bài toán điều khiển đối với một số lớp hệ tuyến tính dương TT TIENG ANH

MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION ——————–o0o——————— MAI THI HONG SOME CONTROL PROBLEMS FOR POSITIVE LINEAR SYSTEMS Speciality: Differential and Integral Equations Code: 9460103 SUMMARY OF DOCTORAL DISSERTATION IN MATHEMATICS HA NOI-2021 The dissertation was written on the basis of the author’s research works carried at Hanoi National University of Education Supervisor: Assoc.Prof LE VAN HIEN Hanoi National University of Education Referee 1: Professor Cung The Anh Hanoi National University of Education Referee 2: Associate Professor Do Duc Thuan Hanoi University of Science and Technology Referee 3: Associate Professor Nguyen Minh Tuan VNU University of Education This dissertation is presented to the examining committee at Hanoi National University of Education, 136 Xuan Thuy Road, Hanoi, Vietnam At the time of , 2021 Full-text of the dissertation is publicly available and can be accessed at: - The National Library of Vietnam - The Library of Hanoi National University of Education INTRODUCTION A Background Positive systems can be used to describe dynamics of various practical models of disciplines from biology, ecology and epidemiology, chemistry, pharmacokinetics to air traffic flow networks, control engineering, telecommunication and chemical-physical processes Theoretically, positive systems possess many elegant properties that have yet no counterpart in general linear systems For instance, by the robustness and monotonicity induced from the positivity, positive systems are highly evolved in designing interval observers, which are relevant in the context of observation of systems for which only a poor model is available, utilized in the problem of state estimations or stability analysis of nonlinear time-delay systems Due to widespread applications and special characteristics, the systems and control theory of linear positive systems has received ever-increasing interest in the past decade In particular, as most relevant issues in the field of analysis and synthesis of positive systems, stability, disturbance attenuation and robustness are essential problems that should be taken into account thoroughly Time-delay is frequently encountered in engineering systems and industrial processes as, for example, transmission lines or telecommunication networks The transportation of resources in logistic networks is another example of positive systems with delays as the resource amount is not only positive but also subject to delays due to traffic jams and other latency aspects The presence of time-delays usually leads to unpredictable system behaviors, degradation of system performance even jeopardize system stability Thus, the study of time-delay systems is essential in the field of control engineering, which has attracted significant research attention in the past two decades A very rich literature with a large number of important results concerning the systems and control theory of time-delay systems has been reported Although the theory of general linear systems is also applicable to positive systems, new challenges often arise in the analysis and synthesis of positive systems since traditional methods (for general linear systems) are often handicapped when dealing with such systems as they are not defined on linear spaces but convex polyhedral cones As a consequence, many properties cannot be preserved under similarity transformation and numerous well-established results for linear systems with or without delays cannot be readily applied to positive systems For instance, it is well known in general linear system theory that if a system is controllable, the poles of the system can be arbitrarily assigned, whereas for positive linear systems, this feature may not be true owing to the positivity constraints on the system matrices Generally speaking, research on positive systems focuses on utilizing the monotonicity induced from positivity to simplify the behavioral analysis of dynamic systems and thus further facilitates the design of such systems to achieve desirable control specifications By this, the system theory for positive linear systems with delays is still one of significant trends in the area of theoretic-control Exogenous disturbances are omnipresent in engineering systems due to many technical issues such as the inaccuracy of data processing, linear approximations or measurement errors Typically, the performance of a dynamical system is characterized by the attenuation ability against the effect of disturbance inputs and is often quantified by certain norms associated with the relation between system responses and disturbance inputs Unlike general systems, a remarkable feature of positive systems is that it allows one to use linear supply rate in the dissipative analysis and naturally results in a number of input-output system properties in terms of linear integral/summation constraints instead of quadratic integral/summation constraints Thus, for positive systems, it is more reasonable to adopt L1 - gain or L∞ -gain as a performance measures rather than the traditional L2 -gain in the analysis and