C_{i} \varGamma_{{12}_i} ], \\ [\check{I}_{i} ] & = [I_{i} ]. \\ \end{aligned}$$

Here, the compact form of (26) can be assumed as follows:

$$\left\{ \begin{aligned} 1. \quad [\check{a}_{1} \check{a}_{2} ]\left[ {\begin{array}{*{20}c} G & 0 \\ 0 & G \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\check{b}_{1} } \\ {\check{b}_{2} } \\ \end{array} } \right] + [\check{c}_{1} \check{c}_{2} ]\left[ {\begin{array}{*{20}c} {K\check{d}_{1} + (P + KDG)\check{b}_{1} } & 0 \\ 0 & {K\check{d}_{2} + (P + KDG)\check{b}_{2} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\check{I}_{1} } \\ {\check{I}_{2} } \\ \end{array} } \right] = \check{S}_{1} \hfill \\ 2. \quad [\check{g}_{1} \check{g}_{2} ]\left[ {\begin{array}{*{20}c} G & 0 \\ 0 & G \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\check{m}_{1} } \\ {\check{m}_{2} } \\ \end{array} } \right] + [\check{c}_{1} \check{c}_{2} ]\left[ {\begin{array}{*{20}c} {K\check{h}_{1} + (P + KDG)\check{m}_{1} } & 0 \\ 0 & {K\check{h}_{2} + (P + KDG)\check{m}_{2} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\check{I}_{1} } \\ {\check{I}_{2} } \\ \end{array} } \right] = \check{S}_{3} \hfill \\ 3. \quad [\check{e}_{1} \check{e}_{2} ]\left[ {\begin{array}{*{20}c} G & 0 \\ 0 & G \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\check{b}_{1} } \\ {\check{b}_{2} } \\ \end{array} } \right] + [\check{f}_{1} \check{f}_{2} ]\left[ {\begin{array}{*{20}c} {K\check{d}_{1} + (P + KDG)\check{b}_{1} } & 0 \\ 0 & {K\check{d}_{2} + (P + KDG)\check{b}_{2} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\check{I}_{1} } \\ {\check{I}_{2} } \\ \end{array} } \right] = \check{S}_{2} \hfill \\ 4. \quad [\check{e}_{1} \check{e}_{2} ]\left[ {\begin{array}{*{20}c} G & 0 \\ 0 & G \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\check{m}_{1} } \\ {\check{m}_{2} } \\ \end{array} } \right] + [\check{f}_{1} \check{f}_{2} ]\left[ {\begin{array}{*{20}c} {K\check{h}_{1} + (P + KDG)\check{m}_{1} } & 0 \\ 0 & {K\check{h}_{2} + (P + KDG)\check{m}_{2} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\check{I}_{1} } \\ {\check{I}_{2} } \\ \end{array} } \right] = \check{S}_{4} \hfill \\ \end{aligned} \right.$$

(27)

Now, by introducing some new variables, (27) is rewritten as

$$\left\{ {\begin{array}{*{20}l} {1. \quad \check{A} \check{X} \check{B} + \check{C} \check{Y} \check{I} = \check{S}_{1} } \hfill \\ {2. \quad \check{G} \check{X} \check{M} + \check{C} \check{Z} \check{I} = \check{S}_{3} } \hfill \\ {3. \quad \check{E} \check{X} \check{B} + \check{F} \check{Y} \check{I} = \check{S}_{2} } \hfill \\ {4. \quad \check{E} \check{X} \check{M} + \check{F} \check{Z} I = \check{S}_{4} } \hfill \\ \end{array} } \right.$$

(28)

where $\begin{aligned} \check{A} = [\check{a}_{1} \check{a}_{2} ], \;\check{B} = [\check{b}_{1}^{\text{T}} \check{b}_{2}^{\text{T}} ]^{\text{T}} ,\; \check{C} = [\check{c}_{1} \check{c}_{2} ] ,\;\check{D} = [\check{d}_{1} \check{d}_{2} ], \;\check{E} = [\check{e}_{1} \check{e}_{2} ],\;\check{F} = [\check{f}_{1} \check{f}_{2} ], \hfill \\ \check{G} = [\check{g}_{1} \check{g}_{2} ],\;\check{M} = [\check{m}_{1}^{\text{T}} \check{m}_{2}^{\text{T}} ]^{\text{T}} ,\;\check{H} = [\check{h}_{1} \check{h}_{2} ],\;\check{I} = [\check{I}_{1}^{\text{T}} \check{I}_{2}^{\text{T}} ]^{\text{T}} , \;\check{X} = {\text{diagonal(}}G ),\; \check{Y} = {\text{diagonal(}}K\check{d}_{i} + (P + KDG)\check{b}_{i} ), \; \check{Z} = {\text{diagonal(}}K\check{h}_{i} + (P + KDG)\check{m}_{i} )\hfill \\ \end{aligned}$ and $\check{I}_{i}$ is a unitary matrix with the proper dimension for $i = 1,2$.

