Robust UIO Design for an Actuator Fault Identification

Abstract

In this paper an actuator robust fault identification scheme is developed, which is based on an observer within $\mathcal {H}_{\infty }$ framework for a class of non-linear systems. The proposed approach is designed in such a way that a prescribed disturbance attenuation level is achieved with respect to the actuator fault estimation error while guaranteeing the convergence of the observer. The effectiveness of the proposed approach is verified with the laboratory multi-tank system.

1 Introduction

During last three decades several efficient Fault Detection and Isolation (FDI) methods for non-linear dynamic systems were developed [6, 10–12]. Such methods allows to reduce the economical losses resulting from the industrial systems malfunction. However, in the last decade the expectations for the industrial systems and fault diagnosis started to change. It was expected that the systems can be operated efficiently despite of existing faults. Such an assumption caused that the scientists focuses on developing the Fault Tolerant Control (FTC) strategies [2, 9, 13, 14, 17, 18].

To achieve this goal the efficient fault estimation methods of the actuators faults which could be used during control process should be elaborated. Such methods allows for the application of the active FTC strategies enabling compensation of the faulty actuator by increasing performance of the other actuator existing in the control system. The problem of the actuators fault estimation can be perceived as the task of estimation of the system unknown inputs and can be solved by the application of the Unknown Input Observer (UIO) [3, 7, 14–16]. Such techniques enable for the state and unknown inputs reconstruction on the basis of mathematical model of the system and measurements from the system inputs and outputs.

In this paper a novel observer synthesis procedure, which is based on the concept of the UIO for the actuators fault detection and estimation, is proposed. The developed approach is a combination of the linear-system strategies [4] for a class of non-linear systems [20]. The UIO is designed in such a way that a prescribed disturbance attenuation level is achieved with respect to the actuator fault estimation error while guaranteeing the convergence of the observer. The resulting design procedures boil down to solving a set of linear matrix inequalities.

The paper is organized as follows. Section 2 describes the design procedure of the robust UIO using $\mathcal {H}_{\infty }$ framework for the actuator fault identification. Section 3 provides a introduction into the structure and parameters of a multi-tank system and contains an illustrative example, which shows the performance of the proposed approach for an actuators fault detection and estimation. The final part of the paper is devoted to conclusions.

2 Methodology of the Robust UIO Design for the Actuators Faults Estimation

The main objective of this section is to provide a detailed design procedure of the robust observer, which can be used for the robust actuator fault diagnosis. As a result the estimate of the actuator fault is obtained. In order to achieve this goal the observer should be designed in such a way that a prescribed disturbance attenuation level is achieved with respect to the actuator fault estimation error while guaranteeing the convergence of the observer.

A dynamic, non-linear system can be represent by the LPV model in a relatively simple way. To design such a model, it is necessary to linearize a non-linear system around a number of operating points. The number of points determines the accuracy of the LPV model. The local system behavior around the operating point is represented by each of these linear models. Let us consider the following discrete-time non-linear model:

$$\begin{aligned} \varvec{x}_{k+1}&=\varvec{h}(\varvec{x}_{k},\varvec{u}_{k})\end{aligned}$$

(1)

$$\begin{aligned} \varvec{y}_{k}&=\varvec{C}\varvec{x}_{k} \end{aligned}$$

(2)

where $\varvec{x}_{} \in \mathbb {R}^n$ is the state vector, $\varvec{y}_{} \in \mathbb {R}^p$ is the output, $\varvec{u}_{} \in \mathbb {R}^m$ is the input vector and $\varvec{h}(\cdot )$ is a non-linear function. Such model can be represented in the form of a discrete-time polytopic LPV model:

$$\begin{aligned} \varvec{x}_{k+1}&=\varvec{A}(h_k)\varvec{x}_{k}+\varvec{B}\varvec{u}_{k}\end{aligned}$$

(3)

$$\begin{aligned} \varvec{y}_{k}&=\varvec{C}\varvec{x}_{k} \end{aligned}$$

(4)

where $\varvec{A}(h_k)$, $\varvec{B}$, $\varvec{C}$ are state-space matrices and $h_k \in \mathbb {R}^l$ is a time-varying parameter vector which ranges over a fixed polytope. The dependence of $\varvec{A}$ on $h_k$ represents a general discrete-time quasi-LPV model. The model (3)–(4) can be written in the following alternative form of the state-space model:

$$\begin{aligned} \varvec{x}_{k+1}&=\varvec{A}\varvec{x}_{k}+\varvec{B}\varvec{u}_{k}+\varvec{g}\left( \varvec{x}_{k}\right) +\varvec{L}_{a}\varvec{f}_{a,k}+\varvec{W}_1\varvec{w}_{k}\end{aligned}$$

(5)

$$\begin{aligned} \varvec{y}_{k+1}&=\varvec{C}\varvec{x}_{k+1}+\varvec{W}_2\varvec{w}_{k+1} \end{aligned}$$