synthesis of positive systems Apart from practical considerations, by the use of L1 - or L∞ -induced gains, many elegant linear programming (LP) characterizations for a number of performance specifications can be formulated To this aspect, while positivity constraints bring more difficulties to the design of controllers, filters or observers, it can help simplify the design procedure in some scenarios It is noted that LP-based approach, an effective convex optimization tool with low computational complexity, is particularly suitable for controller synthesis of positive systems, ranging from stabilization to robust control with optimal performance indices For the design problem, controllers are typically synthesized based on prior analysis results In other words, certain performance-based criteria are first obtained under which the controlled system has desired properties Then, controllers will be designed to make the corresponding closed-loop systems satisfy proposed performance criteria Solutions to analysis and synthesis problems are often involved with the feasibility of matrix equations and inequalities To achieve control specifications, a predominant approach is the use of feedback compensation There are many control strategies such as state-feedback, output-feedback in static and dynamic control schemes, robust control or adaptive control Among those strategies, state-feedback control is widely employed because states of a system can describe dynamic behavior completely However, in many cases, all states of a system cannot be measured, stored and accessed instantly at all time Due to the inaccessibility of system states, an alternative and more realistic method is static output-feedback control In addition, for positive systems, the positivity of closed dynamics is often required which makes design problem of positive systems more complicated and challenging than general systems Although a number results concerning systems and control theory for positive systems have been published, there is still much room involved with open and long standing problems in this area that needs to be further studied and developed For example, the feasibility of design conditions of state-feedback controllers or static output-feedback controllers has not been resolved Some optimal control problems under L1 /ℓ1 - and L∞ /ℓ∞ -gain schemes or guaranteed cost control problem with linear cost functions have not been fully studied This motivates us for the study presented in this thesis B Literature review B1 Static output-feedback control of positive linear systems In practice, system state variables are usually not fully accessible to the control channel and only partial state vector can be measured and used as feedback signal to configure the plant This reveals that the static output-feedback control problem is an essential issue in the systems and control theory In addition, compared with the state-feedback stabilization problem, where full-state vector should be accessible, the static output-feedback stabilization problem cannot be formulated and solved exactly as an LMI problem or via pole placement More precisely, without the positivity constraint on the closed-loop systems, the feasibility of BMIs induced from the static output-feedback is an NP-hard problem The pole placement problem is even harder to handle for positive systems since it is not known how one can choose the desired poles in order to ensure the positivity of the closed-loop system This is one of the main reasons why to this problem LP-based methods prove to be particularly suitable for positive systems However, so far only a few attempts have been successfully established in the literature to provide a complete solution to the static output-feedback control problem of LTI positive systems For instance, the existing design methods are only applicable to multiple-input multiple-output (MIMO) systems with additional restrictions on the controller gain matrix or can only be applied to systems whose output or input matrix has a specific structure The controller gain decomposition method only provides sufficient stabilization conditions, and thus, it cannot help to verify whether a desire static output-feedback controller (SOFC) exists Despite of much effort from researchers has been devoted to the stabilization problem via SOFCs of LTI positive systems, a complete solution to this problem especially for positive time-delay systems is still of significance In particular, the feasibility of existence conditions of static output-feedback stabilization is still a standing problem for LTI positive systems with delays, which will be investigated in this thesis B2 L1 -gain control of positive linear systems with multiple delays The problems of performance analysis and synthesis under L1 - and ℓ1 -gain control schemes have drawn significant research attention in the past few years Typically, in the existing literatures, stability and performance analysis conditions are derived in terms of linear or semidefinite programmings using certain types of co-positive Lyapunov functions The controller synthesis problem under L1 /ℓ1 -gain scheme is often more challenging