Obviously, four equations in (28) can be considered as two sets of coupled SME as

$${\text{SME}} . 1:\left\{ {\begin{array}{*{20}c} {\check{A} \check{X} \check{B} + \check{C} \check{Y} \check{I} = \check{S}_{1} } \\ {\check{E} \check{X} \check{B} + \check{F} \check{Y} \check{I} = \check{S}_{2} } \\ \end{array} } \right.\;{\text{and}}\;{\text{SME}} . 2:\left\{ {\begin{array}{*{20}c} {\check{G} \check{X} \check{M} + \check{C} \check{Z} \check{I} = \check{S}_{3} } \\ {\check{E} \check{X} \check{M} + \check{F} \check{Z} I = \check{S}_{4} } \\ \end{array} } \right.$$

1.3 Appendix 3: Proof 4.5

If matrices $R,\varGamma _{i} = \left[ {\begin{array}{*{20}c} {\varGamma_{11} } & {\varGamma_{12} } \\ {\varGamma_{21} } & {\varGamma_{22} } \\ \end{array} } \right]_{i} \quad {\text{and}}\quad W_{i} = \left[ {\begin{array}{*{20}c} {W_{11} } & {W_{12} } \\ {W_{21} } & {W_{22} } \\ \end{array} } \right]_{i}$ are symmetric positive definite, then obviously, sub-matrices $W_{{11}_i}$, $W_{{22}_i}$, $\varGamma_{{11}_i}$ and $\varGamma_{{22}_i}$ will be symmetric positive definite. Hence, regarding (25) the following relations are achieved:

$$\left\{ {\begin{array}{*{20}c} {\check{g}_{i} \succ 0 \to \check{G}^{\rm T} \check{G} \succ 0} \\ {\begin{array}{*{20}c} {\check{f}_{i} \succ 0 \to \check{F}^{\rm T} \check{F} \succ 0} \\ {\check{m}_{i} \succ 0 \to \check{M} \check{M}^{\rm T} \succ 0} \\ \end{array} } \\ {\check{I}_{i} \succ 0 \to \check{I} \check{I}^{\rm T} \succ 0 } \\ \end{array} } \right.$$

(29)

Now, regarding (29) and using pseudo-inverse definition, the unknown matrix $\check{X}$ in (28.2) is calculated as

$$\begin{aligned} ( {\check{G}^{\rm T} \check{G} } )^{ - 1} \check{G}^{\rm T} \check{G} \check{X} \check{M} \check{M}^{\rm T} ( {\check{M} \check{M}^{\rm T} } )^{ - 1} + ( {\check{G}^{\rm T} \check{G} } )^{ - 1} \check{G}^{\rm T} \check{C} \check{Z} \check{I} \check{M}^{\rm T} ( {\check{M} \check{M}^{\rm T} } )^{ - 1} = ( {\check{G}^{\rm T} \check{G} } )^{ - 1} \check{G}^{\rm T} \check{S}_{3} \check{M}^{\rm T} ( {\check{M} \check{M}^{\rm T} } )^{ - 1} \to \hfill \\ \check{X} = ( {\check{G}^{\rm T} \check{G} } )^{ - 1} \check{G}^{\rm T} (\check{S}_{3} - \check{C} \check{Z} \check{I} )\check{M}^{\rm T} (\check{M} \check{M}^{\rm T} )^{ - 1} \hfill \\ \end{aligned}$$

(30)

