OME CONTROL PROBLEMS FOR POSITIVE LINEAR SYSTEMS.OME CONTROL PROBLEMS FOR POSITIVE LINEAR SYSTEMS.OME CONTROL PROBLEMS FOR POSITIVE LINEAR SYSTEMS.OME CONTROL PROBLEMS FOR POSITIVE LINEAR SYSTEMS.OME CONTROL PROBLEMS FOR POSITIVE LINEAR SYSTEMS.
Background
A control system is an interconnection of components forming a system configu- ration, which provides desired responses by controlling outputs The following figure shows a simple block-diagram of a control system.
Figure 1: Block-diagram of a control system
A control system typically consists of physical components that include input, output, and state variables Inputs can be intentional signals from controllers or sudden disturbances that affect system performance Outputs are the measurable or observable results of the system, which can be regulated The output states are influenced by the inputs, while a state is defined by mathematical functions or physical variables that describe the system's behavior and performance when the inputs are known.
Positive systems, also known as nonnegative systems, are dynamical systems characterized by states and outputs that remain nonnegative when driven by nonnegative inputs, including initial states These systems are crucial in practical applications, such as monitoring liquid levels in tanks, chemical concentrations, and population sizes A common example of positive systems is compartmental networks, where each compartment acts as a conceptual storage tank that holds a specific amount of material in a kinetic process.
A kinetically homogeneous compartment is one where any material entering is immediately mixed with the existing contents, forming a compartmental network This network comprises several homogeneous compartments that facilitate the exchange of nonnegative quantities of materials while adhering to the conservation of mass The compartmental network framework is instrumental in developing models governed by conservation laws Its applications span various fields, including biology, ecology, epidemiology, chemistry, pharmacokinetics, air traffic flow networks, control engineering, telecommunications, and chemical-physical processes.
Positive systems offer a diverse range of applications and unique properties not found in general linear systems Their robustness and monotonicity enhance the design of interval observers, crucial for state estimation and stability analysis of nonlinear time-delay systems Interestingly, many NP-hard problems become significantly simpler within the realm of linear positive systems The increasing interest in the systems and control theory of linear positive systems stems from their widespread applications and distinct characteristics Key issues such as stability, disturbance attenuation, and robustness are critical in the analysis and synthesis of these systems, attracting considerable research attention and yielding numerous recent findings For further insights into stability analysis, controller synthesis, and L1/L∞ control, readers can refer to various studies and references.
Stability theory is a crucial aspect of systems and control theory, consistently ranking as a top research priority Its significance spans various fields, including economics, finance, environmental science, and systems engineering Stability guarantees that a system's steady state remains close to its equilibrium point and often returns to this equilibrium, a concept known as asymptotic stability.
In other words, the equilibrium is insensitive to small perturbation of initial conditions.
Lyapunov stability is a key concept in control systems, encompassing various forms such as asymptotic and exponential stability, which characterize the behavior of equilibrium points as state trajectories converge over time Additionally, other important stability concepts like bounded-input bounded-output stability, input-to-state stability, and finite-time stability play crucial roles in control engineering applications.
Time delays are an inevitable challenge in modeling engineering systems and industrial processes, particularly in multi-agent systems (MASs) where limited communication bandwidth affects information exchange, complicating consensus issues These delays are also prevalent in transmission lines and telecommunication networks, as well as in logistic networks where resource transportation is hindered by traffic jams and other latency factors The impact of time delays can lead to unpredictable behaviors, degraded performance, and potential instability within systems Consequently, the investigation of time-delay systems is crucial in control engineering, garnering significant research interest over the past two decades.
The literature on time-delay systems within systems and control theory is extensive, highlighting significant results While general linear systems theory can be applied to positive systems, unique challenges arise due to the latter's definition within convex polyhedral cones rather than linear spaces This limitation means that many properties do not hold under similarity transformations, making it difficult to apply established results from linear systems to positive ones For example, unlike general linear systems where controllability allows for arbitrary pole assignment, positive linear systems are constrained by positivity requirements on their matrices Consequently, research on positive systems emphasizes leveraging monotonicity derived from positivity to simplify the behavioral analysis and enhance the design of dynamic systems, establishing the theory of positive linear systems with delays as a crucial area in theoretical control.
Exogenous disturbances are prevalent in engineering systems due to issues like data processing inaccuracies and measurement errors The performance of a dynamical system is often measured by its ability to attenuate the effects of these disturbances, commonly quantified using norms that relate system responses to disturbance inputs The H ∞ norm, also known as the L2-induced norm, assesses the maximum gain of a system, reflecting the worst-case norm of regulated outputs against bounded energy inputs In contrast, the H2 norm measures output variance in response to exogenous inputs Positive systems uniquely permit the use of linear supply rates in dissipative analysis, leading to input-output properties defined by linear constraints For instance, analyzing the maximum mass in compartmental networks is more effective using L1-gain or L∞-gain as performance measures rather than the traditional L2-gain These measures facilitate elegant linear programming characterizations for various performance specifications Although positivity constraints can complicate the design of controllers and filters, they can also streamline design processes in certain scenarios Recent advancements highlight that the positivity of dynamic systems can significantly simplify control problems Linear programming, a powerful convex optimization method with low computational demands, is particularly adept at synthesizing controllers for positive systems, addressing a range of challenges from stabilization to robust control with optimal performance indices.
In control system design, controllers are synthesized based on prior analysis to meet specific performance criteria, ensuring the controlled system exhibits desired properties Solutions to these design problems often involve the feasibility of matrix equations and inequalities, with feedback compensation being a predominant approach Various control strategies, including state-feedback and output-feedback, are employed depending on practical situations State-feedback control is favored for its ability to fully describe a system's dynamic behavior; however, in many cases, not all states can be measured or accessed in real-time As a more practical alternative, static output-feedback control is utilized Additionally, for positive systems, maintaining the positivity of closed dynamics complicates the design process, making it more challenging than for general systems.
Despite the publication of numerous results in systems and control theory for positive systems, significant unresolved issues remain that require further investigation Notably, the feasibility of design conditions for state-feedback and static output-feedback controllers is still an open question Additionally, optimal control problems related to L1 and ℓ1 norms continue to pose challenges in the field.
L ∞ /ℓ∞-gain schemes or optimal control problem for positive systems This motivates us for the study presented in this thesis.
Literature review
B1 Static output-feedback control of positive linear systems
In controller design, new challenges emerge when state variables are constrained by convex cones, making it difficult to apply traditional methodologies for general linear systems effectively.
In the literature, various methods have addressed the problem using quadratic or linear Lyapunov functions to establish synthesis conditions through LMIs, iterative LMIs, or LPs Notably, the LP approach tends to offer numerical advantages over the LMI method, as LMIs often involve more decision variables and existing software struggles with larger problems and lacks numerical stability Consequently, the LP-based approach is more commonly employed for deriving analysis and synthesis conditions for positive systems.
The static output-feedback control problem is a critical challenge in systems and control theory, primarily due to the limited accessibility of system state variables Unlike state-feedback stabilization, which requires full-state vector access, static output-feedback cannot be precisely formulated as an LMI problem or solved through pole placement techniques The feasibility of BMIs arising from static output-feedback is NP-hard without a positivity constraint on closed-loop systems, complicating the pole placement for positive systems While LP-based methods have shown promise for positive systems, literature reveals that comprehensive solutions are scarce Notably, existing design methods often impose restrictions, such as being limited to MIMO systems or requiring specific matrix structures Moreover, some approaches only establish sufficient stabilization conditions without confirming the existence of a desired static output-feedback controller Recent work has derived necessary and sufficient conditions for state-feedback stabilization of LTI positive systems without delay, but these findings are not easily applicable to systems with time delays in state and output vectors.
Despite significant research efforts focused on stabilizing LTI positive systems using SOFCs, a comprehensive solution, particularly for positive time-delay systems, remains crucial The investigation into the conditions for static output-feedback stabilization in LTI positive systems with delays continues to be a pertinent issue.
B2 L 1 -gain control of positive linear systems with multiple delays
Stability, performance analysis, and controller synthesis are critical challenges in systems and control theory Over the past twenty years, these issues have been thoroughly investigated, particularly in relation to positive systems, both with and without delays.
The Lyapunov–Krasovskii functional (LKF) method and its variants are commonly employed for analyzing and designing linear systems with delays, particularly in dynamical systems lacking positivity In contrast, for positive systems, the inherent monotonicity provided by positivity plays a crucial role in developing co-positive Lyapunov functionals.
This approach facilitates the formulation of direct comparison techniques, enabling the derivation of essential analysis conditions and effective LP-based synthesis conditions Additionally, it allows for the simplification of Linear Matrix Inequalities (LMIs) using diagonal Lyapunov matrix variables.
The problems of performance analysis and synthesis under L 1-gain and ℓ1-gain control schemes have drawn significant research attention in the past few years [16,
In the existing literature, stability and performance analysis conditions are typically established through linear or semidefinite programming, utilizing specific co-positive Lyapunov functions Moreover, the controller synthesis problem within the L1/ℓ1-gain framework presents greater challenges compared to performance analysis.
To address the design challenge, understanding the exact value of L1-gain is crucial for establishing necessary and sufficient synthesis conditions Research has shown that the ℓ1-induced norm can be precisely determined using the lifting technique and a clear representation of the fundamental matrix This approach successfully resolved the synthesis problem for single-input single-output (SISO) systems However, in more complex scenarios involving multiple-input multiple-output (MIMO) systems, a stringent constraint related to a direction matrix complicates the derivation of necessary conditions for the existence of stabilizing controllers, even in linear time-invariant (LTI) systems without delays.
