Design of unknown input observers for linear systems with unmatched unknown inputs

Abstract

We discuss an algebraic design of unknown input observers (UIOs) for linear systems with unknown inputs that do not satisfy the observer matching condition. Such a condition is often required for the existence of a UIO. To circumvent the restriction imposed by the observer matching condition, auxiliary outputs defined as the higher-order derivatives of the output measurements are introduced so that the observer matching condition is satisfied with respect to unknown inputs. The augmented outputs consisting of both the output measurements and the auxiliary outputs are defined, where the auxiliary outputs are estimated using higher-order sliding-mode exact differentiators based on the output measurements. Through suitable coordinate transformations, a system with the augmented outputs can be partitioned into two interconnected subsystems: an unknown input-free subsystem and an unknown input-dependent subsystem. The state vectors of the second subsystem are described in terms of augmented outputs. The unknown input-free subsystem is then used to design a UIO for estimation of the state vector. The performance of the UIO is verified through simulated numerical examples.

Keywords

Augmented outputs coordinate transformations observer matching condition unknown input observer unmatched unknown inputs

Introduction

It is important to design robust observers for systems driven by unknown inputs. Such a problem arises in systems that are subject to disturbances or that have inaccessible inputs, and in many applications such as feedback control, and fault detection and diagnosis (FDD) (Gaeid and Ping, 2010; Hostetter and Meditch, 1973; Kumar et al., 2012; Marquez and Riaz, 2005; Sotomayor and Odloak, 2005; Zarei and Poshtan, 2011).

One of the most successful robust observer design techniques is the use of the unknown input decoupling principle, in which an observer is designed to be insensitive to unknown inputs. One approach is based on modelling the unknown inputs using the response of a suitably chosen dynamical system (Hostetter and Meditch, 1973). This method, however, increases the dimension of the observer considerably. A more interesting approach can be found in Wang et al. (1975), which use a minimal-order unknown input observer (UIO) structure for linear systems with unknown inputs. The existence conditions for the observer are given in Kudva et al. (1980). With these important works, several approaches for designing reduced-order and full-order UIOs have been proposed; these include using a geometric approach (Bhattacharyya, 1978), using a generalized inverse matrix (Miller and Mukundan, 1982), using the inversion algorithm (Kobayashi and Nakamizo, 1982), using singular value decomposition (Fairman et al., 1984) and using straightforward matrix calculations (Yang and Wilde, 1988). The method described in Yang and Wilde (1988) takes into consideration the observer matching condition, which is a rank condition between the output distribution matrix and the unknown input distribution matrix. Darouach et al. (1994), which extend the result of Yang and Wilde (1988), present the design and the existence conditions of a full-order observer using the generalized inverse matrix, and show that the problem of designing full-order observers for linear systems with unknown inputs can be reduced to that of reduced-order observers. One notable work is described in Park and Stein (1988), wherein unknown input and state estimation algorithms were derived by differentiating the output measurement. Their method is based on singular value decomposition and is applicable when the observer matching condition is satisfied. In Boubaker (2005), the author reviewed both full and reduced-order UIOs with a generalized inverse matrix for linear systems with a matching condition. Transformations based on coordinate systems are used, and existence conditions are provided. FDD has been a prime example of how important such a UIO can be for process monitoring in the presence of unknown inputs (Chen and Saif, 2006; Lum et al., 2009; Park, 2010; Park and Lee, 2004).

In most of the aforementioned studies, UIOs were designed under the assumption that the necessary and sufficient conditions for the construction of UIOs are satisfied in the presence of unknown inputs. That is, the invariant zeros of the system must lie in the open left-half complex plane, and only the outputs are available but not their derivatives. The latter means that the design of observers for systems with unknown inputs requires the system to have relative degree one with respect to the unknown inputs. This restriction allows the decomposition of the state vector into two parts. The first part is not affected directly by the unknown inputs and needs to be observable, and the second part of the vector is completely known. In particular, it is required that the unknown inputs need to match the known outputs, i.e. the so-called observer matching condition is satisfied.

To overcome the restriction of relative degree one with respect to the unknown inputs, the use of higher-order sliding-mode differentiators (Levant, 2003) for exact observer design for linear systems with unknown inputs was studied in Floquet and Barbot (2006). The main idea of this algorithm is to take advantage of some equivalent output injections in order to generate fictitious outputs such that the system can be transformed into a set of block observable triangular forms. The first block of the triangular form is fed by the original inputs of the system while the subsequent blocks are fed by the fictitious outputs. Some similar works can be found in Bejarano et al. (2007), Floquet et al. (2007), Fridman et al. (2007, 2008, 2011), Kalsi et al. (2010) and Zhu (2012) that consider higher-order sliding-mode observers. In Floquet et al. (2007), Kalsi et al. (2010) and Zhu (2012), based on the concept of the relative degree of the output with respect to the unknown input, a kind of auxiliary output vector is defined so that the observer matching condition is met if the auxiliary outputs are included as system outputs. Here, the auxiliary outputs contain some unmeasured information (Zhu, 2012). Kalsi et al. (2010) use high-gain observers as approximate differentiators to obtain the estimates of the auxiliary outputs. The auxiliary outputs generated by the high-gain observers are then used to construct the sliding-mode observer, which was first introduced in paper Walcott and Żak (1987) and later modified for a more general class of systems in Hui and Żak (2005). Floquet et al. (2007) constructs higher-order sliding-mode observers acting as differentiators, specifically by using the super-twisting algorithm to obtain these auxiliary outputs so that the conventional unknown input sliding-mode observer proposed in Edwards and Spurgeon (1998) can be constructed even if the observer matching condition is not satisfied. Zhu (2012) considers the UIO design methods for linear systems with unknown inputs when the observer matching condition is not satisfied. The auxiliary outputs are defined so that the observer matching condition is met when the auxiliary outputs are included as system outputs and a higher-order sliding-mode observer is introduced to estimate the auxiliary outputs. An algebraic unknown input decoupling technique was combined with the design of a kind of observer for a system with the augmented outputs, leading to a UIO that has robustness property in the presence of unmatched unknown inputs.

In this study, we suggest the design of a UIO for linear systems with unknown inputs that do not satisfy the observer matching condition. To circumvent the restriction imposed by the observer matching condition and thereby broaden the class of linear systems for which a UIO can be constructed, our UIO is designed on the basis that the augmented outputs consisting of both the output measurements and the auxiliary outputs are available, where the auxiliary outputs are estimated using higher-order sliding-mode exact differentiators suggested by Floquet et al. (2007). Under suitable coordinate transformations, a system with the augmented outputs can be partitioned into two interconnected subsystems. The two subsystems are an unknown input-free subsystem and an unknown input-dependent subsystem, wherein the state vectors of the second subsystem are described in terms of the augmented outputs. The unknown input-free subsystem is then used to design a UIO whose structure is less complicated than that described in Zhu (2012, equation 9).

The remainder of this paper is organized as follows. The system description and problem statements are given in the next section. Then, the relative degree of the outputs with respect to the unknown inputs and the augmented outputs are defined. Higher-order sliding-mode exact differentiators based on the hierarchical super-twisting algorithm are reviewed, and are used to estimate a sufficient number of output derivatives. With the knowledge of an augmented output distribution matrix and sliding-mode exact differentiators, the design and existence conditions of a UIO are described. We then describe numerical examples to validate the proposed approach and present our conclusions.

