where A, E ∈ ℝ^{n × n}, B ∈ ℝ^{n × m}, and C ∈ ℝ^{p × n} are known matrices with appropriate dimension. x ∈ ℝⁿ, u ∈ ℝ^m, y ∈ ℝ^p are the state vector, the control input and, the measured output respectively.

When unknown inputs are considered the system becomes

where R ∈ ℝ^{n × r} is a known matrix of appropriate dimension and d(t) ∈ ℝ^r the unknown input vector.

The main difference among LTI and descriptor LTI (DLTI) systems, is that matrix E of (2↑) is singular and therefore non-invertible. Two of the most important properties are the solvability and the regularity. If these properties hold, a singular system can be converted to Kronecker-Weistrauss restricted system equivalence form (r.s.e). In this form some properties like controllability, observability and stability are easier to determine.

The regularity is related to the invertibility of the pair pencil (E, A). In [10] the regularity is defined as The system (2↑) is called regular if there exists a scalar γ such thatdet(γE − A) ≠ 0 or equivalently, the polynomial det(sE − A) is not identically zero. In this case the matrix pair (E, A), or the matrix pencil sE − A is regular. \enddefn Solvability is defined as the existence of a unique solution for any given sufficiently differentiable u(t) and any given admissible initial conditions corresponding to the given u(t). [11] The matricial relationship (E, A) is solvable if and only if |sE − A| ≠ 0, or equivalently if there exists an scalar λ ∈ ℂ such that |λE − A| ≠ 0.

If the system is regular, then the following hold: \enddefn [12] Two systems (E, A, B, C) and (Ê Â, B̂, Ĉ) are defined as equivalent, or restricted systems equivalent (r.s.e.), if their order, number of inputs and outputs are equal and there exist two non singular matrices P and Q such that Ê = PEQ, Â = PAQ, B̂ = PB, B̂ = CQ. Where

\enddefnwhen A₁ ∈ ℝⁿ₁ ^{× n}₁, I_n ∈ ℝ^{n × n} is the identity matrix and N is a nilpotent matrix of index r such that N^r = 0 but N^{r − 1} ≠ 0. The equivalent system is:

Fast subsystem

or equivalently

with

= PB, [C₁ C₂] = CQ

Subsystem (5↑) represents the slow response and subsystem (5↑) represents the fast response. The system (8↑) also is called the Kronecker-Weistrauss canonical form.

However, if the system is not regular but

= n + rank E

Then a equivalent form based on QR transformation [13] can be obtained as

is found applying QR decomposition.

3 Descriptor System Package

Due to the lack of special tools for the analysis and observer design for descriptor LTI and LPV systems, It was necessary program some scripts, in order to solve the problem. This functions were developed for SCILAB and MATLAB platforms. The SCILAB version can be downloaded from ATOMS web page [14]. The MATLAB version is under revisions.

It should be noted that currently there are few works about it. For example, in [15] a toolbox for MATLAB is presented which is focused principality on the solution of numerical problems. The functions programmed in the toolbox can be used for i) analysis, ii) observer design and iii) Fault detection for DLTI and LPV systems. These functions are summarized in Table 1↓.

Some restrictions are listed below

• As seen in Table 1↑, the descriptor package can be used like an auxiliary tool in the analysis and observer design for DLTI and DLPV. Some of the observers were used to built bank of observers using a generalized observers schemes (GOS) or dedicated observers schemes (DOS) for DLTI and DLPV[16]. However, the algorithms are limited only to compute the gains of each one of the observers.
• Some functions are only available for SCILAB, due that some requirements are not achievable in MATLAB.
• The YALMIP toolbox [17] is necessary to solve some linear matrix inequalities (LMI) in the MATLAB package. By default the solver lmilab is selected, however this can be changed by sedumi or other.

4 Numerical example

This section shows some examples of how to work with the descriptor systems toolbox. Consider the following descriptor system

In MATLAB the descriptor system is defined as

A=[-0.75 1 0 0 ; -1 -0.85 0 0; 0 -1 -0.75 0; 0 0 0 -1];

B=[1 10; 1 0.5 ; 0 0; 1 0];

R=[1 0. 0 0]’; C=[1 0 1 0; 0 1 0 1; 1 0 1 1];

dsys=dsystem (E,A,B,C,R)

4.1 Equivalent systems

The Kronecker-Weistrauss canonical form (8↑) is computed as

[A1, N, h, B1, B2, C1, C2]=kwrse (E,A,B,C)

C2 =

- 1.1547005 0.

0. 1.

- 1.1547005 1.

C1 =

1.0327956 - 1.

- 0.7745967 0.

1.0327956 - 1.

B2 =

0. 0.

- 1. 0.

B1 =

- 1.2909944 - 0.6454972

- 1. - 10.

h =

1.

