Robust H∞ filter-based functional observer design for descriptor systems: An application to cardiovascular system monitoring

Pabitra K. Tunga; Juhi Jaiswal; Nutan K. Tomar

doi:10.1515/cmb-2024-0013

Article Open Access

Robust H_∞ filter-based functional observer design for descriptor systems: An application to cardiovascular system monitoring

Pabitra K. Tunga , Juhi Jaiswal and Nutan K. Tomar

Published/Copyright: December 10, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Computational and Mathematical Biophysics Volume 12 Issue 1

Abstract

In this article, we present a novel approach for robust state estimation in linear uncertain descriptor systems, with a specific focus on the cardiovascular system (CVS) as a case study. Descriptor systems, which extend traditional state-space representations by incorporating algebraic constraints, present significant challenges in observer design, especially under conditions of uncertainty. To address these challenges, we propose the design of a functional observer using the H ∞ filter approach, capable of managing uncertainties arising from both external disturbances and variations in system parameters. The observer is designed using a unique linear matrix inequality framework, which ensures the stability and performance of the estimation error system. The convergence of the estimation error is rigorously analyzed using the Lyapunov method. The effectiveness of the proposed method is demonstrated through simulations on a CVS model, highlighting its potential to reliably estimate unknown states in the presence of uncertainties.

Keywords: linear systems; descriptor systems (differential-algebraic systems); observers for linear systems; functional observers; parameter uncertainty

MSC 2010: 93B51; 93B53; 93C05

1 Introduction

In control theory, uncertain systems are those where model parameters are either unknown or subject to change over time. These uncertainties can result from various factors, such as modeling inaccuracies, external disturbances, or fluctuations in system components and environmental conditions. These factors can significantly affect a system’s stability, performance, and robustness, making the study of uncertainties crucial, especially in the design of controllers and observers that must function reliably across different scenarios.

Accurate estimation and control of biological systems, such as the cardiovascular system (CVS), are vital in developing new diagnostic and therapeutic techniques. This complex task is essential for creating theoretical tools such as observers and controllers, which can enhance patient outcomes and reduce the mortality rates associated with cardiovascular diseases (CVDs). CVDs remain the leading cause of death worldwide, claiming approximately 17.9 million lives annually [1]. These diseases include a range of heart and blood vessel disorders, such as coronary heart disease, cerebrovascular disease, rheumatic heart disease, and other related conditions [1].

Descriptor systems, also referred to as singular or differential-algebraic systems, extend the traditional state-space representation and are vital for modeling complex dynamical behaviors. Beyond capturing dynamic processes in the form of ordinary differential equations (ODEs), descriptor systems also account for algebraic constraints among physical variables. As a result, they are widely applicable across various fields, including engineering, economics, and biology [7,9,14,22,23,26]. In this study (cf. Section 3), we consider a novel observer-based method for estimating the states of the CVS by reformulating the four-element Windkessel model as a descriptor system [25]. This model is based on the idea that the aorta can be represented using an electrical analogy, specifically as a circuit with a capacitor and a resistor arranged in parallel [31].

Since the state-space systems are represented only by a set of ODEs, the analysis and control of linear state-space models is well-established subject, now routinely covered in many textbooks. However, controlling linear descriptor systems presents unique challenges due to their complex structure and the presence of unmeasurable state variables. In recent years, there has been increasing interest in estimating the states of descriptor systems through state-space models, as researchers seek to develop new methods for observer design for descriptor systems [3,11,12,16,17,24]. Notably, if the observer for a descriptor system is formulated in state-space form, its implementation becomes more straightforward, thanks to the availability of numerous ODE solvers, such as MATLAB’s ode45 solver.

It is important to note that classical full-state observers may estimate all system states, including those that are directly measurable or not particularly useful. In contrast, functional observers focus on estimating only a selected combination of key states, which significantly reduces the computational load compared to full-state observers. Additionally, functional observers can be designed under much less restrictive assumptions than those required for full-order observers. Due to these advantages, functional observer design remains an active area of research, even for standard linear state-space systems [13]. Significant attention has also been given to designing functional observers for linear descriptor systems, reflecting the growing interest in the field [3,11,12,16,17,24,28].

On the other hand, descriptor systems present unique challenges in observer design, especially when parameter uncertainties are present in the underlying mathematical model. These uncertainties can result in inaccurate state estimation and compromise control strategies. To overcome these challenges, advanced techniques like H ∞ observers are used to design robust controllers that ensure stability and performance, even in the face of model inaccuracies and disturbances. The problem of H ∞ filtering for regular descriptor systems was first explored by Xu et al. in [33]. A reduced-order H ∞ observer for such systems was studied under the assumption that the system is free of impulses [32]. Darouach later demonstrated the existence of H ∞ observers for rectangular descriptor systems, provided that the system is impulsive observable [10]. More recently, a sufficient condition has been established for the design of functional H ∞ observers, where the order of the observer is equal to the dimension of the vector to be estimated [29].

The primary contributions of this article include the design of a robust functional observer for linear uncertain descriptor systems using the H ∞ filter approach. Unlike previous studies that only considered external disturbances, this study addresses a broader range of system parameter uncertainties. A unique linear matrix inequality (LMI) is employed to determine the observer’s coefficient matrices. The convergence of the estimation error is rigorously analyzed using the Lyapunov stability theory. The effectiveness of the proposed observer is demonstrated through simulations, which show its ability to efficiently estimate unknown states in a CVS model.