than the performance analysis one To tackle the design problem, toward necessary and sufficient synthesis conditions, a characterization of exact value of L1 -gain plays a key role For example, for discrete LTI systems, an exact value of ℓ1 -induced norm was first obtained by using the lifting technique and an explicit representation of fundamental matrix Then, the synthesis problem of SFCs was completely solved for the case of single-input single-output (SISO) systems However, for more general cases with MIMO systems, a hard constraint involving a so-called direction matrix is imposed which could not help to derive necessary conditions for the existence of stabilizing controllers even for LTI systems without delay For LTI continuous-time systems, necessary and sufficient stabilization conditions subject to an equality constraint For positive systems with delays, the use of solution representations is no longer suitable for obtaining an L1 -gain characterization In addition, a direct approach in the existing literature that can help to formulate such a characterization for time-delay systems is quite scarce Moreover, to deal with the stabilization problem under L1 -gain scheme, the proposed methods are still not effective for the feasibility verification problem To this gap, a systematic approach and a complete solution to L1 -gain control of positive systems with delays are still left open B3 Peak-to-peak gain control of discrete-time positive linear systems For positive system, the applications of linear co-positive Lyapunov functions often lead to analysis and synthesis results that are based on linear settings This stimulates the use of performance indices L∞ -gain and L1 -gain While the analysis and design problems under L1 - and ℓ1 -gain schemes have been extensively studied, the induced L∞ and ℓ∞ theory has received less attention despite the existing results have demonstrated good attenuation for persistent peakbounded disturbances Typically, L∞ -gain analysis results are established by using certain types of co-positive Lyapunov functions, fundamental solution representation or by utilizing the positivity characteristic, which give a characterization of exact value of L∞ -gain of the system However, the obtained characterization is not tractable for the controller design problem Instead, we have to minimize a prescribed bound of the induced ℓ∞ performance rather than to directly minimize the induced ℓ∞ performance of peak-to-peak controllers Moreover, the proposed methods in existing works for obtaining desired controllers subject to a prescribed bound of L1 or L∞ performance index are normally involved some hard constraints related to matrix transformations Thus, they cannot help to derive necessary conditions for the existence of stabilizing controllers under ℓ∞ performance even for LTI systems without delay No attempt has been successfully established in the literature to provide a complete solution in terms of tractable LP conditions to the static output-feedback control problem of positive systems with heterogeneous interval delays C Research topics This thesis is concerned with some control problems for positive linear systems with delays The research includes the methodology development, establishment of performance analysis and the formulation of necessary and sufficient conditions for the existence of desired controllers C1 Static output-feedback control of positive linear systems with time-varying delay Consider the following control systems with delayed measurement output x(t) ˙ = Ax(t) + Ad x(t − δt ) + Bu(t), t ≥ 0, (1) y(t) = Cx(t) + Cd x(t − δt ), x(t) = φ(t), t ∈ [−δ∗ , 0], where x(t) ∈ Rn , u(t) ∈ Rm and y(t) ∈ Rp are the state vector, control input and measured output vectors, respectively An SOFC is designed in the form u(t) = −Ky(t) = −KCx(t) − KCd x(t − δt ), (2) where K ∈ Rm×p is the controller gain The closed-loop system of (1) is presented as x(t) ˙ = (A − BKC)x(t) + (Ad − BKCd )x(t − δt ), t ≥ (3) The objective is to derive testable necessary and sufficient conditions for the existence of an SOFC in the form of (2) that makes the closed-loop system (3) positive and globally asymptotically stable (GAS) Based on some linear optimization techniques, necessary and sufficient stabilization conditions are obtained through LP conditions C2 L1 -gain control of positive linear systems with multiple delays In Chapter 3, we study the L1 -gain control problem for LTI systems of the form m Ak x(t − hk ) + Bw w(t), x(t) ˙ = A0 x(t) + t ≥ 0, k=1 m (4) Ck x(t − τk ) + Dw w(t), z(t) = C0 x(t) + k=1 x(t) = φ(t), t ∈ [−d, 0] In the first part, some stability properties of positive system (4) are discussed Assume that system (4) is globally exponentially stable (GES) The input-output operator is defined as Σ : L1 (R+ , Rnw ) −→ L1 (R+ , Rnz ), w → z, and L1 -gain of system (4) under zero initial condition is Σ (L1 ,L1 ) = sup w L1 =0 z w L1 L1 = sup w L1 =1 z L1 (5) The objective of Chapter is to • Formulate a characterization of L1 -gain Σ • Derive conditions under which Σ (L1 ,L1 ) (L1 ,L1 ) of system (4) < γ for a given γ > • Establish necessary and sufficient conditions for the