N =

0. 0.

0. 0.

A1 =

- 0.85 - 1.2909944

0.7745967 - 0.75

A restricted system equivalent system based on QR transformation (10↑) is determined as

[Ee, Ae, Ae1, Be, Be1, Ce]=qrrse(E,A,B,C)

Ee =

1 0 0 0

0 1 0 0

Ae =

-0.75 1.00 0 0

-1.00 -0.850 0 0

Ae1 =

0 -1.00 -0.75 0

0 0 0 -1.00

Be =

1.00 10.00

1.00 0.50

Be1 =

0 0

1 0

Ce =

0 -1.00 -0.75 0

0 0 0 -1.00

1.00 0 1.00 0

0 1.00 0 1.00

1.00 0 1.00 1.00

4.2 Observability and controllability

In descriptor systems there exists principally three types of controllability/observability: Complete, Reachable and Impulsive. For the case of C-controllability, it is related with the fact that the system can be controlled with any initial conditions [18]. R-controllability means that the system only can be controlled by a reduced set of initial conditions and I-Controllability means that impulsive signals can be controlled.

The controllability in descriptor systems can be found in different ways, e.g. using Laurent expansion, equivalent systems or from direct approaches. In the Laurent expansion approach, the controllability is related with the dimension of (E, A) and the rank of the Laurent expansion matrices. In the restricted system approach the controllability can be found by the analysis of controllability of slow (5↑) and fast (5↑) subsystems [19]. While in the direct form approach the controllability is found by analyzing matrices (E, A, B) (see [20]). Similar definitions are considered for the case of observability.

The observability condition can be tested by the following command

[nC,Co_mat,nR,Ro_mat,nI,Io_mat]=dobsv(E,A,C)

nI =

Iobsv: 'T'

nC =

4

Co_mat =

SlowS: [7x4 double]

FastS: [7x4 double]

nR =

Obsv: 'T'

Rank: 4

Ro_mat =

1.7500 -1.0000 0 0

1.0000 1.8500 0 0

0 1.0000 0.7500 0

0 0 0 1.0000

1.0000 0 1.0000 0

0 1.0000 0 1.0000

1.0000 0 1.0000 1.0000

nI =

Iobsv: 'T'

rank: 6

4.3 Observer design

The observation of the system behavior, in continuous-time, allows us to gain indirect information on the appearance of the unknown signals [21]. These signals can be estimated by an unknown input observer. An observer is defined as an unknown input observer (UIO), if its state estimated error vector e(t) tends to zero asymptotically, regardless of the presence of the unknown input in the system. A popular observer that has been widely studied is the observer proposed in [22]. This observer has been extended to fault diagnosis in LTI and LPV systems in [23, 24]. The observer is based in a Luenberger form as

where N, L₁, L₂ and K are matrices of appropriate dimension. b and d are vectors obtained from

The observer gains can be computed as

poles=[-4,-3,-2 -1]

[N,K,L1,L2,G,test]=darobsv95(dsys,poles)

nI =

Iobsv: 'T'

K =

0.45 -2.84 1.01

0.63 -0.29 0.57

-0.27 0.88 -0.42

-0.05 -0.04 -0.03

L1 =

-3.36 3.27 0.37 6.17 -2.89

-0.75 -0.68 -0.62 -0.75 0.06

1.54 -0.56 -0.70 -0.42 -0.13

-0.12 0.46 0.18 -0.25 -0.27

L2 =

1.78 0.53 0.93 0.12 0.40

-1.14 0.22 -0.30 0.81 -0.55

-0.93 0.08 0.69 -0.51 0.60

0.27 -0.39 -0.09 0.30 0.30

G =

2.75 4.87

0.81 -1.37

-1.40 -4.71

-0.22 -0.98

4.4 Fault detection and isolation based observers

The fault detection can be done with banks of observers by using a generalized observer scheme (GOS) or dedicated observer scheme (DOS) [25].

Figure: DoS and GOS schemes

Considering the proportional unknown input observer (PUIO) proposed in [26]

For the numerical example, if the observer is stabilized in a LMI region “α = − 2,” (see [27]) then the gains for each one of the observers in the GOS bank (for sensor) are computed as

>> lmizone=-10

%'s'= sensors

[N,G1,L,H2,Bo1]=gosbank1(dsys,lmizone,'s')