We use the following notation: 0 and I stand for appropriate dimensional zero and identity matrices, respectively. In a block partitioned matrix, all missing blocks are zero matrices of appropriate dimensions. Sometimes, for more clarity, the identity matrix of size n × n is denoted by I n . The set of complex numbers is denoted by C and C ¯ + ≔ { s ∈ C ∣ Re ( s ) ≥ 0 } . The symbols A ⊤ , A + , ker A , and Row ( A ) denote the transpose, Moore-Penrose inverse (MP-inverse), kernel, and row space of a matrix A , respectively. For any square matrix A , we write A > 0 ( A < 0 ) if, and only if, A is positive (negative) definite. A block diagonal matrix having diagonal elements A 1 , … , A k is represented by blk-diag { A 1 , … , A k } . λ max ( Y ) represents the maximum singular value of Y . L loc 1 and AC loc represent the set of locally Lebesgue integrable functions and the set of locally absolutely continuous functions, respectively.

2 Problem statement and preliminaries

In this study, we consider linear descriptor systems subject to time-varying parameter uncertainty with bounded energy defined by the following system:

(1a) E x ˙ ( t ) = ( A + Δ A ) x ( t ) + F v ( t ) ,

(1b) y ( t ) = C x ( t ) + G v ( t ) ,

(1c) z ( t ) = K x ( t ) + H v ( t ) ,

where E , A ∈ R m × n , F ∈ R m × q , C ∈ R p × n , G ∈ R p × q , K ∈ R r × n , and H ∈ R r × q are the known constant matrices. The vectors x ( t ) ∈ R n , v ( t ) ∈ R q , y ( t ) ∈ R p , and z ( t ) ∈ R r are the semistate vector, the disturbance vector, the output vector, and the functional vector, respectively. Additionally, the system includes structural bounded uncertainty Δ A , which is unknown representing time-varying parameter uncertainty defined as

(2) Δ A ( t ) = M 1 F 1 ( t ) N 1 ,

where M 1 and N 1 are the known real constant matrices, and F 1 ( t ) is an unknown real-valued time-varying matrix that satisfies F 1 ⊤ ( t ) F 1 ( t ) ≤ I for all t ∈ R . Notably, it means that the time-varying parameter uncertainty Δ A is norm-bounded and the uncertain matrix F ( t ) is allowed to depend on the state, as long as F ⊤ ( t ) F ( t ) ≤ I is satisfied [33]. We do not assume that the system is regular or even square. We adopt the following behavioral approach to define solutions of (1):

Definition 1

The tuple ( x , v , y , z ) is said to be a solution of (1) if it belongs to the set

B ≔ { ( x , v , y , z ) ∈ L loc 1 ( R ; R n + q + p + r ) ∣ E x ∈ AC loc ( R ; R m ) and ( x , v , y , z ) satisfies ( 1 ) for almost all t ∈ R } .

The set B is called behavior in [27]. Moreover, this behavior set B has been used to define various observability concepts for (1) in [4] and for proving existence conditions for functional observers in [2,15,18–20,24,28,35].

The purpose of this study is to provide a sufficient condition for the existence of adaptive functional observer using the H ∞ filter technique. A general solution theory, based on behavioral approach, is adopted for descriptor systems, and a rigorous definition is introduced for H ∞ filter-based functional observers. We consider an observer candidate of the form

(3a) w ˙ ( t ) = N w ( t ) + L y ( t ) ,

(3b) z ˆ ( t ) = w ( t ) + M y ( t ) ,

where w ( t ) ∈ R r is the state vector of observer, and z ˆ ( t ) represents the estimate of the functional vector z ( t ) .

Now, we exploit the behavior B to define a robust H ∞ functional observer for system (1).

Definition 2

System (3) is said to be a robust H ∞ functional observer for (1), if for every ( x , v , y , z ) ∈ B and for all allowable parameter variations, there exist w ∈ AC loc ( R ; R q ) and z ˆ ∈ L loc 1 ( R ; R r ) such that ( w , y , z ˆ ) satisfy (3) for almost all t ∈ R , and for all such w , z ˆ , the following properties hold:

If v , z , z ˆ ∈ L loc 2 ( R ; R q + 2 r ) , then
‖ z ˆ − z ‖ 2 ≤ γ 2 ‖ v ‖ 2 + β ,
where γ > 0 and β ≥ 0 .
In the absence of parameter variation and external disturbances,
1. z ˆ ( t ) − z ( t ) → 0 as t → ∞ , i.e., lim t → ∞ ess sup [ t , ∞ ) ∣ ∣ z ˆ ( t ) − z ( t ) ∣ ∣ = 0 .
2. if z ˆ ( 0 ) = z ( 0 ) , then z ˆ = a.e. z , i.e., z ˆ ( t ) = z ( t ) for almost all t ∈ R .

Mathematically, the objective of this article is to determine the parameter matrices N , L , and M so that system (3) becomes a robust H ∞ functional observer for system (1), cf. Definition 2. Notably, the observer is of state-space form where the dynamics is governed only by ODEs. Such observers are easy to implement by using standard ODE solvers, e.g., in MATLAB.

In the remaining section, we recall some fundamental results from matrix theory. These results play an important role in further discussions.

Lemma 1

System X A = B has solution for X if and only if rank A B = rank A . Moreover,

X = B A + − Z ( I − A A + ) ,

where Z is an arbitrary matrix of appropriate dimension.

Lemma 2

[30] Let D , S , and ℱ be the real matrices of appropriate dimensions and ℱ satisfying ℱ ⊤ ℱ ≤ I . Then, for any scalar ε > 0 and vectors x and y, we have

2 x ⊤ D ℱ S y ≤ ε − 1 x ⊤ D D ⊤ x + ε y ⊤ S ⊤ S y .

Lemma 3

[5] The LMI

Q S S ⊤ R > 0 ,

where Q = Q ⊤ and R = R ⊤ is equivalent to,

R > 0 , Q − S R − 1 S ⊤ > 0 .