(6)

where $\varvec{x}_{k} \in \mathbb {X} \subset \mathbb {R}^n$ is the state vector, $\varvec{u}_{k} \in \mathbb {R}^r$ stands for the input, $\varvec{y}_{k} \in \mathbb {R}^m$ denotes the output, $\varvec{f}_{a,k} \in \mathbb {R}^r$ stands for the actuator and $\varvec{L}_{a}$ is its distribution matrix. Moreover, $\varvec{w}_{k} \in l_2$ is an exogenous disturbance vector with $\varvec{W}_1 \in \mathbb {R}^{n \times n}$, $\varvec{W}_2 \in \mathbb {R}^{m \times n}$ being its distribution matrices while:

$$\begin{aligned} l_2=\left\{ \mathbf {w} \in \mathbb {R}^n|\ \Vert \mathbf {w}\Vert _{l_2}< + \infty \right\} ,\quad \Vert \mathbf {w}\Vert _{l_2}&=\left( \sum _{k=0}^{\infty }\Vert \varvec{w}_{k}\Vert ^2\right) ^{\frac{1}{2}} \end{aligned}$$

(7)

As the description of the LPV model is delivered then the robust UIO design procedure can be developed. Following [4], let us assume that the system is observable and the following rank condition is satisfied:

$$\begin{aligned} \text {rank}(\varvec{C}\varvec{L}_a)&=\text {rank}(\varvec{L}_a)=s \end{aligned}$$

(8)

Under the assumption (8) it is possible to obtain:

$$\begin{aligned} \varvec{H}=(\varvec{C}\varvec{L}_a)^+=\left[ (\varvec{C}\varvec{L}_a)^T\varvec{C}\varvec{L}_a\right] ^{-1}(\varvec{C}\varvec{L}_a)^T \end{aligned}$$

(9)

It should be underlined, that the proposed approach is designed for actuator faults estimation only, and hence, the setting $\varvec{f}_{s,k}=\varvec{0}$ is employed in the sequel. Substituting $\varvec{f}_{s,k}=\varvec{0}$ into (6) and multiplying it by matrix $\varvec{H}$, and then substituting (5), it can be shown that:

$$\begin{aligned} \varvec{f}_{a,k}=\varvec{H}(\varvec{y}_{k+1}-\varvec{C}\varvec{A}\varvec{x}_{k}-\varvec{C}\varvec{B}\varvec{u}_{k}-\varvec{C}\varvec{g}\left( \varvec{x}_{k}\right) -\varvec{C}\varvec{W}_1\varvec{w}_{k}-\varvec{W}_2\varvec{w}_{k+1}) \end{aligned}$$

(10)

Finally, by substituting (10) into (5) it can be shown that:

$$\begin{aligned} \varvec{x}_{k+1}=\bar{\varvec{A}}\varvec{x}_{k}+\bar{\varvec{B}}\varvec{u}_{k}+\varvec{G}\varvec{g}\left( \varvec{x}_{k}\right) +\bar{\varvec{L}}\varvec{y}_{k+1}+\varvec{G}\varvec{W}_1\varvec{w}_{k}-\bar{\varvec{L}}\varvec{W}_2\varvec{w}_{k+1} \end{aligned}$$

(11)

where $\varvec{G}=(\varvec{I}_n-\varvec{L}_a\varvec{H}\varvec{C})$, $\bar{\varvec{A}}=\varvec{G}\varvec{A}$, $\bar{\varvec{B}}=\varvec{G}\varvec{B}$, $\bar{\varvec{L}}=\varvec{L}_a\varvec{H}$.

The estimation of the system state $\hat{\varvec{x}}_{k}$ with the corresponding observer:

$$\begin{aligned} \hat{\varvec{x}}_{k+1}=\bar{\varvec{A}}\hat{\varvec{x}}_{k}+\bar{\varvec{B}}\varvec{u}_{k}+\varvec{G}\varvec{g}\left( \hat{\varvec{x}}_{k}\right) +\bar{\varvec{L}}\varvec{y}_{k+1}+\varvec{K}_{a}(\varvec{y}_{k}-\varvec{C}\hat{\varvec{x}}_{k}) \end{aligned}$$

(12)

allows to obtain the estimate of the actuators faults:

$$\begin{aligned} \hat{\varvec{f}}_{a,k}=\varvec{H}(\varvec{y}_{k+1}-\varvec{C}\varvec{A}\hat{\varvec{x}}_{k}-\varvec{C}\varvec{B}\varvec{u}_{k}-\varvec{C}\varvec{g}\left( \hat{\varvec{x}}_{k}\right) ) \end{aligned}$$

(13)