System description and problem statements

Consider the problem of state observation for systems with the unknown inputs:

\dot{x} = A x + B u + E w

(1a)

y = {[y_{1} \dots y_{p}]}^{T} = C x, y_{i} = C_{i} x

(1b)

where $x \in ℜ^{n}$ is the state vector, $u \in ℜ^{m}$ is the known input vector, $w \in ℜ^{l}$ is the unknown input and $y \in ℜ^{p}$ is the output vector. We assumed that $A, B, C$ and $E$ are known real constant matrices of appropriate dimension. Without loss of generality, we further supposed that rank $(C) = p$ and rank $(E) = l$ , where $l \leq p$ .

Most of the UIOs developed so far are designed under the assumption that the necessary and sufficient conditions for the construction of UIOs are satisfied (Floquet et al., 2007).

Remark 1. The necessary and sufficient conditions for the construction of UIOs are as follows:

a) The invariant zeros of the system model given by the triple ( $A, E, C$ ) must lie in the open left-hand complex plane or, equivalently,

rank [\begin{matrix} s I_{n} - A & E \\ C & 0_{p \times l} \end{matrix}] = n + l

(2)

for all s such that $Re (s) \geq 0$

b) rank (C E) = rank (E) = l

(3)

□

We also assumed that the invariant zeros of the system model given by the triple ( $A, E, C$ ) are in the open left-hand complex plane. The condition described in Equation (3) is sometimes called the observer matching condition, which is introduced to realize a UIO decoupled from matched unknown inputs. Given the assumption that Equations (2) and (3) are satisfied, there exists a linear change of coordinates that places the original system into canonical form, and a UIO can be constructed. However, as pointed out by the authors of Bejarano et al. (2007), Floquet et al. (2007), Fridman et al. (2011), Kalsi et al. (2010) and Levant (2003), the observer matching condition described by Equation (3) is sometimes too restrictive for the design of a UIO for a practical system. Many physical systems that can be modelled based on Equation (1) do not satisfy the observer matching condition of Equation (3).

In this study, in order to circumvent the restriction imposed by the observer matching condition (and thus broaden the class of linear systems for which the UIO can be constructed), we first describe the generation of the auxiliary outputs for systems in which the observer matching condition does not hold due to rank( $(C E) < l$ . Here the auxiliary outputs are defined as the higher-order output derivatives of the output measurements. The auxiliary outputs are estimated using higher-order sliding-mode differentiators that are constructed based on the supper-twisting algorithm (Floquet et al., 2007), and the augmented outputs consisting of both the output measurements and the auxiliary outputs are defined. Then, under suitable coordinate transformations, the system with the augmented outputs is partitioned into two interconnected subsystems: an unknown input-free subsystem and an unknown input-dependent subsystem, wherein the state vectors of the second subsystem are described in terms of the augmented outputs. The unknown input-free subsystem is then used to design a UIO that estimates the state vector.

Generating auxiliary outputs for augmented outputs

In this section, the relative degree of the outputs with respect to the unknown inputs and the augmented outputs are defined. Higher-order sliding-mode exact differentiators based on the hierarchical super-twisting algorithm are reviewed and are used below to estimate a sufficient number of output derivatives.

Relative degree and auxiliary outputs

The definition of the relative degree of the ith output $y_{i}$ with respect to the unknown input $w$ (that is to say the number of times the output $y_{i}$ must be differentiated in order to have the unknown input $w$ explicitly appear) is given by Remark 2.

Remark 2. The relative degree of the ith output $y_{i}$ with respect to unknown input $w$ is defined to be the smallest positive integer $r_{i}$ such that

C_{i} A^{r_{i} - 1} E \neq 0, C_{i} A^{j} E = 0, j = 0 \dots r_{i} - 2

(4)

for $i = 1, \dots, p$ □

Based on Remark 2, we can choose integers $γ_{i} (1 \leq γ_{i} \leq r_{i})$ , $i = 1, \dots, p$ such that

C_{a} = [\begin{matrix} C_{1} \\ ⋮ \\ C_{1} A^{γ_{1} - 1} \\ \dots \dots \dots \\ ⋮ \\ \dots \dots \dots \\ C_{p} \\ ⋮ \\ C_{p} A^{γ_{p} - 1} \end{matrix}]

(5)

is of full rank with rank( $(C_{a} E)$ ) = rank( $E$ ), where $\sum_{i = 1}^{p} γ_{i} = γ$ , $p < γ$ . It is proved in Floquet et al. (2007) that the system zeros of the system model given by the triple ( $A, E, C_{a}$ ) are in the open left-hand complex plane if the triple ( $A, E, C$ ) satisfies Equation (2). Thus, under the assumption that rank( $C_{a} E$ ) = rank( $E$ ) and the triple ( $A, E, C$ ) satisfies (2), we can construct a UIO for the following system model:

\dot{x} = A x + B u + E w

(6a)

y_{a} = C_{a} x

(6b)

if the augmented outputs $y_{a} = C_{a} x$ , which consist of both the output measurements and the auxiliary outputs that are defined as the higher-order derivatives of the output measurements, are available. However, the auxiliary outputs in $y_{a}$ are not measurable and thus need to be estimated by observers.

Estimating the auxiliary outputs for the augmented outputs

In this section, based only on knowledge of $y = C x$ , we describe higher-order sliding-mode exact differentiators to estimate the auxiliary outputs in $y_{a}$ independent of the unknown input $w$ once a sliding motion is obtained. To do this, using Equations (4) and (5), we define

y_{i j} = C_{i} A^{j - 1} x, i = 1, \dots, p, j, = 1, \dots, γ_{i}

(7)

Then, from Equations (5) and (6b),

y_{a} = {[y_{a 1}^{T} y_{a 2}^{T} \dots y_{a p}^{T}]}^{T}

(8)

where $y_{a i} = {[y_{i 1} y_{i 2} \dots y_{i (γ_{i} - 1)}]}^{T}$ . Using Equations (5)–(8), we have

y_{a} = C_{a} x = [\begin{matrix} C_{1} \\ ⋮ \\ C_{1} A^{γ_{1} - 1} \\ \dots \dots \dots \\ ⋮ \\ \dots \dots \dots \\ C_{p} \\ ⋮ \\ C_{p} A^{γ_{p} - 1} \end{matrix}] x = [\begin{matrix} y_{a 1} \\ \dots \dots \dots \\ ⋮ \\ \dots \dots \dots \\ y_{a p} \end{matrix}] = [\begin{matrix} y_{11} \\ ⋮ \\ y_{1 γ_{1}} \\ \dots \dots \dots \\ ⋮ \\ \dots \dots \dots \\ y_{p 1} \\ ⋮ \\ y_{p γ_{p}} \end{matrix}]

(9)

With Equation (4), the dynamics of $y_{i j}$ , $i = 1 \dots, p$ and $j = 1, \dots, γ_{i}$ is given by