3 Robust H ∞ functional observer design

Theorem 1

System (3) is an observer for system (1) if the system coefficient matrices satisfy the condition

(4) rank E A C 0 0 C 0 K K 0 = rank E A C 0 0 C 0 K ,

and there exists a matrix T of appropriate dimension and two scalars γ > 0 and β ≥ 0 such that the vectors e = z ˆ − z and e 1 = w − T E x satisfy the system

(5a) e ˙ 1 ( t ) = N e 1 ( t ) + ( L G − T F ) v ( t ) − T Δ A x ( t ) ,

(5b) e ( t ) = e 1 ( t ) + ( M G − H ) v ( t ) ,

and ‖ e ‖ 2 ≤ γ 2 ‖ v ‖ 2 + β .

Proof

Let T ∈ R r × m be a matrix and set the error vectors e 1 ( t ) ≜ w ( t ) − T E x ( t ) and e ( t ) ≜ z ˆ ( t ) − z ( t ) . Then, from (1) and (3), we obtain

(6) e ( t ) = w ( t ) + M y ( t ) − ( K x ( t ) + H v ( t ) ) = e 1 ( t ) + ( T E + M C − K ) x ( t ) + ( M G − H ) v ( t )

and

(7) e ˙ 1 ( t ) = w ˙ ( t ) − T E x ˙ ( t ) = ( N w ( t ) + L y ( t ) ) − T ( ( A + Δ A ) x ( t ) + F v ( t ) ) = N ( e 1 ( t ) + T E x ( t ) ) + L ( C x ( t ) + G v ( t ) ) − T A x ( t ) − T F v ( t ) − T Δ A x ( t ) = N e 1 ( t ) + ( N T E + L C − T A ) x ( t ) + ( L G − T F ) v ( t ) − T Δ A x ( t ) .

From (6) and (7), it is clear that if the parameter matrices N , T , M , and L satisfy the matrix equations

(8a) T E + M C = K ,

(8b) N T E + L C − T A = 0 ,

then the error vectors satisfy the dynamics (5). Notably, equation (8b) is nonlinear in the unknown matrices. By substituting (8a) into (8b), (8b) reduces into a linear one as follows:

(9) T A + P C − N K = 0 ,

where P = N M − L . Clearly, equations (8a) and (9) can be rewritten in the matrix form

(10) [ T M P N ] Σ = Θ ,

where Σ = E A C 0 0 C 0 − K and Θ = K 0 . Now, it follows from Lemma 1 that ∃ matrices T , M , P , and N such that (10), equivalently, (8) holds if and only if the system coefficient matrices satisfy rank condition (4). Moreover, the solution is given by

[ T M P N ] = Θ Σ + − Z ( I − Σ Σ + ) ,

where Z is an arbitrary matrix of appropriate dimension. Notably, [ T M P N ] can be rewritten as

(11) [ T M P N ] = [ T 1 M 1 P 1 N 1 ] − Z [ T 2 M 2 P 2 N 2 ] ,

where

T 1 = Θ Σ + I 0 0 0 , T 2 = ( I − Σ Σ + ) I 0 0 0 , M 1 = Θ Σ + 0 I 0 0 , M 2 = ( I − Σ Σ + ) 0 I 0 0 , P 1 = Θ Σ + 0 0 I 0 , P 2 = ( I − Σ Σ + ) 0 0 I 0 , N 1 = Θ Σ + 0 0 0 I , and N 2 = ( I − Σ Σ + ) 0 0 0 I .

Thus, under rank condition (4), the error dynamics are governed by (5). The remaining proof follows from Definition 2.□

In view of the proof of Theorem 1, it remains to find the matrix Z of appropriate dimension in such a way that system (3) with parameter matrices (11) satisfies all the conditions for robust H ∞ functional observer as defined in Definition 2.

Theorem 2

Assume that system (1) satisfies the rank condition (4). Then, (3) is a robust H ∞ functional observer with system parameter matrices (11) and error dynamics (5) if there exist scalars ε > 0 and γ > 0 , matrices Y 1 and Y 2 > 0 such that

(12a) E ⊤ Y 1 = Y 1 ⊤ E ≥ 0 ,

(12b) Π = Π 11 0 Π 13 0 Π 22 Y 1 ⊤ F Π 13 ⊤ F ⊤ Y 1 H ¯ ⊤ H ¯ − γ 2 I < 0 ,

where

(13a) Π 11 = N ⊤ Y 2 + Y 2 N + ε − 1 ( Y 2 T M 1 ) ( Y 2 T M 1 ) ⊤ + I , Π 13 = Y 2 ( L G − T F ) + H ¯ ,

(13b) Π 22 = A ⊤ Y 1 + Y 1 ⊤ A + ε − 1 ( Y 1 ⊤ M 1 ) ( Y 1 ⊤ M 1 ) ⊤ + 2 ε N 1 ⊤ N 1 , and H ¯ = M G − H .

Proof

Let us choose a candidate of Lyapunov function as

(14) V ( t ) = V 1 ( t ) + V 2 ( t ) ,

where V 1 ( t ) = x ⊤ ( t ) E ⊤ Y 1 x ( t ) , V 2 ( t ) = e 1 ⊤ ( t ) Y 2 e 1 ( t ) , E ⊤ Y 1 = Y 1 ⊤ E ≥ 0 , and Y 2 = Y 2 ⊤ > 0 . Then, using (1a) and (2), we obtain

(15) V ˙ 1 ( t ) = x ˙ ⊤ ( t ) E ⊤ Y 1 x ( t ) + x ⊤ ( t ) E ⊤ Y 1 x ˙ ( t ) = { ( A + Δ A ) x ( t ) + F v ( t ) } ⊤ Y 1 x ( t ) + x ⊤ ( t ) Y 1 ⊤ { ( A + Δ A ) x ( t ) + F v ( t ) } = x ⊤ ( t ) ( A ⊤ Y 1 + Y 1 ⊤ A ) x ( t ) + 2 x ⊤ ( t ) Y 1 ⊤ F v ( t ) + 2 x ⊤ ( t ) Y 1 ⊤ Δ A x ( t ) = x ⊤ ( t ) ( A ⊤ Y 1 + Y 1 ⊤ A ) x ( t ) + 2 x ⊤ ( t ) Y 1 ⊤ F v ( t ) + 2 x ⊤ ( t ) Y 1 ⊤ M 1 F 1 ( t ) N 1 x ( t ) .