It should be underlined that the state estimation error is defined as:

$$\begin{aligned} \nonumber \varvec{e}_{k+1}&=\left( \bar{\varvec{A}}-\varvec{K}_{a}\varvec{C}\right) \varvec{e}_{k}+\varvec{G}\varvec{s}_{k}+(\varvec{G}\varvec{W}_1-\varvec{K}_{a}\varvec{W}_2)\varvec{w}_{k}-\bar{\varvec{L}}\varvec{W}_2\varvec{w}_{k+1}=\\&=\varvec{A}_1\varvec{e}_{k}+\varvec{G}\varvec{s}_{k}+\bar{\varvec{W}}_1\varvec{w}_{k}+\bar{\varvec{W}}_2\varvec{w}_{k+1} \end{aligned}$$

(14)

where

$$\begin{aligned} \varvec{s}_{k}=\varvec{g}\left( \varvec{x}_{k}\right) -\varvec{g}\left( \hat{\varvec{x}}_{k}\right) \end{aligned}$$

(15)

Similarly, the fault estimation error $\varvec{\varepsilon }_{f_{a},k}$ can be calculated according to the following equation:

$$\begin{aligned} \varvec{\varepsilon }_{f_{a},k}=\varvec{f}_{a,k}-\hat{\varvec{f}}_{a,k}=-\varvec{H}\varvec{C}\left( \varvec{A}\varvec{e}_{k}+\varvec{s}_{k}+\varvec{W}_1\varvec{w}_{k}\right) -\varvec{H}\varvec{W}_2\varvec{w}_{k+1} \end{aligned}$$

(16)

Note that both $\varvec{e}_{k}$ and $\varvec{\varepsilon }_{f_{a},k}$ are non-linear with respect to $\varvec{e}_{k}$. To solve this problem, the undermentioned solution is proposed. Using the Differential Mean Value Theorem (DMVT) [19], it can be shown that:

$$\begin{aligned} \varvec{g}\left( \varvec{a}\right) -\varvec{g}\left( \varvec{b}\right) =\varvec{M}_x(\varvec{a}-\varvec{b}) \end{aligned}$$

(17)

with

$$\begin{aligned} \varvec{M}_x=\begin{bmatrix} \dfrac{\partial g_{1}}{\partial x}(\varvec{c_1})\\ \vdots \\ \dfrac{\partial g_{n}}{\partial x}(\varvec{c_n}) \end{bmatrix} \end{aligned}$$

(18)

where $\varvec{c}_1,\ldots , \varvec{c}_n \in \text{ Co }(\varvec{a},\varvec{b})$, $\varvec{c}_i\ne \varvec{a}$ and $\varvec{c}_i\ne \varvec{b}$, $i=1,\ldots ,n$. Assumed that:

$$\begin{aligned} \bar{g}_{i,j}\ge \dfrac{\partial g_{i}}{\partial x_j} \ge \underline{g}_{i,j}, \quad i=1,\ldots ,n, \quad j=1,\ldots ,n \end{aligned}$$

(19)

a gradient can be calculated as follows:

$$\begin{aligned} \dfrac{\partial g_{i}{(\varvec{x}_{k})}}{\partial {\varvec{x}_{k}}} = \left[ \dfrac{\partial g_{i}(x)}{\partial {x_1}},\ldots ,\dfrac{\partial g_{i}(x)}{\partial {x_n}} \right] ^T \end{aligned}$$

(20)

Now, it can be shown that:

$$\begin{aligned} \mathbb {M}_x=\left\{ \varvec{M}\in \mathbb {R}^{n \times n}| \bar{g}_{i,j}\ge m_{x,i,j} \ge \underline{g}_{i,j},\, i,j=1,\ldots ,n \right\} \end{aligned}$$

(21)

Using (17) and assuming that $\varvec{M}_{x,k} \in \mathbb {M}_x$ the term $\varvec{A}_1\varvec{e}_{k}+\varvec{G}\varvec{s}_{k}$ in (14) can be written as:

$$\begin{aligned} \varvec{A}_1\varvec{e}_{k}+\varvec{G}\varvec{s}_{k}=(\bar{\varvec{A}}+\varvec{G}\varvec{M}_{x,k}-\varvec{K}_{a}\varvec{C})\varvec{e}_{k} \end{aligned}$$

(22)

From (22), it can be deduced that the $\varvec{e}_{k+1}$ described by (14) can be converted into following form:

$$\begin{aligned} \varvec{e}_{k+1}&=\varvec{A}_{2}({h_{k}})\varvec{e}_{k}+\bar{\varvec{W}}_1\varvec{w}_{k}+\bar{\varvec{W}}_2\varvec{w}_{k+1}\\ \nonumber \varvec{A}_{2}({h_{k}})&=\tilde{\varvec{A}}({h_{k}})-\varvec{K}_{a}\varvec{C}\end{aligned}$$