{\begin{matrix} {\dot{y}}_{i j} = y_{i (j + 1)} + C_{i} A^{j - 1} B u, j = 1, \dots, γ_{i} - 2 \\ {\dot{y}}_{i (γ_{i} - 1)} = y_{i γ_{i}} + C_{i} A^{γ_{i} - 2} B u = C_{i} A^{γ_{i} - 1} x + C_{i} A^{γ_{i} - 2} B u \\ {\dot{y}}_{i γ_{i}} = C_{i} A^{γ_{i}} x + C_{i} A^{γ_{i} - 1} B u + C_{i} A^{γ_{i} - 1} E w \end{matrix}

(10)

where $x$ and $w$ can be regarded as the unknown inputs. As in Bejarano et al. (2007), Floquet et al. (2007), Kalsi et al. (2010) and Levant (2003), we assume that $x$ and $\dot{x}$ are bounded, and that

| y_{i j} | \leq d_{i j}, i = 1, \dots, p, j = 1, \dots, γ_{i} - 1

(11)

which implies that $u$ is bounded. Consider $ν_{i j}$ , $i = 1, \dots, p$ and $j = 1, \dots, γ_{i} - 1$ , which is defined by the super twisting algorithm (Floquet et al., 2007). Then, for the system described in Equation (10), the following type of second-order sliding-mode observer can be constructed (Floquet et al., 2007):

{\begin{matrix} {\dot{\hat{y}}}_{i 1} = ν_{i 1} + C_{i} B u, ν_{i 1} = η_{i 1} | y_{i 1} - {\hat{y}}_{i 1} |^{\frac{1}{2}} sign (y_{i 1} - {\hat{y}}_{i 1}) + ϕ_{i 1} \\ {\dot{ϕ}}_{i 1} = β_{i 1} sign (y_{i 1} - {\hat{y}}_{i 1}) \\ {\dot{\hat{y}}}_{i j} = ν_{i j} + C_{i} A^{j - 1} B u, ν_{i j} = E_{i (j - 1)} [η_{i j} | {\tilde{y}}_{i j} - {\hat{y}}_{i j} |^{\frac{1}{2}} sign ({\tilde{y}}_{i j} - {\hat{y}}_{i j}) + ϕ_{i j}] \\ {\dot{ϕ}}_{i j} = β_{i j} sign ({\tilde{y}}_{i j} - {\hat{y}}_{i j}) \end{matrix}

(12)

for $j = 2, \dots, γ_{i}$ and $1 \leq i \leq p$ , where

{\begin{matrix} {\tilde{y}}_{i 1} = y_{i 1} \\ {\tilde{y}}_{i j} = {\tilde{η}}_{i j} | {\tilde{y}}_{i (j - 1)} - {\hat{y}}_{i (j - 1)} |^{\frac{1}{2}} sign ({\tilde{y}}_{i (j - 1)} - {\hat{y}}_{i (j - 1)}) + {\tilde{ϕ}}_{i j} \\ {\dot{\tilde{ϕ}}}_{i j} = {\tilde{β}}_{i j} sign ({\tilde{y}}_{i (j - 1)} - {\hat{y}}_{i (j - 1)}) \end{matrix}

(13)

and $E_{i j},$ $j = 1, \dots, γ_{i} - 2$ are defined as

E_{i j} = {\begin{matrix} 1 & i f | {\tilde{y}}_{i k} - {\hat{y}}_{i k} | = 0 for all k \leq j \\ 0 & otherwise \end{matrix}

(14)

and practically,

E_{i j} = {\begin{matrix} 1 & if | {\tilde{y}}_{i k} - {\hat{y}}_{i k} | \leq ε for all k \leq j \\ 0 & otherwise \end{matrix}

(15)

where $ε$ is a small positive constant; ${\tilde{η}}_{i j}$ , ${\tilde{β}}_{i j}$ , $η_{i j}$ and $β_{i j}$ are positive design parameters; and $sign (.)$ denotes the usual sign function. It can be shown that the error between the system described in Equation (10) and the observer states vanishes in a finite time, and that the sliding manifolds are reached one-by-one. At each step, a subdynamic of dimension one is obtained; consequently, no peaking phenomena appear. More precisely, $E_{i j}$ is equal to zero if there exists $k$ ( $k \leq j$ ) such that ${\tilde{y}}_{i k} - {\hat{y}}_{i 1} | > ε$ (by definition ${\tilde{y}}_{i 1} = y_{i 1}$ ); otherwise $E_{i j} = 1$ [see Bejarano et al. (2007), Floquet et al. (2007), Kalsi et al. (2010) and Levant (2003), for more details]. Let $e_{i j} = y_{i j} - {\hat{y}}_{i j}$ . It follows from Equations (10) and (12) that the error dynamics are given by

{\begin{matrix} {\dot{e}}_{i 1} = y_{i 2} - ν_{i 1} \\ {\dot{e}}_{i j} = y_{i (j + 1)} - ν_{i j}, j = 2, \dots, γ_{i} - 1 \end{matrix}

(16)

for $1 \leq i \leq p$ . By choosing sufficiently large ${\tilde{η}}_{i j}$ , ${\tilde{β}}_{i j}$ , $η_{i j}$ and $β_{i j}$ , Equation (16) enters the sliding-mode on the manifold $e_{i 1} = \dots = e_{i (γ_{i} - 1)} = 0$ after a finite time, which implies that

{\dot{e}}_{i 1} = \dots = {\dot{e}}_{i (γ_{i} - 1)} = 0

(17)

Thus, from Equations (7), (16) and (17), by choosing suitable output injections $ν_{i j}$ , the following relations hold after a finite time:

ν_{i j} = C_{i} A^{j} x, j = 1, \dots, γ_{i} - 1

(18)

for $1 \leq i \leq p$ . This means that $y_{s} = C_{a} x$ , i.e. $y_{s} = y_{a}$ after a finite time, where

y_{s} = {[y_{s 1}^{T} \dots y_{s p}^{T}]}^{T}

(19a)

y_{s i} = [\begin{array}{l} y_{s i 1} \\ y_{s i 2} \\ y_{s i 3} \\ ⋮ \\ y_{s i γ_{i}} \end{array}] = [\begin{array}{l} y_{i 1} \\ v_{i 1} \\ v_{i 2} \\ ⋮ \\ v_{i (γ_{i} - 1)} \end{array}], i = 1, \dots, p

(19b)

Design procedure to estimate the auxiliary outputs and form the augmented outputs

Under the assumption that $x$ , $\dot{x}$ and $y_{i j}$ are bounded, the proposed design procedure to estimate the auxiliary outputs and form the augmented outputs can be summarized as follows:

Step 1: Verify whether rank $(C E)$ ≠rank $(E)$ for the given system (Equation 1), where rank $(C) = p$ and rank $(E) = l$ , ( $l \leq p$ ). If rank $(C E)$ ≠rank $(E)$ , progress next steps.

Step 2: Find the relative degree $r_{i}$ of the ith output $y_{i}$ , ( $i = 1, \dots, p$ ) with respect to unknown input $w$ , as shown in Equation (4).

Step 3: With integers $γ_{i}$ ( $1 \leq γ_{i} \leq r_{i}$ ), $i = 1, \dots, p$ , obtain the augmented output matrix $C_{a}$ , as described in Equation (5).

Step 4: Check whether rank( $C_{a} E$ ) = rank( $E$ ). If rank( $C_{a} E$ ) = rank( $E$ ), progress next steps to estimate $y_{i j}$ , $i = 1, \dots, p$ , $j = 1, \dots, γ_{i}$ , as shown in Equation (9).