existence of an SFC that makes the closed-loop system positive, stable and has prescribed L1 -gain performance index C3 Peak-to-peak gain control of discrete-time positive linear systems with interval delays Consider the following discrete-time system with multiple time-varying delays N Aj x(k − dj (k)) + Bw w(k), x(k + 1) = A0 x(k) + k ≥ 0, j=1 N (6) Cj x(k − hj (k)) + Dw w(k), z(k) = C0 x(k) + j=1 x(k) = φ(k), k ∈ Z[−d, 0] The input-output operator of system (6) is defined as Ψ : ℓ∞ (Rnw ) −→ ℓ∞ (Rnz ), w → z, and ℓ∞ -gain of system (6) under zero initial condition as Ψ (ℓ∞ ,ℓ∞ ) = sup w ℓ∞ =0 z w ℓ∞ ℓ∞ = sup w z ℓ∞ (7) ℓ∞ =1 By novel comparison techniques involving steady states of upper and lower scaled systems with peak values of exogenous disturbances, we establish a characterization for ℓ∞ -induced norm of the input-output operator and derive necessary and sufficient conditions subject to ℓ∞ -induced performance with prescribed level On the basis of performance analysis results, and based on a vertex optimization technique, a complete solution to the synthesis problem of an SOFC that minimizes the worst case amplification from disturbances to regulated outputs subject to peak-to-peak gain is addressed D Main contributions The stabilization problem via static output-feedback control is studied for LTI positive systems with a time-varying delay in the state and output vectors A novel approach based on optimization procedures is proposed to derive necessary and sufficient LP-based conditions for the existence of desired controllers A characterization of L1 -induced performance is first established for a class of positive linear systems with multiple delays The performance characterization is then utilized to derive necessary and sufficient conditions for the existence of SFCs subject to L1 -gain of the closed-loop systems The problem of static output-feedback optimal peak-to-peak gain control is addressed for discrete-time positive linear systems with heterogeneous interval delays A new characterization of ℓ∞ -gain is established and a complete solution to the synthesis problem of an SOFC that minimizes the worst case amplification from disturbances to regulated outputs subject to ℓ∞ -gain is presented Chapter PRELIMINARIES In this chapter, we recall some auxiliary results on matrix analysis and stability theory, which will be useful for the presentation of the results in next chapters 1.1 Nonnegative and Metzler matrices 1.2 Lyapunov stability 1.2.1 Stability concepts 1.2.2 Stability and stabilization of LTI systems 1.3 Positive LTI systems 1.3.1 Stability analysis and controller design 1.3.2 L1 -induced performance 1.4 KKM Lemma Chapter STATIC OUTPUT-FEEDBACK CONTROL OF POSITIVE LINEAR SYSTEMS WITH TIME-VARYING DELAY In this chapter, we study the stabilization problem via static output-feedback control for LTI positive systems with a time-varying delay in the state and output vectors By exploiting the induced monotonicity, necessary and sufficient conditions ensuring exponential stability of the closed-loop system are first quoted Then, based on some vertex optimization procedures, necessary and sufficient conditions for the existence of a desired SOFC are obtained The proposed synthesis conditions are presented in the form of LP conditions, which can be effectively solved by various convex algorithms Main content of this chapter is written based on paper [P1] in the List of publications 2.1 Problem formulation Consider the following LTI system with delay x(t) ˙ = Ax(t) + Ad x(t − δt ) + Bu(t), t ≥ 0, (2.1) y(t) = Cx(t) + Cd x(t − δt ), x(t) = φ(t), t ∈ [−δ∗ , 0], where x(t) ∈ Rn is the state vector, u(t) ∈ Rm and y(t) ∈ Rp are the control input and the measured output vectors, respectively In system (2.1), A, Ad ∈ Rn×n , B ∈ Rn×m , C, Cd ∈ Rp×n are given system matrices, δt represents an unknown time-varying delay which satisfies ≤ δt ≤ δ∗ , where δ∗ is a prescribed constant, and φ(t) is the initial condition Definition 2.1.1 System (2.1) is said to be (internally) positive if for any nonnegative 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) and y(t) for all t ≥ Proposition 2.1.1 System (2.1) is positive if and only if A is a Metzler matrix and Ad , B, C and Cd are nonnegative matrices For system (2.1), an SOFC is designed in the form u(t) = −Ky(t) = −KCx(t) − KCd x(t − δt ), (2.2) where K ∈ Rm×p is the controller gain, which will be determined By incorporating the SOFC (2.2), the closed-loop system of (2.1) is presented as x(t) ˙ = (A − BKC) x(t) + (Ad − BKCd ) x(t − δt ), Ac t ≥ (2.3) Adc Our main objective in this chapter is to derive testable necessary and sufficient conditions for the existence of an SOFC in the form of (2.2) that makes the closed-loop system (2.3) positive and globally asymptotically stable (GAS) 2.2 Stability analysis Consider the following scaled system xˆ˙ (t) = Ac xˆ(t) + Adc xˆ(t − δ∗ ), ˆ xˆ(t) = φ(t), t ∈ [−δ∗ , 0] t ≥ 0, (2.4) Necessary and sufficient stability conditions for system (2.4) are given below Theorem 2.2.1 For positive system (2.4), the following statements are equivalent (a) System (2.4) is GAS (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 