In recent research, necessary and sufficient stabilization conditions for LTI systems under an equality constraint were established using L1-gain performance characterization However, for positive systems with delays, existing solution representations are inadequate for L1-gain characterization, and literature lacks direct methods to address this issue in time-delay systems Furthermore, current approaches to stabilization under the L1-gain scheme have proven ineffective for verifying the feasibility of desired controllers, leaving a critical gap in ensuring the existence of suitable L1-gain controllers This highlights the need for a systematic approach and comprehensive solutions to address these challenges.
L 1-gain control of positive systems with delays are still left open.
B3 Peak-to-peak gain control of discrete-time positive linear systems
Exogenous disturbances are prevalent in engineering systems due to various technical issues such as data processing inaccuracies, linear approximations, and measurement errors To effectively manage dynamical systems with uncertainties, robust control theory provides essential tools Two primary paradigms in robust control focus on modeling plant and signal uncertainty related to signal energy (using L2 or ℓ2-norms for continuous or discrete time) and signal peak values, leading to the development of H∞ theory and L1 or L∞ induced theory These frameworks address the challenges of analysis and control in uncertain environments.
The H ∞ performance index focuses on minimizing the worst energy-to-energy (L 2-L 2) gain, a topic that has been extensively explored in the literature Additionally, robust L 2-L ∞ or ℓ2-ℓ∞ stability and controller design aim to minimize the worst energy-to-peak gain from noise input to filtering error output, with significant advancements made for various models of dynamical systems.
The H ∞ and l2-ℓ∞ performance indices are based on the assumption of energy-bounded disturbance inputs; however, real-world engineering systems often experience persistent and amplitude-bounded external disturbances, such as wind shear on aircraft wings or continuous road excitation on vehicle suspension systems Consequently, these control schemes are not suitable for such scenarios Instead, a more appropriate performance index is the worst-case amplification from input disturbance to regulated output, leading to the concept of peak-to-peak gain.
The ℓ∞-induced optimal design aims to minimize the maximum peak-to-peak gain of closed-loop systems affected by bounded amplitude disturbances, making ℓ∞-gain minimization a valuable method for analyzing dynamic systems under persistent disturbances Over the past two decades, ℓ∞ theory has gained significant attention for its theoretical importance and practical applications, serving as a natural counterpart to H∞ stability theory and addressing concerns in control engineering Recent studies have explored various control and filtering challenges in both one-dimensional and two-dimensional systems, often employing Lyapunov or LKF methods to establish sufficient LMI conditions for bounding L∞/ℓ∞-induced gain However, current methods may lead to overly conservative conditions, posing challenges in terms of feasibility and computational complexity for achieving optimal peak-to-peak gain.
In the study of positive systems, the characteristics of positive state trajectories enable the use of linear co-positive Lyapunov functions, which facilitate analysis and synthesis in linear contexts This approach encourages the exploration of performance indices such as L ∞-gain and L 1-gain, essential for addressing analysis and design challenges associated with these indices.
Research topics
This thesis addresses challenges in the systems and control of positive linear systems with delays, focusing on the development of methodologies and the establishment of new findings These results are essential for deriving the necessary and sufficient conditions for the existence of desired controllers The study will cover several key topics related to this area of research.
C1 Static output-feedback control of positive linear systems with time- varying delay
Consider the following control system with delayed measurement output x˙ (t) = Ax(t) + A d x(t − δ t ) + Bu(t), t ≥
In the context of control systems, the state vector x(t) belongs to R^n, while the control input u(t) and measured output y(t) are elements of R^m and R^p, respectively The matrices A and A_d are real matrices of size n×n, B is a real matrix of size n×m, and C and C_d are real matrices of size p×n Additionally, the system experiences an unknown time-varying delay, δt, which is constrained within the interval [0, δ*], where δ* is a defined constant.
A Solid Oxide Fuel Cell (SOFC) is formulated as u(t) = -Ky(t) = -KCx(t) - KC d x(t - δt), where K ∈ R m×p represents the controller gain to be determined By integrating the SOFC into the system, the closed-loop dynamics are expressed as x˙(t) = (A - BKC)x(t) + (A d - BKC d)x(t - δt) for t ≥ 0.
This article aims to establish testable necessary and sufficient conditions for the existence of a Solid Oxide Fuel Cell (SOFC) that ensure the closed-loop system is both positive and globally asymptotically stable (GAS) By utilizing linear optimization techniques, we derive these stabilization conditions through linear programming (LP) criteria A detailed exploration of this topic will be provided in Chapter 2.
C2 L 1 -gain control of positive linear systems with multiple delays
In Chapter 3, we study the problem of L 1-gain control for a class of LTI systems with multiple delays of the following form m x˙ (t) = A 0 x(t) + A k x(t − h k ) + B w w(t), t ≥ 0, k=1 m z(t) = C 0 x(t)+
The state vector x(t) is defined for the interval t ∈ [−d, 0] as x(t) = φ(t), where φ represents the initial condition of the system The regulated output vector z(t) and the exogenous disturbance input vector w(t) are both elements of R^n The matrices A0, Ak, Bw, C0, Ck, and Dw, with k ranging from 1 to m, are specified real matrices of suitable dimensions Additionally, the scalars hk and τk account for time delays in the system.
System (4) with w = 0 is globally exponentially stable (GES) if there exist positive constants α and β such that any solution x(t, φ) of (4) satisfies
The stability of the system is characterized by the inequality ∥x(t, φ)∥≤ β∥φ∥ C e −αt for t ≥ 0, where ∥φ∥ C represents the supremum of the norm of φ over the interval from -d to 0 The system is classified as L 1-stable if it meets two criteria: first, when w = 0, the system exhibits Global Exponential Stability (GES); second, for any nonzero disturbance w in L 1(R+, R n w), the solution x(t, φ) remains bounded within the specified constraints.
∥x(t, φ)∥1 dt < ∞ In the first part, we will show that, for Σ Σ
0 positive system (4), GES and L 1-stability are indeed equivalent Assume that system
(4) is stable (in the sense of GES) The input-output operator is defined as Σ : L 1(R+ , R n w ) −→ L 1(R+ , R n z ), w ›→ z, and L 1-gain of system (4) under zero initial condition is
The objective of Chapter 3 is to
Formulate a characterization of L 1-gain ∥Σ∥ (L 1 ,L 1 ) of system (4)
Derive conditions for L 1-gain performance with prescribed level, that is, whether
Establish necessary and sufficient conditions for the existence of an SFC that makes the closed-loop system positive, stable and has prescribed L 1-gain perfor- mance index.
An alternative method employing Laplace transformation allows for the characterization of the L1-induced norm of the input-output operator, which is essential for establishing necessary and sufficient conditions for achieving L1-induced performance at a specified level Utilizing optimization techniques, a comprehensive solution to the stabilization problem within the framework of L1-gain control is derived through manageable linear programming (LP) conditions.
C3 Peak-to-peak gain control of discrete-time positive linear systems with diverse interval delays
Consider the following discrete-time system with multiple time-varying delays
(6) x(k) = φ(k), k ∈ Z[−d, 0], where x(k) ∈ R n is the state vector, z(k) ∈ R n z and w(k) ∈ R n w are the regulated Σ
0 j j w output and exogenous disturbance input vectors, respectively A 0, A j , B w , C 0, C j and
Let \( D_w \) and \( j \) be real matrices of suitable dimensions, with \( d_j(k) \) and \( h_j(k) \) representing unknown time-varying delays These delays are constrained by the inequalities \( d_j \leq d_j(k) \leq d_j \) and \( h_j \leq h_j(k) \leq h_j \), where \( d_j \), \( d_j \), \( h_j \), and \( h_j \) are positive integers that define the lower and upper bounds of the delays Additionally, let \( d \) be the maximum value of \( d_j \) and \( h_j \) across all \( j \) in the range from 1 to \( N \) The sequence of initial states is denoted as \( \phi(k) \).
System (6) with w = 0 is GES if there exist constants α ∈ (0, 1), β > 0 such that any solution x(k, φ) of (8) satisfies
The inequality ∥x(k, φ)∥∞ ≤ β∥φ∥∞α k, for k ∈ N0, indicates the boundedness of the system's response, where ∥φ∥∞ is defined as the maximum norm over the interval l ∈ Z[−d,0] In Chapter 4, we establish the necessary and sufficient conditions for the positive system (6) with w = 0 to be globally exponentially stable (GES) Furthermore, it is demonstrated that under these conditions, system (6) also exhibits ℓ∞-stability.
∥x(k)∥∞ < ∞ for any disturbance input w ∈ ℓ∞ Assume that system (6) is stable (GES) The input-output operator of system (6) is defined as Ψ : ℓ∞(R n w ) −→ ℓ∞(R n z ), w ›→ z, and ℓ∞-gain of system (6) under zero initial condition as
For a specified γ > 0, system (6) demonstrates ℓ∞-gain performance at level γ if the norm ∥Ψ∥ (ℓ ∞ ,ℓ ∞ ) is less than γ Utilizing innovative comparison methods that incorporate steady states of both upper and lower scaled systems alongside peak values of external disturbances, we provide a characterization for the ℓ∞-induced norm of the input-output operator.