Step 5: Set the initial values ${\hat{y}}_{i j} (0) = 0$ and $ϕ_{i j} (0) = 0$ , $i = 1, \dots, p$ , $j = 1, \dots, γ_{i}$ to realize the sliding-mode observer (12).

Step 6: For $i = 1, \dots, p$ , set $y_{i} = y_{i 1}$ (1b) to ${\tilde{y}}_{i 1} = y_{i 1}$ , and in the case of $γ_{i} \geq 3$ , solve differential equations ${\dot{\tilde{ϕ}}}_{i j}$ (13) with the initial values ${\tilde{ϕ}}_{i j} (0) = 0$ and find ${\tilde{y}}_{i j}$ (13) for $i = 1, \dots, p$ , $j = 2, \dots, γ_{i} - 1$ .

Step 7: With (15), solve differential equations ${\dot{ϕ}}_{i j}$ (12) to find $v_{i j}$ . Then solve ${\dot{\hat{y}}}_{i j}$ (12) to obtain ${\hat{y}}_{i j}$ for $i = 1, \dots, p$ , $j = 1, \dots, γ_{i} - 1$ .

Step 8: Find $y_{s i (j + 1)} = v_{i j}$ for $i = 1, \dots, p$ , $j = 1, \dots, γ_{i} - 1$ and compute $y_{s}$ (19).

The signals $y_{s}$ are directly used to construct a UIO for systems that do not satisfy the observer matching condition described next.

Design and existence conditions of a UIO

Design of the UIO

With the knowledge of an augmented output distribution matrix and sliding-mode exact differentiators addressed previously, the design of our UIO is described.

When rank $(X) = l$ of the matrix $X \in ℜ^{γ \times l}$ , then $X^{†} = {(X^{T} X)}^{- 1} X^{T}$ , which is the left pseudoinverse of $x$ since $X^{†} X = I_{l}$ where $X^{T}$ denotes the transpose of a matrix $x$ . From above, as rank $(C_{a} E) = l$ , $C_{a} E$ possesses ${(C_{a} E)}^{†}$ . We define the square matrix

P = I_{n} - E {(C_{a} E)}^{†} C_{a}, P \in ℜ^{n \times n}

(20)

for the use of a state coordinate transformation. $P$ is a projector satisfying $P^{2} = P$ , i.e. $P$ is idempotent (Ben-Israel and Greville, 2003). Under the assumption that rank $(E) = l$ , there exists a non-singular matrix

T = [N E], N \in ℜ^{n \times (n - l)}

(21)

for a state coordinate transformation of the system described in Equation (6), where $N = P N_{0}$ and $N_{0} \in ℜ^{n \times (n - l)}$ is an arbitrary matrix such that the matrix $[N_{0} E]$ is non-singular.

Lemma 1. If the matrix $[N_{0} E]$ is non-singular, the matrix $T$ is also non-singular.

Proof. The proof is similar to that described in Lee and Bien (1994) and is omitted here for brevity.□

The system described by Equation (6) with unknown inputs can be decomposed into the following form:

\dot{\bar{x}} = \bar{A} \bar{x} + \bar{B} u + \bar{E} w

(22a)

y_{a} = \bar{C} \bar{x}

(22b)

where

\begin{array}{l} x = T \bar{x} = T [\begin{matrix} {\bar{x}}_{1} \\ {\bar{x}}_{2} \end{matrix}], \bar{A} = T^{- 1} A T = [\begin{matrix} {\bar{A}}_{11} & {\bar{A}}_{12} \\ {\bar{A}}_{21} & {\bar{A}}_{22} \end{matrix}], \\ \bar{B} = T^{- 1} B = [\begin{matrix} {\bar{B}}_{1} \\ {\bar{B}}_{2} \end{matrix}], \\ E = T^{- 1} E = [\begin{matrix} 0 \\ I_{l} \end{matrix}], \\ \bar{C} = C_{a} T = [C_{a} N C_{a} E] \end{array}

(23)

with ${\bar{x}}_{1} \in ℜ^{n - l}$ and ${\bar{x}}_{2} \in ℜ^{l}$ . In Equation (22a), the differential equations corresponding to the state subvector ${\bar{x}}_{2}$ involve directly the unknown input $w$ . Dropping the differential equations described in Equation (22a) gives the unknown input-free system

[I_{n - l} 0] \dot{\bar{x}} = [{\bar{A}}_{11} {\bar{A}}_{12}] \bar{x} + {\bar{B}}_{1} u

(24a)

y_{a} = [C_{a} N C_{a} E] \bar{x}

(24b)

On the other hand, when rank $(X) = l$ of the matrix $X \in ℜ^{l \times γ}$ , $X^{‡} = X^{T} {(X X^{T})}^{- 1}$ is the right pseudoinverse of $x$ since $X X^{‡} = I_{l}$ . As rank $(C_{a} E) = l, {(C_{a} E)}^{T}$ possesses ${({(C_{a} E)}^{T})}^{‡}$ . We define the square matrix

M = I_{γ} - {({(C_{a} E)}^{T})}^{‡} {(C_{a} E)}^{T}, M \in ℜ^{γ \times γ}

(25)

for use in the output transformation. $M$ is also a projector satisfying $M^{2} = M$ , i.e. $M$ is idempotent (Ben-Israel and Greville, 2003). As rank $(C_{a} E) = l$ , we choose a non-singular matrix

U = [C_{a} E Q], Q \in ℜ^{γ \times (γ - 1)}

(26)

for algebraic manipulation of the output (Equation 24b) of the system, where $Q = M Q_{0}$ and $Q_{0} \in ℜ^{γ \times (γ - l)}$ is entirely arbitrary as if matrix $[C_{a} E Q_{0}]$ is non-singular.

Lemma 2. If the matrix $[C_{a} E Q_{0}]$ is non-singular, the matrix $U$ is also non-singular.

Proof. The proof is similar to that described in Lee and Bien (1994) and is omitted here for brevity.□

Defining

U^{- 1} U = [\begin{matrix} U_{1} \\ U_{2} \end{matrix}] U_{1} \in ℜ^{l \times γ}, U_{2} \in ℜ^{(γ - 1) \times γ}

(27)

gives

U^{- 1} U = [\begin{matrix} U_{1} \\ U_{2} \end{matrix}] [C_{a} E Q] = [\begin{matrix} U_{1} C_{a} E & U_{1} Q \\ U_{2} C_{a} E & U_{2} Q \end{matrix}] = [\begin{matrix} I_{l} & 0 \\ 0 & I_{γ - 1} \end{matrix}]

(28)

Premultiplying both sides of Equation (24b) by Equation (27) and considering Equation (28), we have

U_{1} y_{a} = U_{1} C_{a} N {\bar{x}}_{1} + {\bar{x}}_{2}

(29a)

U_{2} y_{a} = U_{2} C_{a} N {\bar{x}}_{1}

(29b)

where ${\bar{x}}_{2}$ is given by $y_{a}$ and ${\bar{x}}_{1}$ .