On the other hand, from (5a) and (2), we obtain

(16) V ˙ 2 ( t ) = e ˙ 1 ⊤ ( t ) Y 2 e 1 ( t ) + e 1 ⊤ ( t ) Y 2 e ˙ 1 ( t ) = e 1 ⊤ ( t ) ( N ⊤ Y 2 + Y 2 N ) e 1 ( t ) + 2 e 1 ⊤ ( t ) Y 2 ( L G − T F ) v ( t ) − 2 e 1 ⊤ ( t ) Y 2 T Δ A x ( t ) = e 1 ⊤ ( t ) ( N ⊤ Y 2 + Y 2 N ) e 1 ( t ) + 2 e 1 ⊤ ( t ) Y 2 ( L G − T F ) v ( t ) − 2 e 1 ⊤ ( t ) Y 2 T M 1 F 1 N 1 x ( t ) .

Now, by applying Lemma 2 on the right-hand sides of (15) and (16), we obtain

(17a) V ˙ 1 ( t ) ≤ x ⊤ ( t ) ( A ⊤ Y 1 + Y 1 ⊤ A ) x ( t ) + 2 x ⊤ ( t ) Y 1 ⊤ F v ( t ) + ε − 1 x ⊤ ( t ) Y 1 ⊤ M 1 M 1 ⊤ Y 1 x ( t ) + ε x ⊤ ( t ) N 1 ⊤ N 1 x ( t ) ,

(17b) V ˙ 2 ( t ) ≤ e 1 ⊤ ( t ) ( N ⊤ Y 2 + Y 2 N ) e 1 ( t ) + 2 e 1 ⊤ ( t ) Y 2 ( L G − T F ) v ( t ) + ε − 1 e 1 ⊤ ( t ) Y 2 T M 1 M 1 ⊤ T ⊤ Y 2 e 1 ( t ) + ε x ⊤ ( t ) N 1 ⊤ N 1 x ( t ) .

Therefore, from (17) and (5b), we obtain

(18) V ˙ ( t ) + e ⊤ ( t ) e ( t ) − γ 2 v ⊤ ( t ) v ( t ) = V ˙ 1 ( t ) + V ˙ 2 ( t ) + e ⊤ ( t ) e ( t ) − γ 2 v ⊤ ( t ) v ( t ) ≤ x ⊤ ( t ) ( Π 22 ) x ( t ) + e 1 ⊤ ( t ) ( Π 11 ) e 1 ( t ) + v ⊤ ( t ) ( H ¯ ⊤ H ¯ − γ 2 I ) v ( t ) + x ⊤ ( t ) ( Y 1 ⊤ F + F ⊤ Y 1 ) v ( t ) + e 1 ⊤ ( t ) ( Π 13 ⊤ + Π 13 ) v ( t ) = e 1 x v ⊤ Π 11 0 Π 13 0 Π 22 Y 1 ⊤ F Π 13 ⊤ F ⊤ Y 1 H ¯ ⊤ H ¯ − γ 2 I e 1 x v ,

where Π 11 . Π 13 , Π 22 , and H ¯ are defined same as in (13). Thus, it follows from (18) that if (12a) and (12b) hold, then

V ˙ ( t ) + e ⊤ ( t ) e ( t ) − γ 2 v ⊤ ( t ) v ( t ) ≤ 0 .

Moreover, the increasing property of integral implies

∫ 0 T ( V ˙ ( t ) + e ⊤ ( t ) e ( t ) − γ 2 v ⊤ ( t ) v ( t ) ) d t ≤ 0 ,

equivalently,

(19) V ( T ) − V ( 0 ) + ∫ 0 T e ⊤ ( t ) e ( t ) d t − γ 2 ∫ 0 T v ⊤ ( t ) v ( t ) d t ≤ 0 .

Since V ( T ) ≥ 0 and (19) holds for every finite T ,

(20) ‖ e ‖ 2 ≤ γ 2 ‖ v ‖ 2 + V ( 0 ) ,

i.e., condition ( i ) in Definition 2 holds with β = V ( 0 ) .

Now, it remains to show that condition ( i i ) in Definition 2 also holds. It is clear from (12b) and the properties of negative definite matrices that N ⊤ Y 2 + Y 2 N < 0 in the absence of any external disturbances and parameter uncertainty. Hence, the error dynamics (5) reduces to

e ˙ ( t ) = N e ( t ) ,

which gives that

(21) e ( t ) = exp ( N t ) e ( 0 ) .

Therefore, (21) and stability of matrix N imply

e ( t ) → 0 , for t → ∞ ,
if e ( 0 ) = 0 , then e = a.e. 0 .

This completes the proof.□

Now, we aim to simplify the complex relationships involving the matrix Z . In particular, we are looking at the presence of non-linear term N M in Π , as shown in equation (12b), through the term Z N 2 Z . To address this, we propose the following theorem to establish an LMI.