(23)

defining an LPV polytopic system [1] with:

$$\begin{aligned} \tilde{\mathbb {A}}=\left\{ \tilde{\varvec{A}}({h_{k}}):\quad \tilde{\varvec{A}}({h_{k}})=\sum _{i=1}^N {h_{k}}_i\tilde{\varvec{A}}_i,\, \sum _{i=1}^N {h_{k}}_i=1,\, {h_{k}}_i \ge 0 \right\} \end{aligned}$$

(24)

where $N=2^{n^2}$. Note that this is a general description, which does not take into account that some elements of $\varvec{M}_{x,k}$ may be constant. In such cases, N is given by $N=2^{(n-c)^2}$ where c stands for the number of constant elements of $\varvec{M}_{x,k}$. Similarly, the fault estimation error $\varvec{\varepsilon }_{f_{a},k}$ can be converted into:

$$\begin{aligned} \varvec{\varepsilon }_{f_{a},k}=-\varvec{H}\varvec{C}\left( \varvec{A}_{3}({h_{k}})\varvec{e}_{k}+\varvec{W}_1\varvec{w}_{k}\right) -\varvec{H}\varvec{W}_2\varvec{w}_{k+1} \end{aligned}$$

(25)

$$\begin{aligned} \mathbb {A}_3=\left\{ \varvec{A}_3({h_{k}}):\quad \varvec{A}_3({h_{k}})=\sum _{i=1}^N {h_{k}}_i\varvec{A}_{3,i},\, \sum _{i=1}^N {h_{k}}_i=1,\, {h_{k}}_i \ge 0 \right\} \end{aligned}$$

(26)

The objective of further deliberations is to design the observer (12) in such a way that the state estimation error $\varvec{e}_{k}$ is asymptotically convergent and the following upper bound is guaranteed:

$$\begin{aligned} \Vert \varvec{\varepsilon }_{f}\Vert _{l_2} \le \omega \Vert \mathbf {w}\Vert _{l_2} \end{aligned}$$

(27)

where $\omega >0$ is a prescribed disturbance attenuation level. Thus, on the contrary to the approaches presented in the literature, $\omega$ should be achieved with respect to the fault estimation error but not the state estimation error.

The problem of $\mathcal {H}_{\infty }$ observer design [8, 20] is to determine the gain matrix $\varvec{K}_{a}$ such that:

$$\begin{aligned}&\lim _{k \rightarrow \infty } \varvec{e}_{k}=\varvec{0}\quad \text{ for }\,\, \varvec{w}_{k}=\varvec{0}\end{aligned}$$

(28)

$$\begin{aligned}&\Vert \varvec{\varepsilon }_{f}\Vert _{l_2} \le \omega \Vert \mathbf {w}\Vert _{l_2}\quad \text{ for }\,\, \varvec{w}_{k} \ne \varvec{0},\, \varvec{e}_{0}=\varvec{0} \end{aligned}$$

(29)

To solve the above problem it is necessary to find a Lyapunov function $V_k$:

$$\begin{aligned} \varDelta V_k+\varvec{\varepsilon }_{f_{a},k}^T\varvec{\varepsilon }_{f_{a},k}-\mu ^2\varvec{w}_{k}^T\varvec{w}_{k}-\mu ^2\varvec{w}_{k+1}^T\varvec{w}_{k+1}<0,\, k=0,\ldots \infty \end{aligned}$$

(30)

where $\varDelta V_k=V_{k+1}-V_{k}$, $\mu >0$. Note that the structure of (30) is uncommon in the literature. Indeed, the novelty is that the term $-\mu ^2\varvec{w}_{k+1}^T\varvec{w}_{k+1}$ is introduced. This is caused the fault decoupling procedure (cf.(10)). Indeed, if $\varvec{w}_{k}=\varvec{0}$, ($k=0,\ldots ,\infty$) then (30) boils down to:

$$\begin{aligned} \varDelta V_k+\varvec{\varepsilon }_{f_{a},k}^T\varvec{\varepsilon }_{f_{a},k}<0,\, k=0,\ldots \infty \end{aligned}$$

(31)

and hence $\varDelta V_k<0$, which leads to (28). If $\varvec{w}_{k}\ne \varvec{0}$ for $k=0,\ldots ,\infty$ then the Lyapunov function (30) yields:

$$\begin{aligned} J=\sum _{k=0}^{\infty }\left( \varDelta V_k+\varvec{\varepsilon }_{f_{a},k}^T\varvec{\varepsilon }_{f_{a},k}-\mu ^2\varvec{w}_{k}^T\varvec{w}_{k}-\mu ^2\varvec{w}_{k+1}^T\varvec{w}_{k+1}\right) <0 \end{aligned}$$

(32)

and can be written as:

$$\begin{aligned} J=-V_{0}+\sum _{k=0}^{\infty }\varvec{\varepsilon }_{f_{a},k}^T\varvec{\varepsilon }_{f_{a},k}-\mu ^2\sum _{k=0}^{\infty }\varvec{w}_{k}^T\varvec{w}_{k}-\mu ^2\sum _{k=0}^{\infty }\varvec{w}_{k+1}^T\varvec{w}_{k+1}<0 \end{aligned}$$