Lemma 3. Assuming that $U$ is non-singular, $U^{- 1} = U^{†}$ can be expressed by partitioned matrices of ${(C_{a} E)}^{†}$ and $Q^{†}$ as

U^{- 1} = [\begin{matrix} U_{1} \\ \dots \\ U_{2} \end{matrix}] = [\begin{matrix} {(C_{a} E)}^{†} \\ \dots \dots \dots \\ Q^{†} \end{matrix}]

(30)

Proof. The proof is similar to that given in Park (2010) and is omitted here for brevity.□

From Equations (20), (21) and (30), we note that

\begin{array}{l} U_{1} C_{a} N = {(C_{a} E)}^{†} C_{a} (I_{n} - E {(C_{a} E)}^{†} C_{a}) N_{0} \\ = ({(C_{a} E)}^{†} C_{a} - {(C_{a} E)}^{†} C_{a}) N_{0} = 0_{l \times (n - 1)} \end{array}

(31)

Therefore, Equation (29a) can be stated more simply as follows:

U_{1} y_{a} = {\bar{x}}_{2}

(32a)

U_{2} y_{a} = U_{2} C_{a} N {\bar{x}}_{1}

(32b)

where ${\bar{x}}_{2}$ is obtained from only the augmented outputs, $y_{a}$ . By substituting Equation (32a) into Equation (24a) and combining it with Equation (32b), we obtain

{\dot{\bar{x}}}_{1} = {\bar{A}}_{11} {\bar{x}}_{1} + {\bar{B}}_{1} u + {\bar{A}}_{12} U_{1} y_{a}

(33a)

{\bar{y}}_{a} = {\bar{C}}_{1} {\bar{x}}_{1}

(33b)

where ${\bar{C}}_{1} = U_{2} C_{a} N, {\bar{y}}_{a} = U_{2} y_{a}$ .

Under the assumption that the pair $({\bar{A}}_{11}, {\bar{C}}_{1})$ is observable or detectable, a UIO can be constructed for the unknown input-free system (33) as follows:

\begin{array}{l} {\dot{\hat{\bar{x}}}}_{1} = ({\bar{A}}_{11} - L {\bar{C}}_{1}) {\hat{\bar{x}}}_{1} + {\bar{B}}_{1} u + L {\bar{y}}_{s} + {\bar{A}}_{12} U_{1} y_{s} \\ = ({\bar{A}}_{11} - L {\bar{C}}_{1}) {\hat{\bar{x}}}_{1} + {\bar{B}}_{1} u + (L U_{2} + {\bar{A}}_{12} U_{1}) y_{s} \end{array}

(34)

where ${\hat{\bar{x}}}_{1} \in ℜ^{n - l}, {\bar{y}}_{s} = U_{2} y_{s}$ and $y_{s}$ is given by Equation (19). The matrix $L \in ℜ^{(n - l) \times (γ - l)}$ is a pre-specified observer gain matrix that makes the matrix ${\bar{A}}_{11} - L {\bar{C}}_{1}$ stable. After a finite time, the estimation error dynamics for the state ${\bar{x}}_{1}$ from Equations (33) and (34) become

{\dot{e}}_{{\bar{x}}_{1}} = ({\bar{A}}_{11} - L C_{1}) e_{{\bar{x}}_{1}}

(35)

where $e_{{\bar{x}}_{1}} = {\hat{\bar{x}}}_{1} - {\bar{x}}_{1}$ . If ${\bar{A}}_{11} - L {\bar{C}}_{1}$ is Hurwitz, $e_{{\bar{x}}_{1}}$ will converge exponentially to zero. From Equation (32a), the state can also be estimated, after a finite time, as

\hat{x} = T \hat{\bar{x}} = T [\begin{matrix} {\hat{\bar{x}}}_{1} \\ {\hat{\bar{x}}}_{2} \end{matrix}] = T [\begin{matrix} {\hat{\bar{x}}}_{1} \\ U_{1} y_{a} \end{matrix}]

(36)

Existence conditions of the UIO

The existence conditions for the UIO described by Equation (34) are given as follows.

Theorem 1. If rank $(C) = p$ , rank $(E) = l, l \leq p$ and rank $(C_{a} E) = r a n k (E)$ , but rank $(C E) \neq r a n k (E)$ for the given system described in Equation (1), the following three existence conditions for the UIO (34) are equivalent:

The pair $({\bar{A}}_{11}, {\bar{C}}_{1})$ is detectable (observable);

rank $[\begin{matrix} s I_{n - l} - {\bar{A}}_{11} & - {\bar{A}}_{12} \\ C_{a} N & C_{a} E \end{matrix}] = n, \forall s \in C, R e (s) \geq 0;$

rank $[\begin{matrix} s I_{n} - A & E \\ C_{a} & 0 \end{matrix}] = n + l, \forall s \in C, R e (s) \geq .$

Proof. See Appendix.□

Design procedure for the UIO

From the preceding results for the UIO (Equation 34), the design procedure for estimating the states of the given system (1) is summarized as follows:

Step 1: Check whether the existence conditions of the UIO described in Theorem 1 are satisfied for the given system (Equation 1). If the existence conditions are satisfied, progress next steps.

Step 2: Choose $N_{0}$ such that the matrix $[N_{0} E]$ is non-singular and then find $N = P N_{0}$ with $P$ (Equation 20).

Step 3: Construct $T$ (Equation 21) and then compute $T^{- 1}$ .

Step 4: Select $Q_{0}$ so that the matrix $U$ is non-singular and then find $Q = M Q_{0}$ with $M$ (Equation 25)

Step 5: Calculate $U^{- 1}$ . From Equation (27), select $U_{1}$ and $U_{2}$ .

Step 6: Obtain ${\bar{A}}_{11}$ and ${\bar{B}}_{1}$ from Equation (23) and ${\bar{C}}_{1} = U_{2} C_{a} N$ from Equation (33b).

Step 7: Choose the desired UIO eigenvalues and then compute the UIO gain matrix $L$ that makes the matrix ${\bar{A}}_{11} - L {\bar{C}}_{1}$ stable.

Step 8: Compute ${\bar{A}}_{11} - L {\bar{C}}_{1}$ and $L U_{2} + A_{12} U_{1}$ .

Step 9: Estimate the states of the reduced-order system (33) from the reduced-order UIO (34).

Step 10: Compute the state estimates of the given system (1) from Equation (36).

Numerical examples

In this section, we illustrate the effectiveness of the UIO for systems with unmatched unknown inputs through two numerical examples.