xˆ(t) ≤ β φˆ where φˆ ∞ ˆ = sup−δ∗ ≤t≤0 φ(t) −αt , ∞e t ≥ 0, ∞ (c) There exists a vector ν ∈ Rn , ν ≻ 0, such that ν ⊤ (Ac + Adc ) ≺ (d) There exists a vector η ∈ Rn , η ≻ 0, satisfying the following LP condition (Ac + Adc )η ≺ (2.5) Remark 2.2.1 Let xη (t) be the solution of positive system (2.3) with φ ≡ η, where η ≻ is a vector satisfying condition (2.5), and ψ(t) = xˆη (t) − xη (t) Since xˆη (t − δt ) xˆη (t − δ∗ ), we have ˙ = Ac ψ(t) + Adc (ˆ ψ(t) xη (t − δ∗ ) − xη (t − δt )) Ac ψ(t) + Adc ψ(t − δt ), t ≥ By the positivity of system (2.3), and from (2.6), we have ψ(t) xη (t) xˆη (t), (2.6) which yields t ≥ Proposition 2.2.1 For a time-varying delay δt , the positive system (2.3) is GAS (res GES) if and only if system (2.4) is GAS (res GES), which is equivalent to conditions (c) or (d) given in Theorem 2.2.1 2.3 Controller synthesis In this section, we derive necessary and sufficient checkable conditions for the existence of an SOFC (2.2) that makes the closed-loop system (2.3) positive and stable Hereafter, we assume that system (2.1) is positive Assumption (A): The matrices B and C + Cd have full-column rank and full-row rank, respectively, that is, rank(B) = m, rank (C + Cd ) = 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) Proposition 2.3.1 The stabilization problem of positive system (2.1) is solvable if and only if there exists a matrix K ∈ Rm×p satisfying simultaneously the following conditions Ac = A − BKC is a Metzler matrix, (2.7a) Remark 2.3.5 The result of Theorem 2.3.1 can be extended to multiple-input multiple-output (MIMO) systems, where the matrix B has unity rows, that is, each row of B has at most one non-zero entry Specifically, let us define a∗ θlj = i=j θla∗ aij d∗ : bil = , θlj = bil i∈1,n adij : bil = , j ∈ 1, n, l ∈ 1, m, bil a∗ θ a∗ · · · θ a∗ , θ d∗ = θ d∗ θ d∗ · · · θ d∗ = θl1 l l1 l2 ln l2 ln and the polyhedron ∆l is defined as ∆l = kl ∈ Rp kl⊤ [C Cd ] [θla∗ θld∗ ] (2.16) We have the following result Proposition 2.3.3 Let Assumption (A) hold and assume that the matrix B has unity rows and each column of B is not unity Then, the stabilization problem of positive system (2.1) is solvable if and only if there exist vertices klv of ∆l , l ∈ 1, m, such that the matrix As − BKv Cs is Hurwitz The optimal controller gain matrix is given as v Kv = k1v k2v · · · km ⊤ m Remark 2.3.6 When the matrix B has unity rows, the summation l=1 bil (kl⊤ cj ) is reduced to only one term Thus, we can handle the synthesis process of each vector kl without coupling of B and C This feature simplifies the derivation of Proposition 2.3.3 However, for general MIMO systems, a similar separation is no longer available 2.3.3 Multiple-input single-output systems Let us consider a class of MISO systems as given in (2.1) In this case, the controller gain K = (kl ) ∈ Rm is a column-matrix Let a ˆi = i=j aij : cj = , a ˆdi = cj j∈1,n adij : cdj = , a∗i = min{ˆai , a ˆdi }, i ∈ 1, n, cdj where the operation on empty set will be ignored Then, condition (2.9) holds if and only ˆ defined as if k belongs to the polyhedron ∆ m ˆ = ∆ bil xl ≤ a∗i , i ∈ 1, n m x∈R : (2.17) l=1 ˆ We have the following result Let Sˆ be the set of vertices of ∆ Proposition 2.3.4 Let Assumption (A) hold The stabilization problem of MISO positive systems in the form of (2.1) is solvable if and only if there exists a vertex kv ∈ Sˆ such that the matrix As −Bkv Cs is Hurwitz An optimal controller gain, which assures the fastest convergence rate of the closed-loop system, is an efficient point K = kv∗ of the vector-valued optimization problem m klv (b⊤ l Cs ) max s.t kv ∈ Sˆ and As − Bkv Cs is Hurwitz (2.18) l=1 2.3.4 Multiple-input multiple-output systems In this section, we address the stabilization problem for MIMO systems as given in (2.1) with purely delayed measurements, that is, C = By Assumption (A), Cd has full-row rank 11 (i.e rank(Cd ) = p) We assume that Cd has a particular row echelon form as [C¯d 0p×(n−p)], that is, there exists a nonsingular matrix L such that LCd = [C¯d 0p×(n−p)], where C¯d ∈ Rp×p has full-rank By using the change of variable ¯ C¯ −1 L, K=K d (2.19) ¯ A¯d , where A¯d is the submatrix composed of the first condition (2.9) holds if and only if B K ¯ belongs to the following polyhedron p columns of Ad Thus, each column k¯j of K m ∆j = k¯j ∈ Rm bil k¯lj ≤ adij , i ∈ 1, n l=1 Let S j be the set of vertices of ∆j , j ∈ 1, p Theorem 2.3.2 Consider positive system (2.1) with C = and Cd has its row echelon form [C¯d 0p×(n−p) ] Then, under Assumption (A), the stabilization problem of (2.1) is solvable if and only if there exists a set of vertices k¯jv ∈ S j , j ∈ 1, p, such that the matrix As − ¯ v 0p×(n−p)] is Hurwitz An optimal controller gain is given as K ∗ = K ¯ ∗ C¯ −1 L, where B[K v d ∗ ∗ ∗ ∗ ∗ ¯ ¯ ¯ ¯ ¯ Kv = k1 k2 · · · kp and each kj , j ∈ 1, p, is an efficient point of the problem m v ¯v k¯lj bl s.t k¯jv ∈ S j and As − B[K max 0p×(n−p) ] is Hurwitz l=1 Conclusion of Chapter In this chapter, the problem of static output-feedback control has been investigated for LTI positive systems with time-varying delay in the state and output vectors A novel alternative approach based on vertex