(33)

Bearing in mind that:

$$\begin{aligned} \mu ^2\sum _{k=0}^{\infty }\varvec{w}_{k+1}^T\varvec{w}_{k+1}=\mu ^2\sum _{k=0}^{\infty }\varvec{w}_{k}^T\varvec{w}_{k}-\mu ^2\varvec{w}_{0}^T\varvec{w}_{0} \end{aligned}$$

(34)

inequality (33) can be written as:

$$\begin{aligned} J=-V_{0}+\sum _{k=0}^{\infty }\varvec{\varepsilon }_{f_{a},k}^T\varvec{\varepsilon }_{f_{a},k}-2\mu ^2\sum _{k=0}^{\infty }\varvec{w}_{k}^T\varvec{w}_{k}+\mu ^2\varvec{w}_{0}^T\varvec{w}_{0}<0 \end{aligned}$$

(35)

Knowing that $V_0=0$ for $\varvec{e}_{0}=0$, (35) leads to (29) with $\omega =\sqrt{2}\mu$.

As the general scheme for designing the robust observer is proposed, the following form of the Lyapunov function is assumed [19]:

$$\begin{aligned} V_k=\varvec{e}_{k}^T\varvec{P}({h_{k}})\varvec{e}_{k} \end{aligned}$$

(36)

where $\varvec{P}({h_{k}}) \succ \varvec{0}$. On the contrary to the design approach presented in the literature (see, e.g. [20]) it is not assumed that $\varvec{P}({h_{k}})=\varvec{P}$ is constant. Indeed, $\varvec{P}({h_{k}})$ can be perceived as a parameter-depended matrix [1] of the following form:

$$\begin{aligned} \varvec{P}({h_{k}})=\sum _{i=1}^N{h_{k}}_i\varvec{P}_i \end{aligned}$$

(37)

As a consequence

$$\begin{aligned} \nonumber&\varDelta V_k+\varvec{\varepsilon }_{f_{a},k}^T\varvec{\varepsilon }_{f_{a},k}-\mu ^2\varvec{w}_{k}^T\varvec{w}_{k}-\mu ^2\varvec{w}_{k+1}^T\varvec{w}_{k+1} \\ \nonumber&= \varvec{e}_{k}^T\left( \varvec{A}_{2}({h_{k}})^T\varvec{P}({h_{k+1}})\varvec{A}_2({h_{k}})+\varvec{A}_3({h_{k}})^T\varvec{H}_1\varvec{A}_3({h_{k}})-\varvec{P}({h_{k}})\right) \varvec{e}_{k}\\ \nonumber&+ \varvec{e}_{k}^T\left( \varvec{A}_{2}({h_{k}})^T\varvec{P}({h_{k+1}})\bar{\varvec{W}}_1+\varvec{A}_{3}({h_{k}})^T\varvec{H}_1\varvec{W}_1\right) \varvec{w}_{k}\\ \nonumber&+ \varvec{e}_{k}^T\left( \varvec{A}_{2}({h_{k}})^T\varvec{P}({h_{k+1}})\bar{\varvec{W}}_2+\varvec{A}_{3}({h_{k}})^T\varvec{H}_2\right) \varvec{w}_{k+1}\\ \nonumber&+ \varvec{w}_{k}^T\left( \bar{\varvec{W}}_1^T\varvec{P}({h_{k+1}})\varvec{A}_{2}({h_{k}})+\varvec{W}_1^T\varvec{H}_1\varvec{A}_{3}({h_{k}})\right) \varvec{e}_{k}\\ \nonumber&+ \varvec{w}_{k}^T\left( \bar{\varvec{W}}_1^T\varvec{P}({h_{k+1}})\bar{\varvec{W}}_1+\varvec{W}_1^T\varvec{H}_1\varvec{W}_1-\mu ^2\varvec{I}\right) \varvec{w}_{k}\\ \nonumber&+ \varvec{w}_{k}^T\left( \bar{\varvec{W}}_1^T\varvec{P}({h_{k+1}})\varvec{W}_2+\varvec{W}_1^T\varvec{H}_2\right) \varvec{w}_{k+1}\\ \nonumber&+ \varvec{w}_{k+1}^T\left( \bar{\varvec{W}}_2^T\varvec{P}({h_{k+1}})\varvec{A}_{2,k}+\varvec{H}_2^T\varvec{A}_{3}({h_{k}})\right) \varvec{e}_{k}\\ \nonumber&+ \varvec{w}_{k+1}^T\left( \bar{\varvec{W}}_2^T\varvec{P}({h_{k+1}})\varvec{W}_1+\varvec{H}_2^T\varvec{W}_1\right) \varvec{w}_{k}\\&+ \varvec{w}_{k+1}^T\left( \bar{\varvec{W}}_2^T\varvec{P}({h_{k+1}})\bar{\varvec{W}}_2+\varvec{W}_2^T\varvec{H}^T\varvec{H}\varvec{W}_2-\mu ^2\varvec{I}\right) \varvec{w}_{k+1}<0 \end{aligned}$$