Theorem 3

For given scalars ε > 0 , γ > 0 and matrices Y 1 , Y 2 > 0 the matrix inequality (12b) holds if there exist matrices Y 3 such that

(22) Γ 11 Γ 12 0 0 Γ 15 Γ 12 ⊤ − ε I 0 0 0 0 0 Γ 33 Y 1 ⊤ M 1 Y 1 ⊤ F 0 0 M 1 ⊤ Y 1 − ε I 0 Γ 15 ⊤ 0 F ⊤ Y 1 0 H ¯ 1 ⊤ H ¯ 1 − γ 2 I < 0 ,

where Γ 11 = Y 2 N 1 + N 1 ⊤ Y 2 − ( Y 3 N ¯ 2 + N ¯ 2 ⊤ Y 3 ⊤ ) + I , Γ 12 = Y 2 T 1 M 1 − Y 3 T ¯ 2 M 1 , Γ 33 = A ⊤ Y 1 + Y 1 ⊤ A + 2 ε N 1 ⊤ N 1 , Γ 15 = Y 2 L 1 − Y 3 L ¯ 2 + H ¯ 1 , H ¯ 1 = M 1 G − H , and Y 3 = Y 2 Z 1 .

Proof

Define M ˇ 2 = M 2 G , Z = Z 1 ( I − M ˇ 2 M ˇ 2 + ) , Y 3 = Y 2 Z 1 , and Γ 11 = N ⊤ Y 2 + Y 2 N + I . Then, from (11), we obtain

(23a) [ T M P N ] = [ T 1 M 1 P 1 N 1 ] − Z 1 ( I − M ˇ 2 M ˇ 2 + ) [ T 2 M 2 P 2 N 2 ]

(23b) ≕ [ T 1 M 1 P 1 N 1 ] − Z 1 [ T ¯ 2 M ¯ 2 P ¯ 2 N ¯ 2 ]

(23c) M G = ( M 1 − Z 1 ( I − M ˇ 2 M ˇ 2 + ) M 2 ) G = M 1 G ,

(23d) H ¯ = M G − H = M 1 G − H ≕ H ¯ 1 ,

(23e) Γ 11 = ( N 1 − Z 1 N ¯ 2 ) ⊤ Y 2 + Y 2 ( N 1 − Z 1 N ¯ 2 ) = N 1 ⊤ Y 2 + Y 2 N 1 − ( N ¯ 2 ⊤ Y 3 ⊤ + Y 3 N ¯ 2 ) + I .

In addition,

(24) L G − T F = ( N M − P ) G − T F , = ( N 1 − Z 1 N ¯ 2 ) M 1 G − ( P 1 − Z 1 P ¯ 2 ) G − ( T 1 − Z 1 T ¯ 2 ) F , = ( N 1 M 1 − P 1 ) G − T 1 F − Z 1 { ( N ¯ 2 M 1 − P ¯ 2 ) G − T ¯ 2 F } , ≕ L 1 − Z 1 L ¯ 2 ,

and

Π 13 = Y 2 ( L G − T F ) + H ¯ = Y 2 ( L 1 − Z 1 L ¯ 2 ) + H ¯ 1 = Y 2 L 1 − Y 3 L ¯ 2 + H ¯ 1 ≕ Γ 15 .

Now, Π can be rewritten as

Γ 11 − ( Y 2 T M 1 ) ( − ε I ) − 1 ( Y 2 T M 1 ) ⊤ 0 Γ 15 0 Π 22 Y 1 ⊤ F Γ 15 ⊤ F ⊤ Y 1 H ¯ ⊤ H ¯ − γ 2 I .

Then, Lemma 3 reveals that Π < 0 is equivalent to

Γ 11 Y 2 T M 1 0 Γ 15 ( Y 2 T M 1 ) ⊤ − ε I 0 0 0 0 Π 22 Y 1 ⊤ F Γ 15 ⊤ 0 F ⊤ Y 1 H ¯ 1 ⊤ H ¯ 1 − γ 2 I < 0 .

By applying a similar procedure to the term Π 22 , we can derive that Π < 0 is equivalent to

(25) Γ 11 Γ 12 0 0 Γ 15 Γ 12 ⊤ − ε I 0 0 0 0 0 Γ 33 Y 1 ⊤ M 1 Y 1 ⊤ F 0 0 M 1 ⊤ Y 1 − ε I 0 Γ 15 ⊤ 0 F ⊤ Y 1 0 H ¯ 1 ⊤ H ¯ 1 − γ 2 I < 0 ,

where Γ 33 = A ⊤ Y 1 + Y 1 ⊤ A + 2 ε N 1 ⊤ N 1 . This completes the proof.□

The following theorem provides a sufficient condition in terms of strict LMI for designing robust functional observers for system (1).

Theorem 4

Consider system (1) that satisfies the rank condition (4). Then, (3) is a robust H ∞ functional observer if there exist scalars ε > 0 and γ > 0 , matrices Y > 0 , Y 2 > 0 , Y 3 , and U such that

(26) Γ 11 Γ 12 0 0 Γ 15 Γ 12 ⊤ − ε I 0 0 0 0 0 Γ ¯ 33 Y 1 ⊤ M 1 Y 1 ⊤ F 0 0 M 1 ⊤ Y 1 − ε I 0 Γ 15 ⊤ 0 F ⊤ Y 1 0 H ¯ ⊤ H ¯ − γ 2 I < 0 ,

where Γ ¯ 33 = A ⊤ Y 1 + Y 1 ⊤ A + 2 ε N 1 ⊤ N 1 , Y 1 = Y E + S U , in which S is a full column matrix such that E ⊤ S = 0 .

Proof

Let S be a full column matrix such that E ⊤ S = 0 and Y 1 = Y E + S U . Then, we have

Y 1 ⊤ E = ( Y E + S U ) ⊤ E = E ⊤ Y ⊤ E + U ⊤ S ⊤ E = E ⊤ Y ⊤ E , E ⊤ Y 1 = E ⊤ ( Y E + S U ) = E ⊤ Y E .

Additionally, since Y > 0 , it follows that E ⊤ Y E ≥ 0 . Hence, we have Y 1 such that E ⊤ Y 1 = Y 1 ⊤ E and E ⊤ Y 1 ≥ 0 , which implies that (12a) holds.