ℓ∞-gain characterization is essential for establishing necessary and sufficient conditions for achieving desired performance levels Utilizing performance analysis results and a vertex optimization technique, this approach provides a comprehensive solution to the synthesis problem of solid oxide fuel cells (SOFCs) The solution focuses on minimizing the worst-case amplification of disturbances affecting regulated outputs while adhering to peak-to-peak gain constraints.
Outline of main contributions
This thesis addresses the challenges of stabilization, L1-gain, and ℓ∞-gain control in positive linear systems with Σ delays, utilizing both static output feedback and state feedback methods The key findings of this research are summarized as follows.
This article explores the stabilization problem of linear time-invariant (LTI) positive systems affected by time-varying delays in both state and output vectors It introduces an innovative method utilizing optimization techniques to establish necessary and sufficient linear programming (LP)-based conditions for the existence of effective controllers.
This article presents a characterization of L1-induced performance for a specific category of positive linear systems that incorporate multiple delays The established performance characterization is then employed to formulate both necessary and sufficient conditions for the existence of state feedback controllers (SFCs) that meet the L1-gain requirements of the closed-loop systems.
This article addresses the challenge of static output-feedback optimal peak-to-peak gain control in discrete-time positive linear systems that experience heterogeneous interval delays It introduces a novel characterization of ℓ∞-gain and offers a comprehensive solution for synthesizing a static output-feedback controller (SOFC) aimed at minimizing the worst-case amplification of disturbances to regulated outputs while adhering to ℓ∞-gain constraints.
The aforementioned results have been published in 03 papers in international journals (ISI, Q1-Q2) and have been presented at
The weekly seminar on Differential and Integral Equation, Division of Mathemat- ical Analysis, Faculty of Mathematics and Informatics, Hanoi National University of Education.
Seminar of the Division of Mathematics, Faculty of Informatics Technology, Na- tional University of Civil Engineering.
PhD Annual Conferences, Faculty of Mathematics and Informatics, Hanoi Na- tional University of Education, 2019, 2020.
Workshop Dynamical Systems and Related Topics, Vietnam Institute for Ad- vanced Study in Mathematics (VIASM), Hanoi, December 23-25, 2019.
Workshop Selected Problems in Differential Equations and Control, Vietnam
Insti- tute for Advanced Study in Mathematics (VIASM), Tuan Chau, Ha Long,Novem- ber 5-7, 2020.
Thesis structure
STATIC OUTPUT-FEEDBACK CONTROL OF POSITIVE LINEAR SYS- TEMS
STATIC OUTPUT-FEEDBACK CONTROL OF POSITIVE LINEAR SYSTEMS WITH TIME-VARYING DELAY
This chapter addresses the stabilization of linear time-invariant (LTI) positive systems through static output-feedback control, particularly focusing on systems with time-varying delays in state and output vectors It outlines necessary and sufficient conditions for achieving exponential stability in the closed-loop system by leveraging induced monotonicity The stabilization conditions are expressed using matrix inequalities, which are subsequently reformulated into vertex optimization problems This approach facilitates the derivation of conditions for the existence of a desired static output-feedback controller (SOFC) The synthesis conditions are presented as linear programming criteria, allowing for effective resolution through various convex algorithms The content is primarily based on the findings presented in paper [P1] from the publication list.
Consider the following LTI system with delay x˙ (t) = Ax(t) + A d x(t − δ t ) + Bu(t), t ≥
(2.1) where x(t) ∈ R n is the state vector, u(t) ∈ R m and y(t) ∈ R p are the control input and the measured output vectors, respectively In system (2.1), A, A d ∈ R n×n , B ∈ R n×m , C,
C d ∈ R p×n are given system matrices, δ t represents an unknown time-varying delay which satisfies 0 ≤ δ t ≤ δ∗, where δ∗ is a prescribed constant, and φ(t) is the initial function specifying initial state of the system.
Definition 2.1.1 ([32]) System (2.1) is said to be (internally) positive if for any non- negative input, u(t) ≽ 0, t ≥ 0, and nonnegative initial function, φ(t) ≽ 0, t ∈ [−δ∗ ,
0], the state and output vectors are always nonnegative, that is, x(t) ≽ 0 and y(t) ≽ 0 for all t ≥ 0.
Similar to [79, Chapter 5], we have the following characterization of positivity of system (2.1).
Proposition 2.1.1 System (2.1) is positive if and only if A is a Metzler matrix and
In the design of the SOFC, the control input is defined as u(t) = −Ky(t) = −KCx(t) − KC d x(t − δ t), where K is the controller gain By integrating this SOFC into the system, the closed-loop dynamics are expressed as x˙(t) = (A − BKC)x(t) + (A d − BKC d)x(t − δ t), applicable for t ≥ 0.
In this chapter, we aim to establish testable necessary and sufficient conditions for the existence of a state observer feedback controller (SOFC) that ensures the closed-loop system is positive and globally asymptotically stable (GAS).
This section outlines the necessary and sufficient conditions for ensuring the stability of a positive closed-loop system Specifically, system (2.3) is deemed positive, meaning that x(t) is non-negative for all t ≥ 0 given that φ is also non-negative, if and only if the matrix A c is a Metzler matrix and the matrix A dc is nonnegative.
Since system (2.3) is subject to an unknown time-varying delay δ t , similar to
[60], we consider the following scaled system xˆ˙ (t) = A c xˆ(t) + A dc xˆ(t − δ∗), t ≥ 0, xˆ(t) = φˆ(t), t ∈ [−δ∗ ,
(2.4) where φˆ(t) denotes the initial condition of system (2.4) Necessary and sufficient sta- bility conditions for system (2.4) are given below.
Theorem 2.2.1 Assume that system (2.4) is positive Then, the following statements are equivalent.
(b) System (2.4) is globally exponentially stable (GES), that is, there exist positive constants α, β such that any solution xˆ(t) of (2.4) satisfies
(c) There exists a vector ν ∈ R n , ν ≻ 0, such that ν ⊤ (A c + A dc ) ≺ 0.
(d) There exists a vector η ∈ R n , η ≻ 0, satisfying the following LP condition
Proof The implications (b) ⇒ (a) and (c) ⇒ (d) are obvious We now present a simple proof for the implications (a) to (c) by utilizing Proposition 1.1.2 and (d) to (b).
(a) ⇒ (c): Suppose that there exists a vector ν ∈ R n \{0} such that
Let ν ⊤ (A c + A dc ) ≽ 0. xˆ(t) be the solution of (2.4) with constant initial function φˆ = ν We define a co-positive Lyapunov functional as
= ν ⊤ (A c + A dc )xˆ(t) ≥ 0, as xˆ(t) ≽ 0 for all t ≥ 0 Therefore,
+ A dc φ(s)ds which yields a contradiction as v(xˆ(t)) → 0 when t → ∞ Thus, ν ⊤ (A c + A dc ) has at least one strictly negative entry for any ν ∈ R n \{0} which validates condition (c) according to item (iii) of Proposition 1.1.2.
(d) ⇒ (b): Let η = (η i ) ∈ R n n be a positive vector satisfying (2.5) Then, we have Σ a c c dij Ση j < 0, j=1 for all i ∈ 1, n, where A c = (a c ) and A dc = (a
) For any i ∈ 1, n, the function ij n dij
H i (α) ắ αη i + Σ δ ∗ α c dij Ση j , j=1 is continuous and strictly increasing on the interval [0, ∞), H i (0) < 0 and H i (α) → ∞ as α → ∞ Thus, there exists a unique scalar α c c
∗ > 0 such that H i (α ∗ ) = 0 It is clear i i that H i (α) < 0 for 0 < α < α ∗ Let α ∗ = min i∈1,n α ∗ then, for α ∈ (0, α ∗ ), H i (α) < 0 i i for all i ∈ 1, n, by which
A c + αI n + A dc e δ∗α η ≺ 0 (2.6) Now, for any solution xˆ(t) of (2.4) with initial function φˆ, let xˆ + (t) be the solution of (2.4) with initial function |φˆ| Since φˆ + |φˆ| ≽ 0 and |φˆ| − φˆ ≽ 0, according to the positivity of (2.4), we have xˆ(t) + xˆ + (t) ≽ 0 and xˆ + (t) − xˆ(t) ≽ 0 for all t
On the other hand, let ϵ > 0 be a constant such that |φˆ(t)| ≼ ϵη, t ∈ [−δ∗ , 0] Then, we define the following vector-valued functions ξ(t) = ϵηe −αt , ζ(t) = ξ(t) − xˆ + (t), t ≥ 0.
It is easy to verify from (2.6) that
Therefore, ω(t) ắ ξ˙(t) − (A c ξ(t) + A dc ξ(t − δ∗)) is a positive vector-valued function.
In addition, by the definition of function ζ(t), we have ζ˙(t) = A c ζ(t) + A dc ζ(t − δ∗) + ω(t), which ensures that ζ(t) ≽ 0 for all t ≥ 0 By this, |xˆ(t)| ≼ ϵηe −αt and we have that
.Σ the positive system (2.4) is GES The proof is completed.