optimization techniques has been presented to derive necessary and sufficient conditions for the existence of a desired SOFC The proposed conditions have been formulated as LP problems, which can be effectively solved by various convex algorithms 12 Chapter ON L1 -GAIN CONTROL OF POSITIVE LINEAR SYSTEMS WITH MULTIPLE DELAYS In this chapter, the problem of L1 -gain control is studied for a class of positive linear systems with diverse state and output delays A new alternative approach using Laplace transformation is proposed to establish a characterization of L1 -induced norm of the input-output operator The obtained L1 -induced norm characterization is then utilized to formulate necessary and sufficient conditions subject to L1 -induced performance with prescribed level Based on some vertex optimization techniques, a complete solution to the stabilization problem under L1 -gain control scheme is formulated through tractable LP conditions Main content of this chapter is written based on paper [P2] in the List of publications 3.1 Problem statement Consider the following LTI system with multiple delays m x(t) ˙ = A0 x(t) + Ak x(t − hk ) + Bw w(t), t ≥ 0, (3.1a) Ck x(t − τk ) + Dw w(t), t ≥ 0, (3.1b) k=1 m z(t) = C0 x(t) + k=1 x(t) = φ(t), t ∈ [−d, 0], (3.1c) where x(t) ∈ Rn is the state vector, z(t) ∈ Rnz and w(t) ∈ Rnw are the regulated output and exogenous disturbance input vectors, respectively A0 , Ak , Bw , C0 , Ck and Dw , k ∈ 1, m, are given real matrices with appropriate dimensions hk and τk are known scalars involving time delays and d = max1≤k≤m {hk , τk }, φ ∈ C([−d, 0], Rn ) is the initial condition Assume that system (3.1) is stable (GES) We define the input-output operator Σ : L1 (R+ , Rnw ) −→ L1 (R+ , Rnz ), w → z, and L1 -gain of system (3.1) under zero initial condition as Σ (L1 ,L1 ) = sup w L1 =0 z w L1 L1 = sup w L1 =1 z L1 Definition 3.1.1 For a given scalar γ > 0, system (3.1) is said to have L1 -gain performance at level γ if Σ (L1 ,L1 ) < γ Objectives: Our main objectives in this chapter are as follows (i) Formulate the exact value of L1 -gain Σ (L1 ,L1 ) of system (3.1) (ii) Characterize the L1 -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 prescribed L1 -gain performance index 13 3.2 Stability analysis In this section, we derive necessary and sufficient conditions by which positive system (3.1) with w = is globally exponentially stable (GES) Theorem 3.2.1 Positive system (3.1) with w = is GES if and only if the LP condition m Ak η ≺ A0 + (3.2) k=1 is feasible for a vector η ≻ m Remark 3.2.1 Since As = A0 + k=1 Ak is a Metzler matrix, condition (3.2) holds if and only if the matrix As is Hurwitz By this, condition (3.2) is equivalent to the following one m ⊤ n ∃ˆ η ∈ R , ηˆ ≻ : ηˆ A0 + Ak ≺ (3.3) k=1 Definition 3.2.1 System (3.1) is said to be L1 -stable if the following two requirements hold simultaneously (i) For w = 0, system (3.1) is GAS (ii) For any nonzero disturbance w ∈ L1 (R+ , Rnw ), it holds that x(t, φ) ∈ L1 (R+ , Rn ), that ∞ is, x(t, φ) dt < ∞ Proposition 3.2.1 Positive system (3.1) is L1 -stable if and only if condition (3.3) holds 3.3 L1 -induced performance Assume that system (3.1) is positive Let us now consider system (3.1), where the disturbance w is replaced by |w|, and assume one of the two equivalent conditions (3.2) and (3.3) holds By Proposition 3.2.1, system (3.1) is L1 -stable We denote by ∞ xˆ(p) e−pt x(t)dt Lxu (p) = the Laplace transform of x Then, under zero initial condition, we have m e−hk p Ak xˆ(p) + Bw |w|(p), pˆ x(p) = A0 + k=1 where |w|(p) is the Laplace transform of |w| This identity gives the following representation m −1 −hk p e xˆ(p) = pIn − A0 + Ak Bw |w|(p) (3.4) k=1 In addition, it follows from (3.1b) that zˆ(p) = Zp (pIn − Ap )−1 Bw + Dw |w|(p) m −hk p A k k=1 e by incorporating (3.4), where Ap = A0 + p = and note also that zˆ(0) and Zp = C0 + ∞ = z(t) dt = z 14 L1 , (3.5) m −pτk C k k=1 e Let from (3.5), we obtain z L1 Dw − Cs A−1 s Bw |w|(0) = ≤ Dw − where As = A0 + m k=1 Ak Σ and Cs = C0 + (L1 ,L1 ) = sup w On the other hand, for any w(t) z L1 Cs A−1 s Bw m k=1 Ck By z L1 L1 =1 w L1 , (3.6) (3.6), we can conclude that ≤ Dw − Cs A−1 s Bw (3.7) 0, we have = zˆ(0) = L1 ≤ Dw − Cs A−1 ˆ s Bw w(0) (3.8) and sup w 0, w z L1 =1 sup z L1 =1 w = Σ L1 (L1 ,L1 ) (3.9) Therefore, it follows from (3.8) and (3.9) that sup w 0, w L1 =1 Dw − Cs A−1 ˆ s Bw w(0) By specifying a special w(t), we obtain w = w(0) ˆ L1 Dw − Cs A−1 ˆ s Bw w(0) 1 ≤ Σ = w∗ (L1 ,L1 ) (3.10) = and = Dw − Cs A−1 s Bw In combining with (3.10), the above derivation shows that Dw − Cs A−1 s Bw ≤ Σ (L1 ,L1 ) (3.11) From (3.7) and (3.11) we obtain the following result Theorem 3.3.1 Assume that system (3.1) is positive and stable Then, the value of L1 -gain of (3.1) is obtained as Σ (L1 ,L1 ) = Dw − Cs A−1 (3.12) s Bw , where As = A0 + m k=1 Ak and Cs = C0 + m k=1 Ck Theorem 3.3.2 For a given γ > 0, positive system (3.1) is stable and has L1 -gain performance index γ if and only if there exists a vector µ ∈ Rn , µ ≻ 0, satisfying the following condition ⊤ A⊤ s µ + Cs 1nz ⊤ µ + D ⊤ − γ1 Bw nw w nz ≺ (3.13) 3.4 L1 -gain control In this section, we address the L1 -gain control