By replacing (30) in (28.4), the unknown matrix $\check{X}$ is calculated as

$$\begin{aligned} & (\check{F}^{\text{T}} \check{F} )^{ - 1} \check{F}^{\text{T}} \check{E} (\check{G}^{\text{T}} \check{G} )^{ - 1} \check{G}^{\text{T}} (\check{S}_{3} - \check{C} \check{Z} \check{I} )\check{M}^{\text{T}} (\check{M} \check{M}^{\text{T}} )^{ - 1} \check{M} \check{I}^{\text{T}} (\check{I} \check{I}^{\text{T}} )^{ - 1} + (\check{F}^{\text{T}} \check{F} )^{ - 1} \check{F}^{\text{T}} \check{F} \check{Z} \check{I} \check{I}^{\text{T}} (\check{I} \check{I}^{\text{T}} )^{ - 1} = (\check{F}^{\text{T}} \check{F} )^{ - 1} \check{F}^{\text{T}} \check{S}_{4} \check{I}^{\text{T}} (\check{I} \check{I}^{\text{T}} )^{ - 1} \\ & \quad \to (\check{F}^{\text{T}} \check{F} )^{ - 1} \check{F}^{\text{T}} \check{E} (\check{G}^{\text{T}} \check{G} )^{ - 1} \check{G}^{\text{T}} (\check{S}_{3} - \check{C} \check{Z} \check{I} )\check{I}^{\text{T}} (\check{I} \check{I}^{\text{T}} )^{ - 1} + \check{Z} = (\check{F}^{\text{T}} \check{F} )^{ - 1} \check{F}^{\text{T}} \check{S}_{4} \check{I}^{\text{T}} (\check{I} \check{I}^{\text{T}} )^{ - 1} \\ & \quad \to - \check{Z} + [(\check{F}^{\text{T}} \check{F} )^{ - 1} \check{F}^{\text{T}} \check{E} (\check{G}^{\text{T}} \check{G} )^{ - 1} \check{G}^{\text{T}} \check{C} ]\check{Z} [\check{I} \check{I}^{\text{T}} (\check{I} \check{I}^{\text{T}} )^{ - 1} ] = - (\check{F}^{\text{T}} \check{F} )^{ - 1} \check{F}^{\text{T}} \check{S}_{4} \check{I}^{\text{T}} (\check{I} \check{I}^{\text{T}} )^{ - 1} + (\check{F}^{\text{T}} \check{F} )^{ - 1} \check{F}^{\text{T}} \check{E} (\check{G}^{\text{T}} \check{G} )^{ - 1} \check{G}^{\text{T}} \check{S}_{3} \check{I}^{\text{T}} (\check{I} \check{I}^{\text{T}} )^{ - 1} \\ & \quad \to [(\check{F}^{\text{T}} \check{F} )^{ - 1} \check{F}^{\text{T}} \check{E} (\check{G}^{\text{T}} \check{G} )^{ - 1} \check{G}^{\text{T}} \check{C} ]\check{Z} - \check{Z} = (\check{F}^{\text{T}} \check{F} )^{ - 1} \check{F}^{\text{T}} [\check{E} (\check{G}^{\text{T}} \check{G} )^{ - 1} \check{G}^{\text{T}} \check{S}_{3} - \check{S}_{4} ]\check{I}^{\text{T}} (\check{I} \check{I}^{\text{T}} )^{ - 1} \\ \end{aligned}$$

(31)

Indeed (31) is assumed as a discrete Sylvester equation in the form of $A\check{Z} B - \check{Z} = W$, where

$$A = [(\check{F}^{\text{T}} \check{F} )^{ - 1} \check{F}^{\text{T}} \check{E} (\check{G}^{\text{T}} \check{G} )^{ - 1} \check{G}^{\text{T}} \check{C} ], \quad B = I, \quad W = (\check{F}^{\text{T}} \check{F} )^{ - 1} \check{F}^{\text{T}} [\check{E} (\check{G}^{\text{T}} \check{G} )^{ - 1} \check{G}^{\text{T}} \check{S}_{3} - \check{S}_{4} ]\check{I}^{\text{T}} (\check{I} \check{I}^{\text{T}} )^{ - 1}$$

The Sylvester equation in (31) can be solved and $\check{Z}$ can be computed using MATLAB^®←command “dlyap(A,B,W)”. Hence, by replacing the computed value of $\check{Z}$ in (30), $\check{X}$ can be calculated. In the next step, the calculated parameter $\check{X}$ is replaced in (28.3). In the same way, $\check{Y}$ is calculated as follows:

$$\check{Y} = (\check{F}^{\text{T}} \check{F} )^{ - 1} \check{F}^{\text{T}} \check{S}_{2} \check{I}^{\text{T}} (\check{I} \check{I}^{\text{T}} )^{ - 1} - (\check{F}^{\text{T}} \check{F} )^{ - 1} \check{F}^{\text{T}} \check{E} \check{X} \check{B} \check{I}^{\text{T}} (\check{I} \check{I}^{\text{T}} )^{ - 1}$$

(32)

Using the value of $\check{X}$ and replacing it in (32), $\check{Y}$ will be obtained.