Remark 2.2.1 In the second part of the proof of Theorem 2.2.1, if we take ϵ = 1 and α = 0 then |xˆ(t)| ≼ η for any solution xˆ(t) of system (2.4) with |φˆ(t)| ≼ η, where η
The solution xˆη(t) of equation (2.4), defined with φˆ ≡ η, demonstrates a globally decreasing behavior on the interval [0, ∞) For any fixed τ > 0, the function ζτ(t), which is defined as xˆη(t) − xˆη(t + τ), also satisfies the conditions of system (2.4) with the initial function φˆ ≡ η − xˆη(τ) being non-negative Consequently, ζτ(t) remains non-negative for all t ≥ 0, reinforcing the global decrease of xˆη(t).
Remark 2.2.2 Let x η(t) be the solution of positive system (2.3) with φ ≡ η, where η ≻ 0 is a vector satisfying condition (2.5), and ψ(t) = xˆη(t) − x η(t) In regard to Remark 2.2.1, xˆη(t − δ t ) ≼ xˆη(t − δ∗) Thus, ψ˙(t) = A c ψ(t) + A dc (xˆη(t − δ∗) − x η(t − δ t ))
By the positivity of system (2.3), and from (2.7), we have ψ(t) ≽ 0 which yields x η(t) ≼ xˆη(t), t ≥ 0.
This, together with Remark 2.2.1, provides a simpler proof for essential results of Lemmas 5 and 6 in [60].
Finally, we will show that the stability of system (2.3) for a time-varying delay δ t and that of system (2.4) is actually equivalent.
Proposition 2.2.1 states that for a time-varying delay δt, a positive system (2.3) is globally asymptotically stable (GAS) or globally exponentially stable (GES) if and only if the positive system (2.4) exhibits the same stability properties This equivalence is linked to the conditions (c) or (d) outlined in Theorem 2.2.1.
Remark 2.2.3 In the existing literature, Proposition 2.2.1 is stated for all delay δ t
In the proof of necessity, δ t is considered either zero or δ∗ However, when dealing with system (2.3) being stable for a specific fixed delay δ t, the situation becomes slightly more complex and typically more challenging, as δ t cannot be set to δ∗.
Proof (of Proposition 2.2.1) We define δ = inf t≥0 δ t and consider the following scaled system x¯˙(t) = A c x¯(t) + A dc x¯(t − δ ), t ≥
Let η ≻ 0 be a vector satisfying condition (2.5) and x¯η(t) be the solution of (2.8) with φ¯ ≡ η By similar arguments presented in Remarks 2.2.1 and 2.2.2, we obtain x¯η(t) ≼ x η(t) ≼ xˆη(t), t ≥ 0 (2.9)
By (2.9), the stability of positive systems (2.3), (2.4), (2.8) and conditions (c), (d) of Theorem 2.2.1 are equivalent The proof is completed.
This section focuses on the stabilization problem for system (2.1), aiming to establish necessary and sufficient checkable conditions for the existence of an SOFC (2.2) that ensures the closed-loop system (2.3) is both positive and stable We will operate under the assumption that system (2.1) is positive, and we will introduce an additional assumption to support our analysis.
Assumption (A): The matrices B and C + C d have full-column rank and full-row rank, respectively, that is, rank(B) = m, rank (C + C d ) = p.
Definition 2.3.1 The stabilization problem of system (2.1) is said to be solvable if there exists an SOFC in the form of (2.2) such that the closed-loop system (2.3) is positive and stable (in the sense of GAS or GES).
According to Definition 2.3.1, and based on the result of Theorem 2.2.1, general synthesis conditions of a desired SOFC (2.2) are formulated in the following proposition.
Proposition 2.3.1 The stabilization problem of positive system (2.1) is solvable if and only if there exists a matrix K ∈ R m×p satisfying simultaneously the following conditions
Proof The proof of Proposition 2.3.1 follows from the results of Propositions 2.1.1,
Remark 2.3.1 Let B = b 1 b 2 ããã b m , b j ∈ R n , and K ⊤ = k 1 k 2 ããã k m , k j ∈ R p Then, for any ν ∈ R n , ν ≻ 0, we have
By using the change of variable j j j=1 k ⊤ = 1 z ⊤ , z j ∈ R p , j σ j where σ > 0 is a constant, and under the additional condition that z j ≽ z for j ∈ 1, m, where z is some vector in R p , condition (2.10c) is reduced to ν ⊤ A s − 1 Σ
To get decoupling of design parameters ν and z in (2.11), we take σ = Σ
Then, BK = 1 Σ m b j z ⊤ Thus, condition (2.10b) can be written as σ j=1 j
On the other hand, it is well-known that a matrix M ∈ R n×n is Metzler if and only if there exists a constant ς such that M + ςI n ≽ 0 [92, Lemma 3] Therefore, with
BK = 1 BZ ⊤ , where σ = 1 ⊤ B ⊤ ν, the matrix A c in (2.10a) can be written as σ m
By changing the variable σˆ = ςσ, condition (2.10a) is satisfied if and only if the following one is
In summary, the derived conditions in (2.10) are assured by the following LP-based conditions
Remark 2.3.2 Condition (2.12) provides a sufficient condition for stabilization of sys- tem (2.1) In other words, if (2.12) is feasible then a controller gain matrix K that makes the closed-loop system (2.3) positive and stable is given by
K = 1 (B1 ) ⊤ νZ ⊤ , where ν ∈ R n , ν ≻ 0, and Z ∈ R m×p belong to a feasible solution of (2.12).
Remark 2.3.3 Let us consider a special case of model (2.1), where A d = 0 and C d = 0, the delay-free system Then, the derived conditions in (2.12) are reduced to conditions (3a)-(3c) of Theorem 1 in [92] Thus, the above result can be regarded as an extension of the result of Theorem 1 in [92] Although the proposed conditions in (2.12) are formulated by an LP problem, for which various computational tools and optimization algorithms can be used to search for a desired solution, it still cannot help to solve the verification problem More precisely, ones cannot ensure whether a controller gain matrix K satisfying (2.10) exists based on condition (2.12) On one hand, (2.12) is only a sufficient condition of (2.10) On the other hand, it is actually still in the form BMIs with respect to its variables It is well-known that the feasibility of BMIs is an NP-hard problem Thus, it is of importance to reformulate (2.10c) into a standard LP condition with an optimal controller gain K that satisfies conditions (2.10a)-(2.10b).
We decompose C = c 1 c 2 ããã c n and C d = c d1 c d2 ããã c dn
Thus, conditions (2.10a)-(2.10b) can be written as m
2.3.1 Single-input single-output systems
To demonstrate the design ideas, let us first consider the case of SISO systems.
In this case, we have B = (b i ) ∈ R n is a column-matrix, C and C d are row-vectors with entries c i and c di , i ∈ 1, n, respectively, and K = k is a scalar We define the following constant γ ∗ = min a dij
For convenience, let E i ∈ R n×n denote the diagonal matrix with only iith entry equals one and all other entries equal zero We summarize the derived stabilization conditions in the following proposition.
Proposition 2.3.2 Consider the positive system (2.1) and assume that Assumption (A) is satisfied Then, the following statements hold.
The stabilization problem is solvable if there is an index \(i\) such that \(BC = b_i c_i E_i\), and this is true if and only if the matrix \(A_s - \gamma^* F_i\) is Hurwitz Additionally, a positive vector \(\nu \in \mathbb{R}^n\) exists such that \(\nu^\top A_s \prec \gamma^* \nu^\top F_i\), where \(F_i = b_i E_i (c_i I_n + 1_n C_d)\) The optimal controller gain is determined to be \(k = \gamma^*\).
(ii) Let γ ∗ = min i̸=j , : b i c j ̸= 0, Then, the stabilization problem of (2.1) is a ij solvable if and only if the matrix A s −γ ∗ BC s is Hurwitz, where γ ∗ = min{γ ∗ , γ ∗ }. a d
In this case, the optimal controller gain is obtained as k = γ ∗
In the first case, the initial condition in (2.13) is omitted, while the second condition is satisfied if and only if k is less than or equal to γ ∗ Consequently, we find that kν ⊤ BC s is less than or equal to γ ∗ ν ⊤ (b i c i E i + BC d ) Therefore, the feasibility of condition (2.10c) ensures the feasibility of condition (2.15), and the reverse is also valid when k equals γ ∗ The proof for item (ii) follows similar reasoning.
Remark 2.3.4 Regarding item (i) of Proposition 2.3.2, if C d = 0 then no threshold of k is imposed Therefore, when b i c i ̸= 0, there exists a gain k such that A s − kb i c i E i is
A matrix A is considered Hurwitz if the submatrix A_i, formed by removing the ith row and column from A, is stable However, when C_d is not equal to zero, this criterion does not hold; thus, while the matrix A may be stable, the stabilization problem could lack a feasible solution.
The impact of time-delay on system responses is significant, as demonstrated by a counterexample in the following section It illustrates that while the stabilization problem for system (2.1) can always be solved without delayed measurements, the same problem becomes unsolvable when delayed measurement outputs are introduced.