problem of the system m Ak x(t − hk ) + Bu u(t) + Bw w(t), x(t) ˙ = A0 x(t) + t ≥ 0, (3.14) k=1 m Ck x(t − τk ) + Du u(t) + Dw w(t), z(t) = C0 x(t) + (3.15) k=1 where u(t) ∈ Rnu is the control input and Bu , Du are given real matrices Assume that system (3.14) is positive An SFC will be designed in the form u(t) = −Kx(t), 15 (3.16) where K ∈ Rnu ×n is the controller gain By incorporating the controller (3.16), the closed-loop system of (3.14) can be represented as m x(t) ˙ = Ac x(t) + Ak x(t − hk ) + Bw w(t), (3.17a) Ck x(t − τk ) + Dw w(t), (3.17b) k=1 m z(t) = Cc x(t) + k=1 where Ac = A0 − Bu K and Cc = C0 − Du K Theorem 3.4.1 For a given scalar γ > 0, there exists an SFC (3.16) such that the closedloop system (3.17) is positive, GES and has L1 -gain performance at level γ if and only if the following conditions hold Ac = A0 − Bu K is a Metzler matrix, C c = C − Du K µ 1nz ⊤ As − Bu K Bw − C s − Du K Dw γ1nw for some µ ∈ Rn , µ ≻ 0, where As = A0 + m k=1 Ak (3.18a) 0, (3.18b) ≺0 (3.18c) ⊤ and Cs = C0 + m k=1 Ck Condition (3.18c) belongs to a type of BMIs with respect to variables µ and K Up to date, the feasibility of BMI conditions is still an NP-hard problem To derive necessary and sufficient tractable conditions for the existence of an SFC (3.16), we decompose condition (3.18c) as µ ⊤ 1⊤ nz Bu K ≺ 0, Du (3.19a) ⊤ ⊤ Bw µ + Dw 1nz ≺ γ1nw (3.19b) As − µ ⊤ 1⊤ nz Cs Although condition (3.19a) is still not tractable as it is a BMI with respect to µ and K, it can help to address the underlying control problem when utilizing some optimization techniques presented in Chapter In the later part of this section, by maximizing admissible matrix variable K, the feasibility of conditions (3.18a)-(3.18c) will be derived in the form of tractable LP-based conditions First, a sufficient condition for the existence of K is given as follows Proposition 3.4.1 Assume that, for a given γ > 0, there exist a matrix Z Rnìnu , vectors à, Rn , ≻ 0, and a scalar σ ≥ satisfying the following LP-based condition  ⊤ ⊤ ⊤  1⊤  nu Bu µ + Du 1nz A0 − Bu Z + σIn    ⊤ ⊤ ⊤ ⊤  1nu Bu µ + Du 1nz C0 − Du Z A⊤ µ + C ⊤ ≺χ s s nz   ⊤ µ + D⊤1  Bw  w nz ≺ γ1nw   Z χ1⊤ (3.20) nu Then, the closed-loop system (3.17) is positive, GES and has L1 -gain performance at level γ under SFC (3.16), where the controller gain K is given by K= 1⊤ nu Z ⊤ Bu⊤ µ + Du⊤ 1nz 16 (3.21) Proposition 3.4.1 provides a sufficient synthesis condition of a desired L1 -gain controller for system (3.14) A main question that whether a controller gain K satisfying condition (3.18) exists is still left open First, we say that the problem of L1 -gain control of system (3.14) via state-feedback controllers is solvable if there exists an SFC in the form of (3.16) that makes the closed-loop system (3.17) positive, GES and has L1 -gain performance of prescribed level This issue will be discussed in the remaining of this section The main idea is to maximize admissible gain K from (3.18a)-(3.18b) We first demonstrate this by considering the case of single-input systems In this case, Bu = (bi ) ∈ Rn , Du = (dl ) ∈ Rnz are column-vectors and K = (kj ) is a row-vector Let A0 = (a0ij ) and C0 = (c0lj ), then conditions (3.18a)-(3.18b) can be written as bi kj ≤ a0ij , ∀i = j, (3.22) dl kj ≤ c0lj , ∀l, j Assume that Bu is not unity (i.e Bu has at least two nonzero entries) Let δj∗ = i=j,l a0ij c0lj , : bi = 0, dl = bi dl and δ∗ = [δ1∗ δ2∗ · · · δn∗ ] We have the following result Proposition 3.4.2 For a given γ > 0, the L1 -gain control problem of positive system (3.14) is solvable if and only if there exists a vector ≺ µ ∈ Rn that satisfies the following LP-based condition ⊤ ⊤ ⊤ ⊤ A⊤ s µ + Cs 1nz ≺ µ Bu + 1nz Du δ∗ , (3.23) ⊤ µ + D⊤1 ≺ γ1 Bw n n w z w An optimal controller gain K is obtained as K = δ∗ We now consider the case of multiple-input systems Without loss of generality, assume that Bu Bu has full column rank, that is, rank(Bu ) = nu Then, it is clear that also has full column Du rank We decompose Bu⊤ = b1 b2 · · · bn , Du⊤ = d1 d2 · · · dnz and K = k1 k2 · · · kn As a special case, if all rows of Bu and Du are unitary, we define b δij = ν=j a0νj bνi : bνi = , d δij = l=1,2, ,nz c0lj dli : dli = ∗ = min{δ b , δ d } Note also that the minimum taken on empty set will be omitted If and kij ij ij b d both δij and δij not exist (this is the case only when bjj = 0, bji = for all i = j and dlj = ∗ will be specified as for all l) then kjj a0jj ∗ kjj > bjj Proposition 3.4.3 For a given γ > 0, the L1 -gain control problem of positive system (3.14) is solvable if and only if there exists a vector µ ∈ Rn , µ ≻ 0, satisfying the following LP-based condition ⊤ ⊤ ⊤ ⊤ A⊤ s µ + Cs 1nz ≺ K∗ Bu µ + Du 1nz , (3.24) ⊤ µ + D⊤ Bw w nz ≺ γ1nw , ∗ ) where K∗ = (kij 17 ⊤ ⊤ Remark 3.4.1 When all rows b⊤ ν and dl are unitary, each of the scalar products bν kj and d⊤ l kj contains only one entry of the column-vector kj This feature can help to maximize the gain matrix K from (3.18a)-(3.18b) easily However, for general multiple-input systems, a similar separation is no longer available Bu are nonunitary To Du maximize each column kj of the gain matrix K from (3.18a)-(3.18b), we consider a polytope ∆j defined