It is easy to verify if strict LMI (26) holds for some scalars ε > 0 and γ > 0 , matrices Y > 0 , Y 2 > 0 , Y 3 and U , then there exist matrices Y 1 and Y 2 > 0 such that (12a) and (12b) hold. This completes the proof.□

Based on the aforementioned theorems, we now summarize the observer design procedure in Algorithm 1.

Algorithm 1: Computational steps to construct robust H ∞ functional observer (3) for system (1)

Data Given the system co-efficient matrices

if Check if condition (4) holds. If yes, then

(a) Compute Θ Σ + and I − Σ Σ + by using (10) . (b) Extract N 1 , N 2 , M 1 , and M 2 from Θ Σ + and I − Σ Σ + , respectively, by using (11) . (c) Compute M ˇ 2 = M 2 G and [ T ¯ 2 M ¯ 2 P ¯ 2 N ¯ 2 ] = ( I − M ˇ 2 M ˇ 2 + ) [ T 2 M 2 P 2 N 2 ] , see (23) . (d) Compute L 1 = ( N 1 M 1 − P 1 ) G − T 1 F , L ¯ 2 = { ( N ¯ 2 M 1 − P ¯ 2 ) G − T ¯ 2 F } , and H ¯ 1 = M 1 G − H , see (24) and (23d) . if LMI (22) is feasible for some fixed γ , then (i) Solve for U , Y and Y 2 such that (26) becomes negative definite . (ii) Compute full column matrix S such that E ⊤ S = 0 then compute Y 1 = Y E + S U , (iii) Compute Z 1 = Y − 1 Y 1 and Z = Z 1 ( I − M ¯ 2 M ¯ 2 + ) . (iv) Compute T , M , P , and N from ( 11 ) . (v) Compute L = N M − P . else ∣ The observer can not be computed by this method. end

else

∣ The observer can not be computed by this method. end

4 Simulation

In this section, the proposed robust H ∞ functional observer theory is applied to the four-element Windkessel model in order to estimate one of the cardiovascular variables in the presence of the online fault. To the best of our knowledge, the four-element Windkessel model was introduced by Burathi and Gnudi [6] in 1982. Later, this model has been analyzed for various studies (see [8,21,25,34] and references therein). A schematic depiction of this model is shown in Figure 1 that has been borrowed from [25, Fig. 1].

Figure 1

Four-element Windkessel model. Source from Ref. [25].

The four-element Windkessel model consists of two dynamic elements as pressure by arterial compliance ( P p ), flow across the total arterial inertance ( F L ), and one static element as aortic pressure ( P a ) . Furthermore, F a is representing the flow through the aorta. By taking the semistate vector x ≔ [ P p F L P a ] ⊤ and F a ( t ) = v ( t ) the following equations essentially describe the four-element Windkessel model for the CVS:

(27) C a x ˙ 1 ( t ) = − 1 R x 1 ( t ) + v ( t ) , L x ˙ 2 ( t ) = − Z 0 x 2 ( t ) + Z 0 v ( t ) , x 3 ( t ) = x 1 ( t ) − Z 0 x 2 ( t ) + Z 0 v ( t ) .

Aortic pressure refers to the blood pressure within the aorta, the main artery that carries oxygenated blood from the heart to the rest of the body. Aortic pressure can be measured using sensors, e.g., Edwards Lifesciences’ TruWave Pressure Transducer, and Philips IntelliVue MP Series. Therefore, aortic pressure P a is measurable, and we take this P a as output, i.e., y = P a . On the other hand, arterial compliance is a measure of the elasticity of the arterial walls, particularly how easily they expand and contract in response to a change in the blood pressure. The interpretation of compliance measurements can be more complex. Therefore, we want to estimate the pressure by arterial compliance P p , i.e., z = P p . For more details about four-element WindKessel model, we refer the readers to [8,25]. Hence, system (27) can be written in the form of (1) as follows:

C a 0 0 0 L 0 0 0 0 x ˙ ( t ) = − 1 ⁄ R 0 0 0 − Z 0 0 1 − Z 0 − 1 x ( t ) + 1 Z 0 Z 0 v ( t ) , y ( t ) = [ 0 0 1 ] x ( t ) , z ( t ) = [ 1 0 0 ] x ( t ) .

We assume that there is a problem in the peripheral resistance R . Notably, peripheral resistance changes mainly due to the narrowing or widening of blood vessels – narrowing increases resistance, while widening decreases it. Other factors such as blood thickness, artery stiffness, and body responses to stress or exercise also affect resistance. This can be expressed as

1 ⁄ R = 1 ⁄ R 0 + Δ R ,

where R 0 is the nominal resistance value, and Δ R is the bounded uncertainty with ∣ Δ R ∣ ≤ 0.05 . Then, the perturbed system matrix A can be written as

A = A 0 + Δ A ,

where A 0 = − 1 ⁄ R 0 0 0 − Z 0 0 1 − Z 0 − 1 and Δ A = Δ R 0 0 0 0 0 0 0 0 . For simulation purposes, we take the parameters of the four-element Windkessel model as

(28) C a = 1.5 ml/mmH , L = 0.01 mmHg s 2 , R = 0.95 mmHg s/mL , and Z 0 = 0.033 mmHg s/mL .

With the parameters (28), it is straightforward to verify that the system satisfies the rank condition (4). Now, by following the steps of Algorithm 1, we obtain the observer coefficient matrices as

T = [ 0.7294 0.4361 − 0.4671 ] , M = − 0.0337 , P = − 0.4184 , N = − 1.1888 , and L = 0.4585 .

Hence, we obtain the robust H ∞ functional observer (3) as follows:

w ˙ ( t ) = − 1.1888 w ( t ) − 0.4585 y ( t ) , z ˆ ( t ) = w ( t ) − 0.0337 y ( t ) .