2.3.2 Single-input multiple-output systems
We now turn our attention to SIMO systems in the form of (2.1) In this case, the controller gain K = k i
⊤ is a row-vector of dimension p (i.e k ∈ R p ) We first assume that B = (b i ) is not unity (i.e at least two entries of B are nonzero) Similar to Proposition 2.3.2, we define the constants δ ∗ = min a ij
Then, condition (2.13) holds if and only if k ⊤
Revealed by condition (2.16), we define a polyhedron ∆ as follows
ON L 1 -GAIN CONTROL OF POSITIVE LINEAR SYSTEMS WITH MUL-
ON L 1 -GAIN CONTROL OF POSITIVE LINEAR SYSTEMS WITH MULTIPLE DELAYS
This chapter addresses the L1-gain control problem for positive linear systems with varying state and output delays It begins by presenting a necessary and sufficient stability condition through exponential and L1-stability concepts, introducing a novel approach using Laplace transformation to characterize the L1-induced norm of the input-output operator This characterization is then applied to establish conditions for achieving L1-induced performance at a specified level Utilizing vertex optimization techniques, the chapter formulates a comprehensive solution for the stabilization problem under the L1-gain control scheme, expressed through manageable linear programming (LP) conditions The effectiveness of the proposed control method is demonstrated through numerical examples and simulations, with the content derived from paper [P2] in the List of publications.
Consider the following LTI system with multiple delays m x˙ (t) = A 0 x(t) + A k x(t − h k ) + B w w(t), t ≥ 0, (3.1a) k=1 m z(t) = C 0 x(t)+ C k x(t − τ k )+ D w w(t), t ≥ 0, (3.1b) k=1 x(t) = φ(t), t ∈ [−d, 0], (3.1c) where x(t) ∈ R n is the state vector, z(t) ∈ R n z and w(t) ∈ R n w are the regulated output and exogenous disturbance input vectors, respectively A 0, A k , B w , C 0, C k and
Given real matrices D w (k = 1, 2, , m) with appropriate dimensions, the scalars h k and τ k represent time delays, and d is defined as the maximum of h k and τ k for k ranging from 1 to m The function φ belongs to the space C([−d, 0], R n ) and serves as the initial condition for initializing the system state over the interval [−d, 0] To specify this initial condition, we denote the corresponding solution of equation (3.1) as x(t, φ), where φ is an element of C([−d, 0], R n ).
Definition 3.1.1 ([32]) System (3.1) is said to be positive if its state and output tra- jectories x(t), z(t) are always nonnegative for any φ(t) ≽ 0 and w(t) ≽ 0.
Proposition 3.1.1 ([12, 41]) System (3.1) is positive if and only if the following con- dition holds
Definition 3.1.2 System (3.1) with w = 0 is said to be globally exponentially stable (GES) if there exist positive constants α and β such that any solution x(t, φ) of (3.1) satisfies the inequality
Assume that system (3.1) is stable (GES) We define the input-output operator Σ : L 1(R+ , R n w ) −→ L 1(R+ , R n z ), w ›→ z, and L 1-gain of system (3.1) under zero initial condition as
The L1-gain, or L1-induced norm, is essential for evaluating the performance of positive systems by analyzing their input and output signals A well-defined system gain provides a quantitative measure of system performance, particularly in assessing the impact of disturbances and uncertainties The L1-gain is crucial for robust stability analysis, ensuring that systems can withstand external disturbances In controller, filter, or observer design, accurately determining the L1-induced norm of closed-loop systems can be challenging Therefore, it is more practical to create controllers that ensure the stability of the closed-loop system while adhering to specified L1-gain thresholds, leading to a clear definition of L1-gain performance.
Definition 3.1.3 For a given scalar γ > 0, system (3.1) is said to have L 1-gain per- formance at level γ if ∥Σ∥ (L 1 ,L 1 ) < γ.
Objectives: Our main objectives in this chapter are as follows
(i) Formulate the exact value of L 1-gain ∥Σ∥ (L 1 ,L 1 ) of system (3.1).
(ii) Characterize the L 1-gain performance index.
(iii) Establish necessary and sufficient conditions for the existence of a state feedback controller (SFC) that makes the closed-loop system positive, stable and has pre- scribed L 1-gain performance index.
In this section, we derive necessary and sufficient conditions by which positive system (3.1) with w = 0 is GES.
Lemma 3.2.1 Let x(t, φ1) and x(t, φ2) be any two solutions of positive system (3.1).
If φ1(s) ≼ φ2(s) for s ∈ [−d, 0] then x(t, φ1) ≼ x(t, φ2) for all t ∈ [0, ∞).
Proof It can be verified directly from (3.1) that x(t, φ) = x(t, φ2) − x(t, φ1), t ≥ 0, is a solution of (3.1a) with initial function φ = φ2 − φ1 Since φ(s) ≽ 0 for s ∈ [−d, 0], by the positivity of system (3.1a), we have x(t, φ) ≽ 0 for t ≥ 0 Thus, x(t, φ1) ≼ x(t, φ2) The proof is completed.
Remark 3.2.1 For any φ ∈ C([−d, 0], R n ), by Lemma 3.2.1, we have
Therefore, to derive stability conditions for positive system (3.1), it is only necessary to consider initial conditions that belong to C + = C([−d, 0], R n ).
Lemma 3.2.2 Consider positive system (3.1) with w = 0 If there exists a nonzero vector ξ ∈ R such that A 0 + Σ k
+ m then any solution x(t, φ) of (3.1) satisfies x(t, φ) ≽ ξ for t ∈ [0, ∞) provided that φ(s) ≽ ξ for s ∈ [−d, 0].
Proof By Lemma 3.2.1, we have x(t, φ) ≽ x(t, ξ), t ∈ [0, ∞), where x(t, ξ) is the solution of (3.1) with constant initial condition φ(s) = ξ, s ∈ [−d,
A k Σξ can be regarded as a nonnegative input vector.
Since system (3.4) is positive and e ξ(s) = 0 for s ∈ [−d, 0], we can conclude that e ξ(t) ≽ 0 for t ∈ [0, ∞).
It is deduced from Lemma 3.2.2 that if positive system (3.1) is GES then, for any ξ ∈ R n n \ {0}, there exists at least a component of the vector A 0 + Σ m
A k Σξ that is strictly negative Similar to [10, 45], and by utilizing the KKM Lemma, we obtain the following necessary stability condition.
Proposition 3.2.1 Positive system (3.1) with w = 0 is GES only if there exists a vector η ∈ R n , η ≻ 0, such that
Proof We define a mapping Γ : R n → R n , x ›→ Γ(x), as Γ(x) .A 0 + Σ k
By Lemma 3.2.2, for any x ∈ R \ {0}, there exists an i ∈ 1,n such that Γ i (x) < 0, where Γ(x) = (Γ i (x)) We now apply KKM Lemma to show the existence of a vector η ≻ 0 satisfying (3.5) To this end, we define the set
Ω i = ∆ n By the continuity of Γ, Ω1 , , Ω n are open subsets (in the induced topology) of ∆ n Let F ∈ F(∆ n ) be an arbitrary face of ∆ n and x ∈ F Then, x j = 0 m m k= 1 m m
1 for all j ∈/ v F On the other hand, Γ j (x) ≥ 0 Thus, x ∈/ Ω j , ∀j ∈/ v F Consequently, x ∈ 0 i∈v F
≠ ∅ It is obvious that η ≻ 0 for any η ∈ 1 n Ω i and Γ(x) ≺ 0 This completes the proof of
We are now in a position to derive necessary and sufficient stability conditions for positive system (3.1) as stated in the following theorem.
Theorem 3.2.1 Positive system (3.1) with w = 0 is GES if and only if the LP con- dition (3.5) is feasible for a vector η ≻ 0.
Proof Necessity follows from Proposition 3.2.1 We now prove Sufficiency Indeed, according to (3.5), the inequality
.A 0 + αI n + Σ k=1 e α h k A k Ση ≺ 0 (3.6) holds for 0 < α < α∗ with a sufficiently small α∗ > 0 Let x(t, φ) be a solution of
(3.1) Since |φ(s)| ≼ ∥φ∥ C 1 n for s ∈ [−d, 0], as revealed by (3.6), we consider the following scaled function ψ(t) = ∥φ∥ C e n
−αt η, t ∈ [−d, ∞), (3.7) η+ where η+ = min1≤i≤n η i It is easy to verify that m ψ˙(t) ≽ A 0ψ(t) + A k ψ(t − h k ), t ≥ 0 (3.8) k=1
Since |φ(s)| ≼ ψ(s) for s ∈ [−d, 0], by Lemma 3.2.1, we have |x(t, φ)| ≼ ψ(t), t ≥ 0, and therefore,
This shows that positive system (3.1) is GES The proof is completed.
A k is a Metzler matrix, condition (3.5) holds if and only if the matrix A s is Hurwitz (see, [41, Proposition 2]) By this, condition
(3.5) is equivalent to the following one
Next, we give the following definition. i= 1 n Σ k= 1 m
Definition 3.2.1 System (3.1) is said to be L 1-stable if the following two requirements hold simultaneously.
(ii) For any nonzero disturbance w ∈ L 1(R+ , R ⊤ n w ), it holds that x(t, φ) ∈ L 1(R+ , R n ), that is, ∫ ∞
In the following, we will show that positive system (3.1) is L 1-stable if and only if condition (3.9) holds.
Proposition 3.2.2 Positive system (3.1) is L 1 -stable if and only if condition (3.9) holds.
Proof Necessity is derived by similar arguments of Proposition 3.2.1 To prove Suf- ficiency, let ηˆ ≻ 0 be a vector satisfying condition (3.9) Consider the following co- positive Lyapunov function
Similar to (3.6), it follows from (3.9) that there exists an α > 0 such that
For any t f > 0, by integrating both sides of (3.11), we obtain t f ( ( t f
This section presents a characterization of the L1-gain for positive systems Building on this characterization, we derive necessary and sufficient LP-based conditions to achieve a specified level of L1-gain performance.