(38)

where $\varDelta V_k=V_{k+1}-V_k$, $\varvec{H}_1=\varvec{C}^T\varvec{H}^T\varvec{H}\varvec{C}$ and $\varvec{H}_2=\varvec{C}^T\varvec{H}^T\varvec{H}\varvec{W}_2$. Defining $\varvec{v}_{k}=\left[ \varvec{e}_{k}^T,\, \varvec{w}_{k}^T,\,\varvec{w}_{k+1}^T\right] ^T$, inequality (38) takes the following form:

$$\begin{aligned} \varDelta V_k+\varvec{\varepsilon }_{f_{a},k}^T\varvec{\varepsilon }_{f_{a},k}-\mu ^2\varvec{w}_{k}^T\varvec{w}_{k}-\mu ^2\varvec{w}_{k+1}^T\varvec{w}_{k+1}=\varvec{v}_{k}^T\varvec{M}_V\varvec{v}_{k}<0 \end{aligned}$$

(39)

where $\varvec{M}_V$ is given by:

$$\begin{aligned} \begin{array}{rcl} \varvec{M}_V&{}= \left[ \begin{array}{c} \varvec{A}_{2}({h_{k}})^T\varvec{P}({h_{k+1}})\varvec{A}_{2}({h_{k}})+\varvec{A}_{3}({h_{k}})^T\varvec{H}_1\varvec{A}_{3}({h_{k}})-\varvec{P}({h_{k}}) \\ \bar{\varvec{W}}_1^T\varvec{P}({h_{k+1}})\varvec{A}_{2}({h_{k}})+\varvec{W}_1^T\varvec{H}_1\varvec{A}_{3}({h_{k}}) \\ \bar{\varvec{W}}_2^T\varvec{P}({h_{k+1}})\varvec{A}_{2}({h_{k}})+\varvec{H}_2^T\varvec{A}_{3}({h_{k}}) \end{array} \right. \\ &{}\left. \begin{array}{c} \varvec{A}_{2}({h_{k}})^T\varvec{P}({h_{k+1}})\bar{\varvec{W}}_1+\varvec{A}_{3}({h_{k}})^T\varvec{H}_1\varvec{W}_1 \\ \bar{\varvec{W}}_1^T\varvec{P}({h_{k+1}})\bar{\varvec{W}}_1+\varvec{W}_1^T\varvec{H}_1\varvec{W}_1-\mu ^2\varvec{I} \\ \bar{\varvec{W}}_2^T\varvec{P}({h_{k+1}})\varvec{W}_1+\varvec{H}_2^T\varvec{W}_1 \end{array} \right. \\ &{}\left. \begin{array}{c} \varvec{A}_{2}({h_{k}})^T\varvec{P}({h_{k+1}})\bar{\varvec{W}}_2+\varvec{A}_{3}({h_{k}})^T\varvec{H}_2 \\ \bar{\varvec{W}}_1^T\varvec{P}({h_{k+1}})\varvec{W}_2+\varvec{W}_1^T\varvec{H}_2 \\ \bar{\varvec{W}}_2^T\varvec{P}({h_{k+1}})\bar{\varvec{W}}_2+\varvec{W}_2^T\varvec{H}^T\varvec{H}\varvec{W}_2-\mu ^2\varvec{I} \\ \end{array} \right] \end{array} \end{aligned}$$

(40)

The above results can be written in the form of theorem describing the developed framework of observer design:

Theorem 1

For a prescribed disturbance attenuation level $\mu >0$ for the fault estimation error (16), the $\mathcal {H}_{\infty }$ observer design problem for the system (5)–(6) and the observer (12) is solvable if there exists matrices $\varvec{P}_i\succ \varvec{0}$ $(i=1,\ldots ,N$ and $j=1,\ldots ,N)$, $\varvec{U}$ and $\varvec{N}$ such that the following LMIs are satisfied:

$$\begin{aligned}&\left[ \begin{array}{cccc} \varvec{A}_{3,i}^T\varvec{H}_1\varvec{A}_{3,j}-\varvec{P}_i &{} \varvec{A}_{3,i}^T\varvec{H}_1\varvec{W}_1 &{} \varvec{A}_{3,i}^T\varvec{H}_3 &{} \varvec{A}_{2,i}\varvec{U}^T\\ \varvec{W}_1^T\varvec{H}_1\varvec{A}_{3,i} &{} \varvec{W}_1^T\varvec{H}_1\varvec{W}_1-\mu ^2\varvec{I} &{} \varvec{W}_1^T\varvec{H}_2 &{} \bar{\varvec{W}}_1^T\varvec{U}^T\\ \varvec{H}_2^T\varvec{A}_{3,i} &{} \varvec{H}_2^T\varvec{W}_1 &{} \varvec{W}_2^T\varvec{H}^T\varvec{H}\varvec{W}_2-\mu ^2\varvec{I} &{} \bar{\varvec{W}}_2^T\varvec{U}^T \\ \varvec{U}\varvec{A}_{2,i} &{} \varvec{U}\bar{\varvec{W}}_1 &{}\varvec{U}\bar{\varvec{W}}_2 &{} \varvec{P}_j-\varvec{U}-\varvec{U}^T \end{array} \right] \,\,\prec \,\,\varvec{0} \end{aligned}$$

(41)

for $i=1,\ldots ,N$ and $j=1,\ldots ,N$ where:

$$\begin{aligned} \varvec{U}\varvec{A}_{2,i}&=\varvec{U}(\tilde{\varvec{A}}_i-\varvec{K}_{a}\varvec{C})=\varvec{U}\tilde{\varvec{A}}_i-\varvec{N}\varvec{C},\end{aligned}$$

(42)

$$\begin{aligned} \varvec{U}\bar{\varvec{W}}_1&=\varvec{U}(\varvec{G}\varvec{W}_1-\varvec{K}_{a}\varvec{W}_2)=\varvec{U}\varvec{G}\varvec{W}_1-\varvec{N}\varvec{W}_2 \end{aligned}$$

(43)

Proof

The following two lemmas can be perceived as the generalization of those presented in [1].

Lemma 1

The following statements are equivalent

• 1. There exists $\varvec{X}\succ 0$ such that:

  $$\begin{aligned} \varvec{V}^T\varvec{X}\varvec{V}-\varvec{W}\prec 0 \end{aligned}$$

  (44)

• 2. There exists $\varvec{X}\succ 0$ such that:

  $$\begin{aligned} \left[ \begin{array}{cc} -\varvec{W}&{} \varvec{V}^T\varvec{U}^T \\ \varvec{U}\varvec{V}&{} \varvec{X}-\varvec{U}-\varvec{U}^T \end{array} \right] \prec 0 \end{aligned}$$

  (45)

Subsequently, observing that the matrix (40) must be negative definite and writing it as:

$$\begin{aligned}&\left[ \begin{array}{c} \varvec{A}_{2}({h_{k}})^T\\ \bar{\varvec{W}}_1^T\\ \bar{\varvec{W}}_2^T\\ \end{array} \right] \varvec{P}({h_{k+1}}) \left[ \begin{array}{ccc} \varvec{A}_2({h_{k}}) &{} \bar{\varvec{W}}_1 &{} \bar{\varvec{W}}_2\\ \end{array} \right] \end{aligned}$$

(46)

$$\begin{aligned}&+\left[ \begin{array}{ccc} \varvec{A}_{3}({h_{k}})^T\varvec{H}_1\varvec{A}_{3}({h_{k}})-\varvec{P}({h_{k}}) &{} \varvec{A}_{3}({h_{k}})^T\varvec{H}_1\varvec{W}_1 &{} \varvec{A}_{3}({h_{k}})^T\varvec{H}_3\\ \varvec{W}_1^T\varvec{H}_1\varvec{A}_{3}({h_{k}}) &{} \varvec{W}_1^T\varvec{H}_1\varvec{W}_1-\mu ^2\varvec{I} &{} \varvec{W}_1^T\varvec{H}_2 \\ \varvec{H}_2^T\varvec{A}_{3}({h_{k}}) &{} \varvec{H}_2^T\varvec{W}_1 &{} \varvec{W}_2^T\varvec{H}^T\varvec{H}\varvec{W}_2-\mu ^2\varvec{I} \end{array} \right] \prec 0 \end{aligned}$$

(47)

and then applying Lemma 1 leads to (41), which completes the proof.

Finally, the design procedure boils down to solving LMIs (41) and then (cf. (42)–(43)) $\varvec{K}_{a}=\varvec{U}^{-1}\varvec{N}$. It can be also observed that the noticed design problem can be treated as minimization task, i.e.

$$\begin{aligned} \mu ^*= \min _{\mu >0, \varvec{P}_1 \succ \varvec{0}, \varvec{U}, \varvec{N}} \mu \end{aligned}$$

(48)

under (41).