The true and estimated values of the functional vector z are plotted in Figure 2 by taking the initial conditions x ( 0 ) = [ − 1 1 − 1 ] ⊤ , w ( 0 ) = 0.5 . In this setup, we introduce a disturbance vector to simulate external influences or uncertainties affecting the CVS. The disturbance is randomly generated between − 0.1 and 0.1 within the time interval [ 10 , 15 ] , as shown in Figure 3.

$Figure 2 Time responses of the actual and estimated P p {P}_{p} .$

Figure 2

Time responses of the actual and estimated P p .

Figure 3

Disturbances vector.

This randomness helps to create realistic conditions that the system might face, challenging the robust H ∞ filter to handle these uncertainties. We then check how well the filter can still accurately estimate the arterial pressure despite these disturbances. Figure 2 shows the time responses of the actual and estimated pressure by arterial compliance P p in the CVS. The solid red line represents the actual pressure by arterial compliance. The dashed blue line shows the estimated pressure by arterial compliance using the H ∞ filter-based observer. Initially, the estimate starts higher than the actual pressure, but it quickly converges to the true value as time progresses. By around 8 s, the estimated pressure aligns closely with the actual pressure, demonstrating the observer’s ability to accurately track the pressure by arterial compliance despite any disturbances or uncertainties. As time continues, both the actual and estimated pressures stabilize, indicating that the estimation error is minimized and the observer is performing effectively.

5 Conclusion

In this article, we have introduced a robust approach to state estimation for linear uncertain descriptor systems, with a specific application to the CVS. The complexity of descriptor systems, coupled with the challenges posed by uncertainties in model parameters and external disturbances, necessitates advanced observer design techniques. By employing the H ∞ filter approach and formulating the observer design through the LMI framework, we have developed a functional observer capable of maintaining stability and performance even in the presence of significant uncertainties.

Our rigorous analysis of the estimation error using the Lyapunov stability theory assures the convergence of the proposed observer, while simulations on a CVS model demonstrate its practical effectiveness. The ability of this observer to reliably estimate unknown states under uncertain conditions underscores its potential for enhancing the control of biological systems and improving outcomes in medical applications, particularly in the management and treatment of CVDs. This research contributes to the broader field of control theory by advancing the design of observers for descriptor systems, addressing a gap in the literature related to uncertain systems. Future work could explore the extension of this approach to nonlinear systems and real-time implementation in clinical settings, further bridging the gap between theoretical developments and practical applications.

Funding information: Pabitra Tunga acknowledged funding by Council of Scientific and Industrial Research, New Delhi, India, for the award of SRF through Grant No. 09/1023(0032)/2019-EMR-I. Nutan K. Tomar acknowledged funding by the Science and Engineering Research Board, New Delhi, via Grant No. CRG/2023/008861.
Author contributions: Pabitra K. Tunga: conceptualization, writing – original draft, software, investigation, validation and writing – review and editing. Juhi Jaiswal: validation and writing, structuring the draft and editing. Nutan K. Tomar: conceptualization, supervision, writing – original draft, validation and review and editing.
Conflict of interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.
Ethical approval: This research did not require ethical approval.
Data availability statement: Data sharing is not applicable to this article as no new data were created or analyzed in this study.

References

[1] Cardiovascular diseases (CVDs). (2021, June 11). Retrieved from https://www.who.int/news-room/fact-sheets/detail/cardiovascular-diseases-(cvds).Search in Google Scholar

[2] Berger, T. (2019). Disturbance decoupled estimation for linear differential-algebraic systems. International Journal of Control, 92(3), 593–612. 10.1080/00207179.2017.1363411Search in Google Scholar

[3] Berger, T. & Reis, T. (2019). ODE observers for DAE systems. IMA Journal of Mathematical Control and Information, 36(4), 1375–1393. 10.1093/imamci/dny032Search in Google Scholar

[4] Berger, T., Reis, T., & Trenn, S. (2017). Observability of linear differential-algebraic systems: A survey. Switzerland: Springer. 10.1007/978-3-319-46618-7_4Search in Google Scholar

[5] Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Philadelphia: SIAM. 10.1137/1.9781611970777Search in Google Scholar

[6] Burattini, R. & Gnudi, G. (1982). Computer identification of models for the arterial tree input impedance: Comparison between two new simple models and first experimental results. Medical and Biological Engineering and Computing, 20, 134–144. 10.1007/BF02441348Search in Google Scholar PubMed

[7] Campbell, S. L. (1982). Singular systems of differential equations II, vol. 61. Pitman, London. Search in Google Scholar

[8] Catanho, M., Sinha, M., & Vijayan, V. (2012). Model of aortic blood flow using the Windkessel effect, University of California of San Diago, San Diago. Search in Google Scholar

[9] Dai, L. (1989). Singular control systems. Berlin Heidelberg: Springer. 10.1007/BFb0002475Search in Google Scholar

[10] Darouach, M. (2009). Hinfty unbiased filtering for linear descriptor systems via LMI. IEEE Transactions on Automatic Control, 54(8), 1966–1972. 10.1109/TAC.2009.2023962Search in Google Scholar

[11] Darouach, M. (2013). Observers and observer-based control for descriptor systems revisited. IEEE Transactions on Automatic control, 59(5), 1367–1373. 10.1109/TAC.2013.2292720Search in Google Scholar

[12] Darouach, M., Amato, F., & Alma, M. (2017). Functional observers design for descriptor systems via LMI: Continuous and discrete-time cases. Automatica, 86, 216–219. 10.1016/j.automatica.2017.08.016Search in Google Scholar

[13] Darouach, M. & Fernando, T. (2022). Functional detectability and asymptotic functional observer design. IEEE Transactions on Automatic Control, 68(2), 975–990. 10.1109/TAC.2022.3151732Search in Google Scholar

[14] Duan, G.-R. (2010). Analysis and design of descriptor linear systems, vol. 23. New York: Springer Science & Business Media. 10.1007/978-1-4419-6397-0_3Search in Google Scholar