Assuming that condition (3.2) is met, we can conclude that system (3.1) is positive Now, let’s examine system (3.1) with the disturbance w replaced by its nonnegative counterpart |w|, while also considering one of the two equivalent conditions.
(3.5) and (3.9) holds By Proposition 3.2.2, system (3.1) is L 1-stable We denote by
∞ the Laplace transform of x Then, under zero initial condition, we have pxˆ(p) .A 0 + Σ k=1 e −h k p A k Σxˆ(p) + B w |^ w|(p), where |w|(p) is the Laplace transform of |w| This identity gives the following repre- sentation xˆ(p) = ΣpI n − A 0 Σ k
In addition, it follows from (3.1b) that zˆ(p) .C 0 + Σ k=1 e − pτ k C k Σxˆ(p) + D w |^ w|(p) by incorporating (3.13), where A p = A 0+Σ m e − hk p A k and Z p = C 0+Σ m e −pτ k C k Let p = 0 and note also that
C s = C 0 + Σ m C k By (3.15), we can conclude that
On the other hand, for any nonnegative disturbance w(t) ≽ 0, similar to (3.13) and (3.14), we have
Therefore, it follows from (3.17) and (3.18) that sup (
We now define an input w(t) e −t w ∗, where w ∗ ≽ 0 is a constant vector which will be determin ed later
By selecting w ∗ = e j , jth vector of the standard basis of
In combining with (3.19), the above derivation shows that
From (3.16) and (3.20) we obtain the following result.
Theorem 3.3.1 Assume that system (3.1) is positive and stable Then, the value of
L 1 -gain of (3.1) is obtained as
Remark 3.3.1 The derivation process of Theorem 3.3.1 provides a direct approach to the problem of obtaining a characterization of L 1-gain of time-delay system (3.1) First, Proposition 3.2.2 shows that, under equivalent conditions (3.5) and (3.9), system (3.1) is L 1-stable, and hence, the Laplace transformations x(p) and z(p) of x(t) and z(t) in (3.13), (3.14) are well-defined for p ∈ C with Rep ≥ 0 On the other hand, as a key feature that lays behind the positivity, system (3.1) is stable for any delays h k if and only if the corresponding delay-free system (i.e h k = 0) is stable Thus, it follows from (3.5) that det pI n − A 0 + Σ m e − hk p A k Σ ̸= 0 for p in a neighborhood of p = 0, and the matrix H(p) = pI n − A 0 + Σ m e − hk p A k Σ is invertible Based on this fact, a characterization of L 1-induced norm of the input-output operator Σ is formulated as the result presented in Theorem 3.3.1.
Remark 3.3.2 It is also noted that the result of Theorem 3.3.1 is still valid for the case of unknown but constant delays However, it should be mentioned here that this result cannot be simply extended to the case of time-varying delays due to the nature of the use of Laplace transform method This issue still poses much challenging and indeed is left open A hidden key behind is that exogenous disturbances belonging to
L 1 class are not confined with constant threshold vectors and thus the monotonicity of compared systems cannot established.
On the basis of the result of Theorem 3.3.1, a necessary and sufficient condition for L 1-gain performance with prescribed level is given as follows.
Theorem 3.3.2 For a given γ > 0, positive system (3.1) is stable and has L 1 -gain k= 1 k=
1 1 s 1 performance index γ if and only if there exists a vector à ∈ R n , à ≻ 0, satisfying the following condition Σ
Proof Necessity: Let ηˆ ≻ 0 be a vector satisfying condition (3.9) For an ϵ > 0, we define à = ϵηˆ − (C s A −1 ) ⊤ 1 n ≻ 0 s z t h e n
On the other hand, for a given γ > 0, by Theorem 3.3.1, ∥Σ∥ (L 1 ,L 1 ) < γ if and only if
Therefore, ν = γ1 nw − (D w − C s A −1 B w ) ⊤ 1 nz ≻ 0 By this
(3.24) w w z w w for sufficiently small ϵ Combining (3.23) and (3.24) we obtain condition (3.22).
Sufficiency: Since C ⊤ 1 nz ≻ 0, condition (3.22) implies (3.9) and thus system (3.1) is GES In addition, by defining the vector λ = A ⊤ à + C ⊤ 1 nz ≺ 0, we have s s à = (A −1 ) ⊤ λ − C ⊤ 1 n Σ.
Substituting to (3.22), we then obtain
For any disturbance w with ∥w∥ L 1 = 1, by (3.15) and (3.25), we have
(3.26) due to the fact that λ ⊤ A −1
In this section, we apply the L1-gain performance results from the previous analysis to tackle the L1-gain control problem for the system described by the equation m x˙(t) = A0 x(t) + Ak x(t − hk) + Bu u(t) + Bw w(t), for t ≥ 0 The regulated output vector is defined as m z(t) = C0 x(t) + Ck x(t − τk) + Du u(t) + Dw w(t) Here, u(t) represents the control input, while Bu and Du are specified real matrices This framework allows us to systematically address the control challenges associated with the given system dynamics.
A system is considered positive if, for nonnegative inputs \( u(t) \) and \( w(t) \), as well as a nonnegative initial condition \( \phi(t) \), the state and output trajectories remain nonnegative This aligns with Proposition 3.1.1, indicating that system (3.27) is positive if and only if matrix \( A_0 \) is a Metzler matrix.
, are nonnegative matrices In the remaining of this section, we assume that system
For system (3.27), an SFC will be designed in the form u(t) = −Kx(t), (3.29) where K ∈ R n u ×n is the controller gain By incorporating the controller (3.29), the closed-loop system of (3.27) can be represented as m x˙(t) = A c x(t) + A k x(t − h k ) + B w w(t), (3.30a) k=
The control target is to design an SFC (3.29) that makes the closed-loop system
The system is characterized as positive and stable, adhering to the prescribed L1-gain performance level Building on the findings from Proposition 3.1.1 and Theorem 3.3.2, a comprehensive stabilization criterion is established in the subsequent theorem.
Theorem 3.4.1 For a given scalar γ > 0, there exists an SFC (3.29) such that the closed-loop system (3.30) is positive, GES and has L 1 -gain performance at level γ if and only if the following conditions hold
Proof The proof can follow the results of Proposition 3.1.1 and Theorem 3.3.2, and thus is omitted here.
Condition (3.31c) pertains to a specific category of BMIs concerning the variables à and K It is widely recognized that the feasibility of BMI conditions remains an NP-hard problem To advance towards our primary objective of establishing necessary and sufficient tractable conditions for the existence of the desired SFC (3.29), we will decompose condition (3.31c).
Condition (3.32a) remains intractable as it presents a BMI concerning variables à and K; however, it provides insights into the control problem when applying optimization techniques from Chapter 2 This section will later derive the feasibility of conditions (3.31a)-(3.31c) by maximizing the admissible matrix variable K, resulting in tractable LP-based conditions Initially, a sufficient condition for the existence of K will be presented.
Proposition 3.4.1 Assume that, for a given γ > 0, there exist a matrix Z ∈ R n×n u , k= 1 k=
1 n z C s n z D u Σ Σ Σ Σ vectors à, χ ∈ R n , à ≻ 0, and a scalar σ ≥ 0 satisfying the following LP-based condition
Then, the closed-loop system (3.30) is positive, GES and has L 1 -gain performance at level γ under SFC (3.29), where the controller gain K is given by
Proof With the controller gain (3.34), we have
Thus, condition (3.32a) is satisfied On the other hand, a matrix M ∈ R n×n is Metzler if and only if there exists a scalar σ ≥ 0 such that M + σI n ≽ 0 (see, [41]) Therefore, conditions (3.31a), (3.31b) are also fulfilled.
Proposition 3.4.1 outlines a sufficient synthesis condition for achieving a desired L1-gain controller for system (3.27) However, the existence of a controller gain K that satisfies condition (3.31) remains an open question We define the L1-gain control problem for system (3.27) via state-feedback controllers as solvable if there exists a state-feedback controller (SFC) in the form of (3.29) that ensures the closed-loop system (3.30) is positive, globally exponentially stable (GES), and meets the prescribed L1-gain performance level This section will explore this issue further, focusing on maximizing the admissible gain K as indicated in conditions (3.31a) and (3.31b), starting with the case of single-input systems.
B u = (b i ) ∈ R n , D u = (d l ) ∈ R n z are column-vectors and K = (k j ) is a row-vector.
Let A 0 = (a 0 ) and C 0 = (c 0 ), then conditions (3.31a)-(3.31b) can be written as ij lj 0 ij
Assume that B u is not unity (i.e B u has at least two nonzero entries) Let
⊤ Σ j i̸=j,l b i d l and δ∗ = [δ ∗ ∗ ããã δ ∗ ] We have the following result.
Proposition 3.4.2 For a given γ > 0, the L 1 -gain control problem of positive system
(3.27) is solvable if and only if there exists a vector 0 ≺ à ∈ R n that satisfies the following LP-based condition
The condition (3.35) is satisfied if and only if K is less than or equal to δ∗ Furthermore, because the term à ⊤ 1 ⊤ ΣB u Σ K increases with K, a controller gain K that meets the requirements of conditions (3.31a)-(3.31b) and (3.32a) exists if and only if à⊤ 1 ⊤ ΣAsΣ is less than à⊤ 1 ⊤ ΣBu Σδ.