C =

0 1 0 1

1 0 0 0

C =

1 0 1 0

1 0 0 0

C =

1 0 1 0

0 1 0 1

N(:,:,1) =

-0.0001 0.0000 0.0000 0.0000

-0.6667 -0.5668 -0.0001 -0.0001

0.8889 0.7555 -0.0000 0.0000

0.3333 0.2833 0.0000 -0.0001

N(:,:,2) =

-0.2401 -0.2040 -0.0000 0

-0.6800 -0.5780 -0.0000 0

0.4800 0.4080 -0.0000 0

0 0 0 -0.0000

N(:,:,3) =

-0.8889 -0.7556 -0.0000 0.0000

-0.6667 -0.5667 -0.0000 -0.0001

0.8888 0.7555 -0.0001 0.0000

0.3333 0.2833 0.0000 -0.0001

G1(:,:,1) =

0 0

0.6667 0

-0.8889 0

-0.3333 0

G1(:,:,2) =

0.2400 0

0.6800 0

-0.4800 0

0 0

G1(:,:,3) =

0.8889 0

0.6667 0

-0.8889 0

-0.3333 0

L(:,:,1) =

0 -0.0000 -0.0000 0.0000

0.0000 -0.1889 -0.1889 -0.6667

-0.0000 0.2519 0.2519 0.8889

-0.0000 0.0944 0.0944 0.3333

L(:,:,2) =

0.0077 0 0.0058 -0.2458

0.0218 0 0.0163 -0.6963

-0.0154 0 -0.0115 0.4915

0 0 0 0

L(:,:,3) =

-1.1852 -0.6469 -0.8889 -0.6469

-0.8889 -0.4852 -0.6667 -0.4852

1.1852 0.6469 0.8889 0.6469

0.4444 0.2426 0.3333 0.2426

H2(:,:,1) =

0 0 0 1.0000

-0.0000 0.3333 0.3333 0

-1.3333 -0.4444 -0.4444 0

0.0000 -0.6667 0.3333 0

H2(:,:,2) =

0.2400 0 0.1800 0.8200

-0.3200 0 -0.2400 0.2400

-0.4800 0 0.6400 -0.6400

0 -1.0000 0 0

H2(:,:,3) =

1.3333 0.4444 1.0000 0.4444

-0.0000 0.3333 -0.0000 0.3333

-1.3333 -0.4444 0.0000 -0.4444

0.0000 -0.6667 0.0000 0.3333

Bo1(:,:,1) =

0 0

1 0

Bo1(:,:,2) =

0 0

1 0

Bo1(:,:,3) =

0 0

1 0

where (:, :, j), with j = 1, 2, 3, indicates the number of the observer. The function selects automatically the output matrices.

5 Descriptor LPV systems

Continuous descriptor LPV systems (DLPV), in polytopic form, under unknown inputs (noise, disturbances, etc) are usually defined as

where A_i, E ∈ ℝ^{n × n}, B_i ∈ ℝ^{n × m}, C ∈ ℝ^{p × n} and R_i ∈ ℝ^r are known matrices with appropriate dimension. x(t), u(t), y(t) and d(t) are the state vector, the control input, the measured output and the unknown input vector respectively. The parameter θ(t) is assumed to be bounded and lies into a hypercube such that

θ(t) ∈ Γ = {θ|\underbarθ_i(t) ≤ θ(t) ≤ θ_i(t)}, ∀t > 0

The functions ρ_iθ(t) are the weighting function that permits the commutation among the models. The polytopic DLPV system is scheduled through the following convex set

ρ_i(θ(t)) = 1

5.1 Numerical example

Consider the numerical example presented in [28], the gains of the PIUIO can be computed as

%% descriptor-LPV system

dsys=dsystem(E,Ai,Bi,C,Ri)

alpha=3; %LMI zone

% observer gains

[N,G,L,Ti,H2,Phi]=lpvpiuiobsv(dsys,alpha)

where (A, B, R ) are in polytopic form ∀i ∈ {1, 2, ..h} and, alpha (α) is to define a zone in the left part of the complex plan bounded with a line of abscissa ( − α) where α ∈ ℝ ⁺ .

The resulting matrices are not presented here because the sizes are very large. However, others observers can be achieved similar to the examples in Sections 4.3↑ and 4.4↑ (see the Table 1↑).

6 CONCLUSIONS

This paper presents a set of functions that have been written in order to develop a descriptor systems toolbox for SCILAB and MATLAB. The toolbox contains a set of tools dedicated to analyze descriptor systems, from LTI extended to LPV. These tools include the implementation of algorithms for the analysis of controllability, observability, stability and to find the fundamental matrices of the Laurent expansion. Some functions to design state-observers are proposed. Some of these observers are considered for fault detection schemes, for the construction of manual or automatic banks of observers. The toolbox can also be used for discretization of continuous systems by the method proposed in [29] (see Table 1↑).

It is very important to note that this paper is not a survey and therefore, the examples presented here are only to demonstrate the usefulness of the toolbox. Therefore, we recommend a detailed review of the references.

The SCILAB version of the toolbox is available to the community in the SCILAB files exchange web page [30]. The toolbox was created on the basis of free software, so anyone can improve it and modify it freely. The MATLAB package is under revisions and can be requested by email to the authors.

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