[15] Jaiswal, J., Berger, T., & Tomar, N. K. (2023). Partial detectability and generalized functional observer design for linear descriptor systems. arXiv: http://arXiv.org/abs/arXiv:2301.10109. Search in Google Scholar

[16] Jaiswal, J., Berger, T., & Tomar, N. K. (2024). Existence conditions for functional ODE observer design of descriptor systems revisited. Journal of the Franklin Institute, 361(8), 106848. 10.1016/j.jfranklin.2024.106848Search in Google Scholar

[17] Jaiswal, J., Berger, T., & Tomar, N. K. (2024). Partial causal detectability of linear descriptor systems and existence of functional ODE estimators. arXiv: http://arXiv.org/abs/arXiv:2405.07968. Search in Google Scholar

[18] Jaiswal, J., Gupta, M. K., & Tomar, N. K. (2021). On functional observers for descriptor systems. In: 2021 American Control Conference (ACC), pp. 4093–4098. IEEE. 10.23919/ACC50511.2021.9482902Search in Google Scholar

[19] Jaiswal, J. & Tomar, N. K. (2021). Existence conditions for ODE functional observer design of descriptor systems. IEEE Control Systems Letters, 6, 355–360. 10.1109/LCSYS.2021.3076024Search in Google Scholar

[20] Jaiswal, J., Tunga, P. K., & Tomar, N. K. (2022). Functional ODE observers for a class of descriptor systems. In: 2022 Eighth Indian Control Conference (ICC), pp. 97–102. IEEE. 10.1109/ICC56513.2022.10093626Search in Google Scholar

[21] Kind, T., Faes, T. J., Lankhaar, J.-W., Vonk-Noordegraaf, A., & Verhaegen, M. (2010). Estimation of three-and four-element Windkessel parameters using subspace model identification. IEEE Transactions on Biomedical Engineering, 57(7), 1531–1538. 10.1109/TBME.2010.2041351Search in Google Scholar PubMed

[22] Kumar, A. & Daoutidis, P. (1999). Control of nonlinear differential algebraic equation systems with applications to chemical processes, vol. 397. Florida: CRC Press. Search in Google Scholar

[23] Kunkel, P. (2006). Differential-algebraic equations: Analysis and numerical solution, vol. 2. Switzerland: European Mathematical Society. 10.4171/017Search in Google Scholar

[24] Lan, J. & Patton, R. J. (2015). Robust fault-tolerant control based on a functional observer for linear descriptor systems. IFAC-papersonline, 48(14), 138–143. 10.1016/j.ifacol.2015.09.447Search in Google Scholar

[25] Noreddine, C., Sadek, B. A., Driss, E.-J., Ismail, B., et al. (2022). Design of observer for cardiovascular anomalies detection: an LMI approach. In: 2022 International Conference on Intelligent Systems and Computer Vision (ISCV), pp. 1–6. IEEE. 10.1109/ISCV54655.2022.9806114Search in Google Scholar

[26] Riaza, R. (2008). Differential-algebraic systems: Analytical aspects and circuit applications. Singapore: World Scientific. 10.1142/9789812791818Search in Google Scholar

[27] Trenn, S. (2013). Solution concepts for linear DAEs: A survey. Surveys in differential-algebraic equations I. Springer, Berlin, Heidelberg; pp. 137–172. 10.1007/978-3-642-34928-7_4Search in Google Scholar

[28] Tunga, P. K., Jaiswal, J., & Tomar, N. K. (2023). Functional observers for descriptor systems with unknown inputs. IEEE Access, 11, 19680–19689. 10.1109/ACCESS.2023.3249099Search in Google Scholar

[29] Tunga, P. K. & Tomar, N. K. (2023). Hinfty filter based functional observers for descriptor systems. In: 2023 62nd IEEE Conference on Decision and Control (CDC), pp. 7681–7686. IEEE. 10.1109/CDC49753.2023.10383889Search in Google Scholar

[30] Wang, Y., Xie, L., & De Souza, C. E. (1992). Robust control of a class of uncertain nonlinear systems. Systems & Control Letters, 19(2), 139–149. 10.1016/0167-6911(92)90097-CSearch in Google Scholar

[31] Westerhof, N., Lankhaar, J.-W., & Westerhof, B. E. (2009). The arterial Windkessel. Medical & Biological Engineering & Computing, 47(2), 131–141. 10.1007/s11517-008-0359-2Search in Google Scholar PubMed

[32] Xu, S. & Lam, J. (2007). Reduced-order Hinfty filtering for singular systems. Systems & Control Letters, 56(1), 48–57. 10.1016/j.sysconle.2006.07.010Search in Google Scholar

[33] Xu, S., Lam, J., & Zou, Y. (2003). H/sub/spl infin//filtering for singular systems. IEEE Transactions on Automatic Control, 48(12), 2217–2222. 10.1109/TAC.2003.820149Search in Google Scholar

[34] Zervides, C. & Hose, D. R. (2006). A simple computational model-based validation of guytonas closed circuit analysis of the heart and the peripheral circulatory system. In: 2005 IEEE Engineering in Medicine and Biology 27th Annual Conference, pp. 5486–5489. IEEE. 10.1109/IEMBS.2005.1615725Search in Google Scholar PubMed

[35] Zhang, J., Wang, Z., Chadli, M., & Wang, Y. (2022). On prescribed-time functional observers of linear descriptor systems with unknown input. International Journal of Control, 95(11), 3137–3147. 10.1080/00207179.2021.1959066Search in Google Scholar

Received: 2024-08-30

Revised: 2024-09-14

Accepted: 2024-09-16

Published Online: 2024-12-10

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/cmb-2024-0013

Keywords for this article

linear systems; descriptor systems (differential-algebraic systems); observers for linear systems; functional observers; parameter uncertainty

Creative Commons

BY 4.0