This, together with (3.32b), is recast as the condition (3.36) The optimal controller gain is thus K = δ∗.
We now consider the case of multiple-input systems Without loss of generality, assume that B u has full column rank, that is, rank(B u ) = n u Then, it is clear that
B u also has full column rank We decompose
An optimal controller gain K is obtained as K δ∗ u s u
2 u u and K = k 1 k 2 ããã k n As a special case, if all rows of B u and D u are unitary, we define b = min ij ν̸=j
PEAK-TO-PEAK GAIN CONTROL OF DISCRETE-TIME POSITIVE LIN-
PEAK-TO-PEAK GAIN CONTROL OF DISCRETE-TIME POSITIVE LINEAR SYSTEMS WITH DIVERSE INTERVAL DELAYS
This chapter investigates peak-to-peak gain control through static output feedback for discrete-time positive linear systems with varying interval delays It focuses on time-varying delays in state and regulated output vectors, which are heterogeneous and bounded within specific ranges, potentially including nonzero lower bounds Using innovative comparison techniques involving the steady states of upper and lower scaled systems and peak values of external disturbances, the chapter establishes a characterization of the ℓ∞-induced norm of the input-output operator, also referred to as ℓ∞-gain This characterization is then applied to derive necessary and sufficient conditions for achieving ℓ∞-induced performance at a specified level Furthermore, leveraging performance analysis results and a vertex optimization approach, the chapter presents a comprehensive solution to the synthesis problem of a static output feedback controller that minimizes the worst-case amplification of disturbances to regulated outputs while adhering to ℓ∞-gain constraints The main content is based on the findings presented in paper [P3] from the list of publications.
Consider the following discrete-time system with multiple time-varying delays
N z(k) = C 0 x(k)+ C j x(k − h j (k)) + D w w(k), (4.1b) j=1 x(k) = φ(k), k ∈ Z[−d, 0], (4.1c) Σ Σ where x(k) ∈ R n is the state vector, z(k) ∈ R n z and w(k) ∈ R n w are the regulated output and exogenous disturbance input vectors, respectively A 0, A j , B w , C 0, C j and
Given matrices \( D_w \) and \( j \) with dimensions \( 1 \) to \( N \), the unknown time-varying delays \( d_j(k) \) and \( h_j(k) \) are constrained within known bounds, specifically \( d_j \leq d_j(k) \leq d_j \) and \( h_j \leq h_j(k) \leq h_j \) Here, \( d_j \), \( d_j \), \( h_j \), and \( h_j \) are positive integers that define the lower and upper limits of these delays, with \( d \) representing the maximum delay across all indices The initial state sequence is denoted as \( \phi(k) \), and to clarify the initial condition, the solution corresponding to this initial sequence is represented as \( x(k, \phi) \).
Definition 4.1.1 System (4.1) is said to be positive if its state and output trajectories x(k), z(k) are always nonnegative (i.e x(k) ∈ R n , z(k) ∈ R n z , k ∈ N0) for any
Similar to [18, 40], we have the following positivity characterization.
Proposition 4.1.1 System (4.1) is positive if and only if the following condition holds
Definition 4.1.2 System (4.1) with w = 0 is said to be globally exponentially stable (GES) if there exist constants α ∈ (0, 1), β > 0 such that any solution x(k, φ) of (4.1) satisfies the following inequality
In the following section, we will derive the necessary and sufficient conditions for the positive system (4.1) with w = 0 to achieve Global Exponential Stability (GES) Additionally, we will demonstrate that under these established conditions, system (4.1) exhibits ℓ∞-stability, indicating its robustness over time.
∥x(k)∥∞ < ∞ for any disturbance input w ∈ ℓ∞ Assume that system (4.1) is stable (GES) We define the input-output operator of system (4.1) as Ψ : ℓ∞(R n w ) −→ ℓ∞(R n z ), w ›→ z, k= 0 and ℓ∞-gain of system (4.1) under zero initial condition as
Definition 4.1.3 For a given γ > 0, system (4.1) is said to have ℓ∞-gain performance of level γ if ∥Ψ∥ (ℓ ∞ ,ℓ ∞ ) < γ.
Our main objective in chapter is to formulate the exact value of ℓ∞-induced norm
In this study, we analyze the ℓ∞-gain performance index of system (4.1) and provide a characterization of its properties We then apply our findings to ℓ∞-gain control, establishing the necessary and sufficient conditions for the existence of a static output-feedback controller This controller ensures that the closed-loop system is positive, stable, and meets the specified ℓ∞-gain performance index.
In this section, we briefly formulate conditions by which positive system (4.1) is
In system (4.1) with w = 0, and under the conditions AΣ ≽ 0 and A O ≽ 0, the system is deemed positive It can be verified that the relationship |x(k, φ)| ≼ x(k, |φ|) holds for all k ∈ N0, leading to the conclusion that ∥x(k, φ)∥∞ ≤ ∥x(k, |φ|)∥∞ This indicates that the inequality (4.3) is satisfied for all initial sequences φ if and only if it is true for any non-negative φ(k) Additionally, it is noteworthy that if there exists a nonzero vector v ∈ R, certain implications may arise.
=1 A j Σv ≽ v, (4.5) then x(k, v) ≽ v for all k ∈ N0, where x(k, v) is the solution of (4.1) with constant initial sequence φ(k) = v This observation reveals that if positive system (4.1) is GES then for any v ∈ R n n \ {0}, the vector (A 0 + Σ N
A j )v must include at least one strictly negative component By applying a method akin to the KKM Lemma, we can derive a necessary stability condition.
Proposition 4.2.1 Positive system (4.1) (with w = 0) is GES if and only if the following LP condition is feasible for a vector 0 ≺ η ∈
Since the matrices A 0 and A j , j ∈ 1,N, are nonnegative, it is well known that condition (4.6) is feasible if and only if the spectral radius ρ(A 0 + Σ N
A j ) is strictly less than one We now summarize some stability criteria in the following theorem.
Theorem 4.2.1 For positive system (4.1) with w = 0, the following statements are equivalent.
(ii) There exists a vector η ∈ R n n , η ≻ 0, such that η⊤.
(iii) There exists a vector v ∈ R n , v ≻ 0, such that
(iv) The matrix A 0 + Σ N A j is Schur stable, that is, the spectral radius of the matrix
Remark 4.2.1 The stability conditions derived in (4.7)-(4.9) are delay-independent, that is, positive system (4.1) is GES for any delays d j (t) satisfying (4.2) if and only if the corresponding delay-free system is GES However, the magnitudes of delays have an impact on the exponential convergence rate of solutions of (4.1) More precisely, from (4.8) we get the following exponential estimate
∥x(k, φ)∥∞ ≤ C v ∥φ∥∞α k , where a maximum possible decay rate can be obtained from the following generic procedure
4.3 Peak-to-peak gain characterization
This section examines system (4.1) under zero initial conditions, represented as φ = 0 We denote the solution of system (4.1) in relation to the input disturbance w as x(k, w) Furthermore, we will assume that system (4.1) is positive throughout this discussion The subsequent technical lemmas will be utilized to demonstrate the ℓ∞ performance of system (4.1) when subjected to ℓ∞ input disturbances.
Lemma 4.3.1 If w 1(k) ≼ w 2(k) then x(k, w 1) ≼ x(k, w 2) for all k ≥ 0.
Proof It is fact that x(k, w 2) − x(k, w 1) is a solution of (4.1) with nonnegative input w 2(k) − w 1(k) ≽ 0 Thus, x(k, w 2) − x(k, w 1) ≽ 0 for all k ≥ 0.
As a consequence of Lemma 4.3.1, for any w, we have
Therefore, |x(k, w)| ≼ x(k, |w|) for all k ∈ N0 Moreover, for a w ∈ ℓ∞(R n w ), we have
Associated with system (4.1)-(4.2), we consider the following upper scaled system
Lemma 4.3.2 Let x + (k, w) be a solution of (4.11) Then, for any positive integer q, x + (k + q, w) ≽ x + (k, w) for all k ≥ 0. Σ Σ
Proof Let x + (k) = x + (k + q, w), then x + (k) is a solution of (4.11a) with initial q q condition φ + (l) = x + (l, w) ≽ 0, l ∈ Z[q − d, q], as system (4.11) is positive We now define e(k) = x + (k) − x + (k, w) It follows from (4.11a) that
By the positivity of system (4.12), e(k) ≽ 0, which ensures that x + (k+q, w) ≽ x + (k, w) for all k ≥ 0.
Based on Lemma 4.3.2, the following key lemmas are obtained.
Lemma 4.3.3 For a given w ∈ ℓ∞(R n w ), w(k) ≽ 0, let x(k), x + (k) and z(k), z + (k) be the state and output trajectories of systems (4.1) and (4.11) with zero initial condition, respectively It holds that x(k) ≼ x + (k) and z(k) ≼ z + (k) for all k ≥ 0.
Proof Let e + (k) = x + (k) − x(k), k ≥ 0 It follows from (4.1a) and (4.11a) that
In addition, w − w(k) ≽ 0 Therefore, e + (k) ≽ 0, k ≥ 0, since system (4.13) is positive.
On the other hand, from (4.1b) and (4.11b), we also have
Similar to (4.13), x + (k − h j ) − x + (k − d j (k)) ≽ 0 and w − w(k) ≽ 0 By this, it can be deduced from (4.14) that z + (k) − z(k) ≽ 0 for all k ≥ 0 The proof is completed.