Example 1: UIO design and simulation results

We first consider a system (Zhu, 2012) described by

\begin{array}{l} A = [\begin{array}{l} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ - 1 & - 5 & - 10 & - 10 & - 5 \end{array}], \\ B = E = [\begin{array}{l} 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}], C = [\begin{array}{l} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}] \end{array}

(37)

where $x = {[x_{1} x_{2} x_{3} x_{4} x_{5}]}^{T}$ . It is easy to check that the system’s rank( $(C E)$ )≠rank( $E$ ) because $C E = 0$ , where rank( $(C E)$ )=0 and rank( $E$ )=1. Therefore, standard UIO approaches cannot be applied to this system. The relative degrees of the first output $y_{1}$ and the second output $y_{2}$ with respect to unknown input $w$ are $r_{1} = 2$ and $r_{2} = 3$ , respectively. Thus, we choose $γ_{1} = 2$ and $γ_{2} = 3$ such that

C_{a} = [\begin{matrix} C_{1} \\ C_{1} A \\ C_{2} \\ C_{2} A \\ C_{2} A^{2} \end{matrix}] = [\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ - 1 & - 5 & - 10 & - 10 & 5 \end{matrix}]

(38)

is of full rank with rank( $C_{a} E$ )=rank( $E$ )=1, where rank( $C_{a}$ )=5. We used the sliding-mode observer to estimate the auxiliary outputs $y_{12} = C_{1} A x, y_{22} = C_{2} A x$ and $y_{23} = C_{2} A^{2} x$ , where

y_{a} = C_{a} x = [\begin{matrix} C_{1} \\ C_{1} A \\ C_{2} \\ C_{2} A \\ C_{2} A^{2} \end{matrix}] x = [\begin{matrix} y_{11} \\ y_{12} \\ y_{21} \\ y_{22} \\ y_{23} \end{matrix}]

(39)

To produce to the second, fourth and fifth outputs in $y_{a}$ , the following sliding-mode observer can be constructed:

{\begin{array}{l} {\dot{\hat{y}}}_{11} = v_{11} + C_{1} B u, v_{11} = η_{11} {| y_{11} - {\hat{y}}_{11} |}^{\frac{1}{2}} sign (y_{11} - {\hat{y}}_{11}) + ϕ_{11} \\ {\dot{ϕ}}_{11} = β_{11} sign (y_{11} - {\hat{y}}_{11}) \\ {\dot{\hat{y}}}_{21} = v_{21} + C_{2} B u, v_{21} = η_{21} {| y_{21} - {\hat{y}}_{21} |}^{\frac{1}{2}} sign (y_{21} - {\hat{y}}_{21}) + ϕ_{21} \\ {\dot{ϕ}}_{21} = β_{21} sign (y_{21} - {\hat{y}}_{21}) \\ {\dot{\hat{y}}}_{22} = v_{22} + C_{2} A B u, v_{22} = E_{21} [η_{22} {| {\tilde{y}}_{22} - {\hat{y}}_{22} |}^{\frac{1}{2}} sign ({\tilde{y}}_{22} - {\hat{y}}_{22}) + ϕ_{22}] \\ {\dot{ϕ}}_{22} = β_{22} sign ({\tilde{y}}_{22} - {\hat{y}}_{22}) \end{array}

(40a)

where

{\begin{matrix} {\tilde{y}}_{11} = y_{11} \\ {\tilde{y}}_{21} = y_{21} \\ {\tilde{y}}_{22} = {\tilde{η}}_{22} {| {\tilde{y}}_{21} - {\hat{y}}_{21} |}^{\frac{1}{2}} sign ({\tilde{y}}_{21} - {\hat{y}}_{21}) + {\tilde{ϕ}}_{22} \\ {\dot{\tilde{ϕ}}}_{22} = {\tilde{β}}_{22} sign ({\tilde{y}}_{21} - {\hat{y}}_{21}) \end{matrix}

(40b)

E_{21} = {\begin{matrix} 1 & if | {\tilde{y}}_{21} - {\hat{y}}_{21} | \leq ε \\ 0 & otherwise \end{matrix}

(40c)

The design parameters for the sliding-mode observer were selected as $ε = 0.01, β_{11} + β_{22} = 2, η_{11} = η_{22} = 6, β_{21} = {\tilde{β}}_{22} = 5$ and $η_{22} = {\tilde{η}}_{21} = 10$ . Moreover, the function ‘sign’ was approximated using a saturation function:

s a t (\cdot) = \frac{.}{| \cdot | + δ}

(41)

where the threshold $δ > 0$ was chosen to be $δ = 0.01$ . The auxiliary outputs are generated as

y_{s} = [\begin{array}{l} y_{s 11} \\ y_{s 12} \\ y_{s 21} \\ y_{s 22} \\ y_{s 23} \end{array}] = [\begin{array}{l} y_{11} \\ v_{11} \\ y_{21} \\ v_{21} \\ v_{22} \end{array}]

(42)

Now, with Equation (42), we can design the UIO. Choosing

N_{o} = [\begin{array}{l} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] Q_{o} = [\begin{array}{l} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]

(43)

gives the non-singular matrices $T$ and $U$ . Also, we can check that $U_{1} C_{a} N = 0 .$ . As a result, the reduced-order system is obtained as

{\dot{\bar{x}}}_{1} = {\bar{A}}_{11} {\bar{x}}_{1} + {\bar{B}}_{1} u + {\bar{A}}_{12} U_{1} y_{a}

(44a)

{\bar{y}}_{a} = {\bar{C}}_{1} {\bar{x}}_{1}

(44b)

where

\begin{array}{l} {\bar{A}}_{11} = [\begin{matrix} - 0.1923 & - 1.9231 & - 1.9231 & - 0.9615 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ - 0.0385 & - 0.3846 & - 0.3846 & - 0.1923 \end{matrix}] \\ {\bar{A}}_{12} = [\begin{array}{l} 1 \\ 0 \\ 0 \\ - 5 \end{array}], U_{1} = [\begin{array}{l} 0 & 0.0385 & 0 & 0 & - 0.1923 \end{array}] \\ {\bar{B}}_{1} = [\begin{array}{l} 0 \\ 0 \\ 0 \\ 0 \end{array}], {\bar{C}}_{1} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ - 1 & - 10 & - 10 & - 5 \end{matrix}] \end{array}

It is easy to verify that the pair $({\bar{A}}_{11}, {\bar{C}}_{1})$ is observable. Therefore, by choosing the desired eigenvalues of $({\bar{A}}_{11} - L {\bar{C}}_{1})$ as ${- 1 - 2 - 3 - 4}$ , we can design the UIO:

{\dot{\hat{\bar{x}}}}_{1} = ({\bar{A}}_{11} - L {\bar{C}}_{1}) {\hat{\bar{x}}}_{1} + {\bar{B}}_{1} u + (L U_{2} + {\bar{A}}_{12} U_{1}) y_{s}

(45)

Computer simulations were performed. We assumed that the initial condition was $x (0) = {[0.5 0.5 0.5 - 0.5 - 0.5]}^{T}$ and the input $U$ was set to be zero. The unknown input $w$ consisted of a sawtooth signal with an amplitude of 2 and frequency of 1 Hz. The true and estimated values of the auxiliary outputs $y_{12}$ , $y_{22}$ and $y_{23}$ are shown in Figure 1. Figure 2 shows the state estimation performance. In the figures, the estimates are shown with dotted lines. These figures illustrate that the proposed scheme for the system with unmatched unknown inputs is successful in estimating the auxiliary outputs and system states.

Figure 1.

True and estimated values of the auxiliary outputs $y_{12}$ , $y_{22}$ and $y_{23}$ .

Figure 2.

True and estimated values of the states $x_{1}, x_{2}, x_{3}, x_{4}, and x_{5}$ .