Fig. 1

Multi-tank system

3 Fault Identification of the Multi-tank System

The objective of this section is to provide the reliable experimental results for proposed approach in the actuator fault estimation simultaneously with the state estimation. The multi-tank system [5] presented in Fig. 1, designed to reflect the behaviour and dynamics of full scale multi-tank industrial systems (e.g. hydroelectric power-plants, reservoir, etc.) was used in the laboratory conditions to show proposed approach at work. The multi-tank system consists of three separate tanks placed in series one under another, equipped with drain valves and level sensors based on a hydraulic pressure measurement. Each of them has a different cross-section in order to reflect system nonlinearities. The bottom tank is a water reservoir for the system. A variable speed water pump is used to fill the top tank. Due to gravity, the water flows through valves and tanks. The considered multi-tank system has been designed to operate with an external, PC-based digital controller. The control computer communicates with the level sensors, valves and a pump by a dedicated I/O board with the power interface. The I/O interface is controlled by the real-time software, which can operate in Matlab/Simulink environment.

The distribution matrices $\varvec{W}_1$ and $\varvec{W}_2$ should express the influence and magnitude of disturbances $\varvec{w}_{k}$ onto the state and output equations (5)–(6), respectively. To obtain appropriate ratio between the elements of $\varvec{W}_1$ and $\varvec{W}_2$, series of constant liquid level measurements was performed for the top tank. Next, the mean was removed, which represents the constant level of the liquid. Subsequently the disturbances where analyzed. The standard deviation of the disturbance is equal $1.75\cdot 10^{-4}$ (obtained for 1000 measurements). Similar results were obtained for the sensors in the middle and bottom tanks. The term $\varvec{W}_1\varvec{w}_{k}$ (cf. (5)) will represent the inaccuracy of the pump with respect to a desired control action. After a similar experiments like for the sensors, it was derived that the maximum magnitude of $\varvec{W}_1\varvec{w}_{k}$ is approximately five (5) times larger than that of $\varvec{W}_2\varvec{w}_{k}$. Thus resulting in the following values of the distribution matrices:

$$\begin{aligned} \varvec{W}_1=\text {diag}(0.05,0,0),\quad \varvec{W}_2=0.01\varvec{I}_m \end{aligned}$$

(49)

Subsequently the UIO design procedure was performed. As a result of solving the problem (41), the following couple was obtained:

$$\begin{aligned} \mu =0.55;\quad \varvec{K}_a=\begin{bmatrix} 0.1089&0\\ 0.0004&1.7107\\ 0&0.9473 \end{bmatrix} \end{aligned}$$

(50)

Next, to proceed to the experiment itself, the following initial conditions for the system $\varvec{x}_{0}=[0.001,0.001,0.001]^T$ and the observer $\hat{\varvec{x}}_{0}=[0.2,0.1,0.1]^T$ were assumed, while the input is $\varvec{u}_{k}=0.009$. Moreover, following actuator fault scenario was used:

$$\begin{aligned} \begin{array}{l} \varvec{f}_{a,k}=\left\{ \begin{array}{ll} -0.001, &{}\text{ for } 30000 \le k \le 50000\\ 0, &{}\text{ otherwise } \end{array} \right. \end{array} \end{aligned}$$

(51)

Initially the case when $\hat{\varvec{x}}_{0}=\varvec{x}_{0}$ ($\varvec{e}_{0}=\varvec{0}$) was considered. Figure 2 clearly indicates that condition (29) is satisfied, which means that attenuation level $\mu =0.55$ is achieved. Subsequently the case where $\varvec{w}_{k}=\varvec{0}$ and $\hat{\varvec{x}}_{0} \ne \varvec{x}_{0}$ was taken into consideration. Figure 3 clearly shows that (28) is satisfied as well. Finally, Fig. 4 shows the fault and its estimate for the nominal case ($\hat{\varvec{x}}_{0} \ne \varvec{x}_{0}$ and $\varvec{w}_{k} \ne \varvec{0}$). At the same time observer estimates unavailable state $x_2$. This appealing property is depicted in Fig. 5, it is easy to observe that estimate converges to (and then tracks) the state. Thus making example complete.

Fig. 2

Evolution of $\varDelta V_k+\varvec{\varepsilon }_{f_{a},k}^T\varvec{\varepsilon }_{f_{a},k}-\mu ^2\varvec{w}_{k}^T\varvec{w}_{k}-\mu ^2\varvec{w}_{k+1}^T\varvec{w}_{k+1}$

Fig. 3

Evolution of $\Vert e_{k}\Vert$

Fig. 4

Actuator fault and its estimate

Fig. 5

State of the second tank and its estimate

4 Conclusions

In this paper a novel approach allowing for the robust actuators fault estimation on the basis of the polytopic LPV model and UIO is proposed. In particular, the complete methodology of observer design with the robust $\mathcal {H}_{\infty }$ approach is proposed to settle the problem of robust fault diagnosis. The proposed UIO is designed in such a way that prescribed disturbance attenuation level is achieved with respect to the actuator fault estimation error while guaranteeing the convergence of the observer. The final part of the paper is concerned with an exhaustive case study regarding the fault estimation of the multi-tank system with the application of the proposed approach. The achieved results clearly show the performance and quality of the proposed method, which confirms its practical usefulness.