To establish a lower bound for x(k) and z(k), similar to (4.11), we consider the following scaled system where w ∈
N z − (k) = C 0 x − (k)+ C j x − (k − h j )+ D w w , (4.15b) j=1 x − (k) = 0, k ∈ Z[−d, 0], (4.15c) is a constant vector such that w(k) ≽ w.
Lemma 4.3.4 For a given w ∈ ℓ∞(R n w ) and a vector w ∈ R n w such that w(k) ≽ w, let x(k), x − (k) and z(k), z − (k) be the state and output trajectories of systems (4.1) and (4.15) with zero initial condition, respectively It holds that
Proof The proof is similar to Lemma 4.3.3 and is omitted.
Next, we assume one of the stability conditions derived in Theorem 4.2.1 holds Then, it can be deduced from (4.9) that the matrix I n − (A 0 + A d ) is invertible, where
An equilibrium point of system (4.11a), we mean a vector x ∗ ∈ R n which satisfies the algebraic equation
It can be verified that the unique equilibrium point of (4.11a) is given as x ∗ = (I − A 0 − A d ) −1 B w w.
Lemma 4.3.5 For any w ∈ ℓ∞(R n w ), the solution x(k, w) of (4.1) with zero initial condition satisfies
In particular, x(k, w) also belongs to ℓ∞(R n ).
Proof Since |w(k)| ≼ w, by Lemmas 4.3.1 and 4.3.3, we have
|x(k, w)| ≼ x(k, |w|) ≼ x + (k, w), k ≥ 0 (4.19) Let e + (k) = x ∗ − x + (k, w), k ≥ 0, and e + (k) = x ∗, k ∈ Z[−d, 0] It follows from
System (4.20) is positive and e + (k) = x ∗ ≽ 0 for k ∈ Z[−d, 0] Thus, e + (k) ≽ 0 for all
The above estimation also yields
Remark 4.3.1 By similar lines used in the proof of Theorem 4.2.1, it can be shown under equivalent conditions (4.7)-(4.9) that system (4.20) is GES Thus, the state trajectory x + (k, w) of (4.11a) increases exponentially to x ∗ More precisely, there exists an α ∈ (0, 1) such that
By this, we obtain the following interval estimate 1 − α k Σ x ∗ ≼ x + (k, w) ≼ x ∗ , k ≥ 0.
We now establish the following main result. j= 1
Theorem 4.3.1 Assume that system (4.1) is positive and GES The exact value of
ℓ∞ -gain of system (4.1) is obtained as
C j Proof For any w ∈ ℓ∞(R n w ), by Lemma 4.3.3, we have
To validate, we notice from the proof of Lemma 4.3.5 that x + (k) ≼ x ∗ , x + (k − h j ) ≼ x ∗ , k ≥ 0, where x ∗ is defined in (4.17) This, together with (4.11b), gives z + (k) ≼ (C 0 + C d )x ∗ + D w w
It follows from (4.22) that |z(k)| ≼ χ∥w∥ℓ ∞ by which we readily obtain
Taking supremum in k, from (4.23) we obtain ∥Ψ∥ℓ ∞ ≤ ∥Λ∥∞ as desired.
For a given w 0 ∈ ℓ∞(R n w ) with ∥w 0∥ℓ ∞ > 0, we specify input disturbances w(k) and w in (4.1) and (4.10) as w(k) = w = ∥w 0∥ℓ ∞ 1 nw , k ≥ 0.
∞ where z − (k) is determined by (4.15b) In addition, with the input w , the unique equilibrium point of (4.15a) is given by x ∗ = (I n − A 0 − A d ) −1 B w w.
Similar to the proof of Lemma 4.3.5 and Remark 4.3.1, we can conclude under condition (4.9) that there exists a scalar αˆ ∈ (0, 1) such that
Note also from (4.15c) and (4.24) that x − (k − d j ) ≽ 1 − αˆ max(k−dj ,0) Σ x ∗ , k ≥ 0 (4.25) Combining (4.15b) and (4.24) we then obtain
Σ where θˆ k = 1 − αˆ k and θˆ j = 1 − αˆ max(k− dj ,0) Observe that the right hand side of (4.26) is increasing in k Therefore,
It follows from (4.27) and the fact ∥z∥ℓ ∞ ≥ sup k≥0 ∥z − (k)∥∞ that
The results in Claims 1 and 2 show that ∥Ψ∥ (ℓ ∞ ,ℓ ∞ ) = ∥Λ∥∞, by which we obtain (4.21) The proof is completed.
Remark 4.3.2 Theorem 4.3.1 provides a characterization of ℓ∞-gain of positive sys- tems with diverse interval delays in the state and output vectors as presented in (4.1).
This paper introduces a novel and systematic approach for estimating the operator norm of input-output mapping, diverging from existing methods that utilize lifting techniques and solution representations.
By leveraging the monotonicity of state trajectories in systems (4.11) and (4.15), we establish upper and lower bounds for the state and output vectors of system (4.1) Additionally, we derive the ℓ∞-gain characterization of system (4.1) based on the exponential convergence properties to equilibria of (4.11) and (4.15) The method outlined in Theorem 4.3.1 is applicable to general positive systems, with or without delays Importantly, traditional methods based on solution representation are ineffective for various time-varying delay systems due to the complexities of the infinite-dimensional solution space, highlighting the advantages of the approach presented in Theorem 4.3.1.
Theorem 4.3.1 serves as a crucial and effective resource for tackling controller design challenges related to ℓ∞-gain performance Specifically, for a scalar γ > 0, known as the attenuation level, our goal is to establish clear necessary and sufficient conditions that ensure the ℓ∞-gain meets the criterion ∥Ψ∥ (ℓ ∞ ,ℓ ∞ ) < γ.
Theorem 4.3.2 For a given γ > 0, positive system (4.1) is stable and has ℓ∞ -gain performance at level γ if and only if there exists a vector η ∈ R n , η ≻ 0, that satisfies the following LP-based conditions
Proof (Necessity) Let v ≻ 0 be a vector satisfying condition (4.8) For a given ϵ > 0, we define η = ϵv + (I n − A 0 − A d ) −1 B w 1 n ≻ 0, then
On the other hand, by Theorem 4.3.1, ∥Ψ∥ (ℓ ∞ ,ℓ ∞ ) < γ if and only if χ = (C 0 + C d ) (I n − A 0 − A d ) −1 B w + D w
Therefore, ζ = γ1 nw − χ ≻ 0 and we have
C j η + D w 1 nw − γ1 nz ≺ 0 (4.28b) for sufficiently small ϵ which, together (4.29), yields (4.28).
(Sufficiency) Since B w 1 nw ≽ 0, condition (4.28a) implies (4.8) and thus system (4.1) is GES In addition, let η˜ = (A 0 + A d − I n )η + B w 1 nw ≺ 0, we have η = (A 0 + A d − I n ) −1 (η˜ − B w 1 n )
It follows from (4.28b) that γ1 nz ≻ (C 0 + C d )η + D w 1 nw
(A 0 + A d − I n ) −1 η˜ ≻ 0, from (4.23) and (4.30), we obtain ∥z∥ℓ ∞ ≤ ∥χ∥∞ < γ The proof is completed.
The following alternative performance conditions can be obtain by similar argu- ments used in the proof of Theorem 4.3.2.
Theorem 4.3.3 For a given γ > 0, positive system (4.1) is stable and has ℓ∞ -gain performance at level γ if and only if the following LP conditions are feasible for a vector ν ∈ R n , ν ≻ 0 ν ⊤ A 0 +Σ j
4.4 Static output-feedback peak-to-peak gain control
In this section, we consider the problem of ℓ∞-gain static output-feedback control of the following system
. n z where y(k) ∈ R ⊤ n o is the measurement output, u(k) ∈ R n u is the control input, B, D,
E, F are real matrices with appropriate dimensions and other vectors and matrices are specified as system (4.1) Assume that system (4.32) is positive (i.e all system matrices are nonnegative) We now focus on the existence of an SOFC of the form u(k) = −Ky(k), k ≥ 0, (4.33) that makes the resulting closed-loop system positive, stable and has prescribed ℓ∞-gain performance, where K ∈ R n u ×n y is the controller gain which will be determined By incorporating (4.33), the closed-loop system of (4.32) is presented as x(k + 1) = A c x(k)+ Σ
For a given γ > 0, by Proposition 4.1.1 and Theorems 4.3.2, 4.3.3, system (4.34) is positive, stable and has ℓ∞-gain performance at level γ if and only if
C 0 D w Σ − ΣB Σ KH ≽ 0 (4.35) and one of the following two LP-based conditions is feasible
The primary objective is to determine the existence of a controller gain K that meets the criteria outlined in equation (4.35) and satisfies one of the conditions specified in (4.36) or (4.37) To address this challenge, we apply the principles of vertex optimization to system (4.32).
A 0 + A d − I n B w C 0 + C d D w following we illustrate the ideas by considering the case of single-input systems.
Matrix transformations are frequently utilized in the literature to derive the desired controller gain K However, this approach only provides sufficient conditions, which can lead to conservativeness and does not fully address the problem due to additional constraints.
1 n z Σ = Z ∈ R c n o , (4.38) then, subject to (4.38), conditions (4.35) and (4.37) hold if and only if