Example 2: UIO design and simulation results

In this example, we consider a linearized vertical takeoff and landing (VTOL) helicopter model presented in De Farias et al. (2000) and Huang et al. (2011). The simplified dynamics of this VTOL aircraft in the vertical plane can be described as Equation (1), where

\begin{array}{l} x = [\begin{matrix} horizontal velocity [knots] \\ vertical velocity [knots] \\ pitch rate [deg / s] \\ pitch angle [deg] \end{matrix}], \\ u = [\begin{matrix} collective pitch control \\ longitudal cyclic pitch angle \end{matrix}] \\ A = [\begin{array}{l} - 0 : 0336 & 0 : 0271 & 0 : 0188 & - 0 : 4555 \\ 0 : 0482 & - 1 : 0100 & 0 : 0024 & - 4 : 0208 \\ 0 : 1002 & 0 : 2855 & - 0 : 7070 & 1 : 3229 \\ 0 & 0 & 1 & 0 \end{array}], \\ B = [\begin{array}{l} 0 : 4422 & 0 : 1761 \\ 3 : 0447 & - 7 : 5922 \\ - 5 : 5200 & 4 : 9900 \\ 0 & 0 \end{array}] C = [\begin{array}{l} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}] \end{array}

(46a)

The collective pitch control is used for the vertical motion, and the longitudinal cyclic pitch input is used to control the horizontal velocity of the aircraft. The given dynamic equation is computed for typical loading and flight conditions of the VTOL helicopter at the airspeed of 135 knots. As the airspeed changes, the dynamical equation of the model changes, too. It is assumed that the most significant parameter perturbation occurs in the matrix $A$ as the airspeed changes:

Δ A = [\begin{array}{l} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0.2 & 0 & 1.2 \\ 0 & 0 & 0 & 0 \end{array}]

(46b)

The additive unknown input term $E W$ can be thus represented as

E = {[0 0 1 0]}^{T}, w = [0 0.2 0 1.2] x

(46c)

With the matrices of (46), we have rank $(C E) \neq rank (E)$ because $C E = 0$ , where rank $(C E) = 0$ and rank $(E) = 1$ . Therefore, standard UIO approaches cannot be applied to this system. The relative degrees of the first output $y_{1}$ and the second output $y_{2}$ with respect to the unknown input $w$ are $r_{1} = 2$ and $r_{2} = 2$ , respectively. Thus, we choose $γ_{1} = 2$ and $γ_{2} = 2$ such that

C_{a} = [\begin{matrix} C_{1} \\ C_{1} A \\ C_{2} \\ C_{1} A \end{matrix}] = [\begin{array}{l} 1 & 0 & 0 & 0 \\ - 0.0336 & 0.0271 & 0.0188 & - 0.4555 \\ 0 & 1 & 0 & 0 \\ 0.0482 & - 1.0100 & 0.0024 & - 4.0208 \end{array}]

(47)

is of full rank with rank $(C_{a} E)$ = rank $(E) = 1$ , where rank $(C_{a}) = 4$ . We employ two separate sliding-mode observers to estimate the auxiliary outputs $y_{12} = C_{1} A x$ and $y_{22} = C_{2} A x$ , respectively, where

y_{a} = C_{a} x = [\begin{matrix} C_{1} \\ C_{1} A \\ C_{2} \\ C_{2} A \end{matrix}] x = [\begin{matrix} y_{11} \\ y_{12} \\ y_{21} \\ y_{22} \end{matrix}]

(48)

To generate the second and fourth outputs in $y_{a}$ , the sliding-mode observer is designed:

\begin{array}{l} {\dot{\hat{y}}}_{11} = v_{11} + C_{1} B u, v_{11} = η_{11} {| y_{11} - {\hat{y}}_{11} |}^{\frac{1}{2}} sign (y_{11} - {\hat{y}}_{11}) + ϕ_{11} \\ {\dot{ϕ}}_{11} = β_{11} sign (y_{11} - {\hat{y}}_{11}) \\ {\dot{\hat{y}}}_{21} = v_{21} + C_{2} B u, v_{21} = η_{21} {| y_{21} - {\hat{y}}_{21} |}^{\frac{1}{2}} sign (y_{21} - {\hat{y}}_{21}) + ϕ_{21} \\ {\dot{ϕ}}_{21} = β_{21} sign (y_{21} - {\hat{y}}_{21}) \end{array}

(49)

where

y_{s} = [\begin{matrix} y_{s 11} \\ y_{s 12} \\ y_{s 21} \\ y_{s 22} \end{matrix}] = [\begin{matrix} y_{11} \\ v_{11} \\ y_{21} \\ v_{21} \end{matrix}]

(50)

The design parameters for generating the auxiliary outputs were selected as $ε = 0.01$ , $β_{11} = 1$ , $η_{11} = 2$ , $β_{21} = 2$ and $η_{21} = 4$ , and the function ‘sign’ is approximated by a saturation function as in Equation (41). Now, with Equation (46), we can design the UIO. Choosing

N_{0} = [\begin{array}{l} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}], Q_{0} = [\begin{array}{l} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]

(51)

gives the non-singular matrices $T$ and $U$ , where $γ = 4$ . Also, we can check that $U_{1} C_{a} N = 0$ . As a result, the reduced-order system is obtained as

{\dot{\bar{x}}}_{1} = {\bar{A}}_{11} {\bar{x}}_{1} + {\bar{B}}_{1} u + {\bar{A}}_{12} U_{1} y_{a}

(52a)

{\bar{y}}_{a} = {\bar{C}}_{1} {\bar{x}}_{1}

(52b)

where

\begin{array}{l} {\bar{A}}_{11} = [\begin{matrix} - 0.0066 & 0.1273 & 0.4978 \\ 0.0516 & - 0.9972 & - 3.8991 \\ 1.4365 & 5.3300 & 50.7052 \end{matrix}] \\ {\bar{A}}_{12} = [\begin{matrix} 0.0188 \\ 0.0024 \\ 1 \end{matrix}], U_{1} = [\begin{matrix} 0 & 52.3385 & 0 & 6.6815 \end{matrix}] \\ {\bar{B}}_{1} = [\begin{matrix} 0.4422 & 0.1761 \\ 3.0447 & - 7.5922 \\ 0 & 0 \end{matrix}] \\ {\bar{C}}_{1} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0.0525 & - 1.0135 & - 3.9627 \end{matrix}] \end{array}

Here, the pair $({\bar{A}}_{11}, {\bar{C}}_{1})$ is observable. Therefore, by choosing the desired eigenvalues of $({\bar{A}}_{11} - L {\bar{C}}_{1})$ as ${- 10 - 14 - 18}$ , we can design the UIO:

{\dot{\hat{\bar{x}}}}_{1} = ({\bar{A}}_{11} - L {\bar{C}}_{1}) {\hat{\bar{x}}}_{1} + {\bar{B}}_{1} u + (L U_{2} + {\bar{A}}_{12} U_{1}) y_{s}

(53)

Computer simulations were carried out under the assumption that the initial conditions are $x (0) = {[0.1 0 0 0]}^{T}$ for the system. The open loop poles of the nominal system are located at {0.2817±j0.0977, −0.3332, −1.9809}, and the system is thus unstable. The unstable system is here stabilized by using a proper control law for asymptotic regulation of the system. Figure 3 represents the estimated and true values of the auxiliary outputs and Figure 4 shows the state estimation performance. These figures illustrate that we can design a UIO, which can asymptotically converge to the actual states within some desired accuracy, even in the presence of unmatched unknown inputs.

Figure 3.

True and estimated values of the auxiliary outputs $y_{12}$ and $y_{22}$ .

Figure 4.

True and estimated values of the states $x_{1}, x_{2}, x_{3}, and x_{4}$ .

Conclusion

In this paper, an algebraic method for designing UIOs for a class of linear systems, which cannot satisfy the design conditions of conventional UIOs, has been considered. First, the auxiliary outputs were defined so that the observer matching condition is met when the auxiliary outputs are included as system outputs and a higher-order sliding-mode observer was introduced to estimate the auxiliary outputs, where the auxiliary outputs denote the higher-order derivatives of the output measurements. Then, an algebraic unknown input decoupling technique based on suitable coordinate transformations was combined with the design of a reduced-order UIO for a system with the augmented outputs, which were defined as consisting of both the output measurements and the auxiliary outputs. The designed UIO was verified through simulation studies performed in terms of numerical examples.

Footnotes

Appendix: proof of Theorem 1

1) First, the equivalence of conditions (a) and (b) of Theorem 1 is proved. Condition (a) means

(54)

rank [\begin{matrix} s I_{n - l} - {\bar{A}}_{11} \\ {\bar{C}}_{1} \end{matrix}] = n - l, \forall s \in C, R e (s) \geq 0

It is only necessary to prove that statement (b) is equivalent to Equation (54). Let

(55a)

S = [\begin{matrix} I_{n - l} & 0 \\ 0 & U^{- 1} \end{matrix}] \in ℜ^{(n - l + γ) \times (n - l + γ)}

where $U^{- 1}$ is defined by Equation (27). It is true that $S$ is a non-singular matrix with rank $(S) = n - l + γ$ and rank $(S) > n$ under the assumption $γ > n$ . Then,

(55b)

\begin{array}{l} rank [\begin{matrix} s I_{n - l} - {\bar{A}}_{11} & - {\bar{A}}_{12} \\ C_{a} N & C_{a} E \end{matrix}] = \\ rank S [\begin{matrix} s I_{n - l} - {\bar{A}}_{11} & - {\bar{A}}_{12} \\ C_{a} N & C_{a} E \end{matrix}] = rank [\begin{matrix} s I_{n - l} - {\bar{A}}_{11} & - {\bar{A}}_{12} \\ U^{- 1} C_{a} N & U^{- 1} C_{a} E \end{matrix}] \end{array}

Substituting $U^{- 1}$ (Equation 27) into Equation (55b) and considering Equation (28) yields $U_{1} C_{a} N = 0_{l \times (n - l)}$ (Equation 31) and ${\bar{C}}_{1} = U_{2} C_{a} N$ (Equation 33b). Then, we have

(55c)

rank [\begin{matrix} s I_{n - l} - {\bar{A}}_{11} & - {\bar{A}}_{12} \\ [\begin{matrix} 0_{l \times (n - l)} \\ {\bar{C}}_{1} \end{matrix}] & [\begin{matrix} I_{l} \\ 0_{(γ - l) \times l} \end{matrix}] \end{matrix}] = l + rank [\begin{matrix} s I_{n - l} - {\bar{A}}_{11} \\ {\bar{C}}_{1} \end{matrix}]

The equivalence of Equation (55c) and condition (b) means that

(55d)

rank [\begin{matrix} s I_{n - l} - {\bar{A}}_{11} \\ {\bar{C}}_{1} \end{matrix}] = n - l

Thus, based on Equations (55d) and (54), condition (b) is equivalent to condition (a).

2) The equivalence of conditions (a) and (c) of Theorem 1 is now proved. Let

(56a)

\begin{array}{l} W = [\begin{matrix} T^{- 1} & 0 \\ 0 & U^{- 1} \end{matrix}] \in ℜ^{(n + γ) \times (n + γ)}, \\ V = [\begin{matrix} T & 0 \\ - V_{1} & I_{l} \end{matrix}] \in ℜ^{(n + l) \times (n + l)} \end{array}

where $V_{1} = [\begin{matrix} - {\bar{A}}_{21} & s I_{l} & - {\bar{A}}_{22} \end{matrix}]$ , and the matrices $- {\bar{A}}_{21}$ and $- {\bar{A}}_{22}$ are defined by Equation (23) and $T$ and $U^{- 1}$ are defined by Equations (21) and (27), respectively. Here, $W$ is the non-singular matrix with rank $(W) = n + γ$ , and rank $(W) > n + l$ under the assumption that $γ > l$ . Also, because $V$ is a non-singular matrix with rank $(V) = n + l$ , we have

(56b)

\begin{array}{l} rank [\begin{matrix} s I_{n} - A & E \\ C_{a} & 0 \end{matrix}] = rank W [\begin{matrix} s I_{n} - A & E \\ C_{a} & 0 \end{matrix}] V \\ = rank [\begin{matrix} T^{- 1} (s I_{n} - A) T - T^{- 1} E V_{1} & Τ^{- 1} E \\ U^{- 1} C_{a} T & 0_{γ \times l} \end{matrix}] \\ = rank [\begin{matrix} s I_{n} - \bar{A} - \bar{E} V_{1} & \bar{E} \\ U^{- 1} \bar{C} & 0_{γ \times l} \end{matrix}] \\ = rank [\begin{matrix} [\begin{matrix} s I_{n - l} - {\bar{A}}_{11} & - {\bar{A}}_{12} \\ 0_{l \times (n - l)} & 0_{l \times l} \end{matrix}] & \bar{E} \\ U^{- 1} \bar{C} & 0_{γ \times l} \end{matrix}] \end{array}

From $U^{- 1}$ in Equation (27) and $\bar{E}$ and $\bar{C}$ in Equation (23), Equation (56b) becomes

(56c)

rank [\begin{matrix} [\begin{matrix} s I_{n - l} - {\bar{A}}_{11} & - {\bar{A}}_{12} \\ 0_{l \times (n - l)} & 0_{l \times l} \end{matrix}] & [\begin{matrix} 0 \\ I_{l} \end{matrix}] \\ \begin{matrix} [\begin{matrix} U_{1} \\ U_{2} \end{matrix}] [C_{a} N & C_{a} E] \end{matrix} & 0_{γ \times l} \end{matrix}]

Considering Equation (28), $U_{1} C_{a} N = 0_{l \times (n - l)}$ (Equation 31) and ${\bar{C}}_{1} = U_{2} C_{a} N$ (Equation 33b), Equation (56c) becomes

(56d)

\begin{array}{l} rank [\begin{matrix} [\begin{matrix} s I_{n - l} - {\bar{A}}_{11} & - {\bar{A}}_{12} \\ 0_{l \times (n - l)} & 0_{l \times l} \end{matrix}] & [\begin{matrix} 0 \\ I_{l} \end{matrix}] \\ [\begin{matrix} 0_{l \times (n - l)} & I_{l} \\ {\bar{C}}_{1} & 0_{(γ - l) \times l} \end{matrix}] & 0_{γ \times l} \end{matrix}] \\ = 2 l + rank [\begin{matrix} s I_{n - l} - {\bar{A}}_{11} \\ {\bar{C}}_{1} \end{matrix}] \end{array}

From the equivalence of Equation (56d) and condition (c), we obtain

(56e)

rank [\begin{matrix} s I_{n - l} - {\bar{A}}_{11} \\ {\bar{C}}_{1} \end{matrix}] = n - l

By Equations (56e) and (54), condition (c) is equivalent to condition (a). This theorem indicates that existence conditions (a), (b) and (c) are equivalent to each other.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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