as   b⊤  i kj ≤ aij (i = j)  (3.25) ∆j = kj ∈ Rnu b⊤ j kj ≥ ajj   ⊤ dl kj ≤ clj In general case, we assume that all columns of the matrix H = It is easy to verify that (3.25) defines a nonempty convex bounded polyhedron Let Vj be the set of vertices of ∆j For a given vector µ ∈ Rn , µ ≻ 0, the function ̺µ : Rnu → R, k → ̺µ (k) = k ⊤ Bu⊤ µ + Du⊤ 1nz , is continuous Thus, ̺µ (k) attains its maximum in the compact ∆j Lemma 3.4.1 There exists at least a vertex kv ∈ Vj such that ̺µ (kv ) = max {̺µ (k) : k ∈ ∆j } Theorem 3.4.2 For a given γ > 0, the problem of L1 -gain control of positive system (3.14) is solvable if and only if there exist vertices kjv ∈ Vj of polytope ∆j , j ∈ 1, n, such that the following LP-based condition is feasible for a vector ≺ µ ∈ Rn ⊤ ⊤ ⊤ ⊤ A⊤ s µ + Cs 1nz ≺ Kv Bu µ + Du 1nz , ⊤ µ + D⊤ Bw w nz ≺ γ1nw , (3.26) where Kv = k1v k2v · · · knv Remark 3.4.2 The existing methods for solving L1 -gain control problem in the literature only give sufficient LP-based synthesis conditions according to some equality constraints or matrix transformations Different from those results, Proposition 3.4.3 and Theorem 3.4.2 in this paper provide a complete solution in the sense of necessary and sufficient tractable conditions to the problem of L1 -induced control of positive systems with multiple delays Conclusion of Chapter In this chapter, the problem of L1 -gain control has been studied for positive linear systems with multiple delays A characterization of L1 -induced performance has been established and utilized to derive necessary and sufficient conditions for the existence of SFCs subject to L1 -gain of the closed-loop systems 18 Chapter PEAK-TO-PEAK GAIN CONTROL OF DISCRETE-TIME POSITIVE LINEAR SYSTEMS WITH DIVERSE INTERVAL DELAYS In this chapter, the problem of peak-to-peak gain control via static output-feedback is studied for discrete-time positive linear systems with diverse interval delays By novel comparison techniques involving steady states of upper and lower scaled systems with peak values of exogenous disturbances, a characterization of ℓ∞ -induced norm of the input-output operator (also known as ℓ∞ -gain) is established The obtained ℓ∞ -gain characterization is then utilized to derive necessary and sufficient conditions subject to ℓ∞ -induced performance with prescribed level On the basis of performance analysis results, and based on a vertex optimization technique, a complete solution to the synthesis problem of an SOFC that minimizes the worst case amplification from disturbances to regulated outputs subject to ℓ∞ -gain is addressed Main content of this chapter is written based on paper [P3] in the List of publications 4.1 Problem formulation Consider the following discrete-time system with multiple time-varying delays N x(k + 1) = A0 x(k) + Aj x(k − dj (k)) + Bw w(k), k ≥ 0, (4.1a) j=1 N Cj x(k − hj (k)) + Dw w(k), z(k) = C0 x(k) + (4.1b) j=1 x(k) = φ(k), k ∈ Z[−d, 0], (4.1c) where x(k) ∈ Rn is the state vector, z(k) ∈ Rnz and w(k) ∈ Rnw are the regulated output and exogenous disturbance input vectors, respectively A0 , Aj , Bw , C0 , Cj and Dw , j ∈ 1, N, are given matrices of appropriate dimensions Assume that positive system (4.1) is stable (GES) We define the input-output operator of system (4.1) as Ψ : ℓ∞ (Rnw ) −→ ℓ∞ (Rnz ), w → z, and ℓ∞ -gain of system (4.1) under zero initial condition as Ψ (ℓ∞ ,ℓ∞ ) = sup w ℓ∞ =0 z w ℓ∞ ℓ∞ = sup w z ℓ∞ (4.2) ℓ∞ =1 Definition 4.1.1 For a given γ > 0, system (4.1) is said to have ℓ∞ -gain performance of level γ if Ψ (ℓ∞,ℓ∞ ) < γ 4.2 Stability analysis In this section, we briefly formulate conditions by which positive system (4.1) is GES Some stability criteria in the following theorem Theorem 4.2.1 For positive system (4.1) with w = 0, the following statements are equivalent (i) System (4.1) is GES 19 (ii) There exists a vector η ∈ Rn , η ≻ 0, such that N η ⊤ Aj − η ⊤ ≺ A0 + (4.3) j=1 (iii) There exists a vector v ∈ Rn , v ≻ 0, such that N A0 + Aj v − v ≺ (4.4) j=1 N j=1 Aj (iv) The matrix A0 + N j=1 Aj is Schur stable, that is, the spectral radius of the matrix A0 + satisfies N Aj ρ A0 + < (4.5) j=1 4.2.1 Peak-to-peak gain characterization Associated with system (4.1), we consider the following upper scaled system N + Aj x+ (k − dj ) + Bw w, + x (k + 1) = A0 x (k) + k ≥ 0, (4.6a) j=1 N + Cj x+ (k − hj ) + Dw w, + z (k) = C0 x (k) + (4.6b) j=1 x+ (k) = 0, where w ∈ ℓ∞ (Rnw ) and w = w k ∈ Z[−d, 0], (4.6c) ℓ∞ 1nw Lemma 4.2.1 For a given w ∈ ℓ∞ (Rnw ), w(k) 0, let x(k), x+ (k) and z(k), z + (k) be the state and output trajectories of systems (4.1) and (4.6) with zero initial condition, respectively It holds that x(k) x+ (k) and z(k) z + (k) for all k ≥ To establish a lower bound for x(k) and z(k), we consider the following scaled system N − − x (k + 1) = A0 x (k) + Aj x− (k − dj ) + Bw w, (4.7a) Cj x− (k − hj ) + Dw w, (4.7b) j=1 N − − z (k) = C0 x (k) + j=1 x− (k) = 0, k ∈ Z[−d, 0], where w ∈ Rn+w is a constant vector such that w(k) (4.7c) w Lemma 4.2.2 For a given w ∈ ℓ∞ (Rnw ) and a vector w ∈ Rn+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 (4.7) with zero initial condition, respectively It holds that x− (k) x(k), z − (k) 20 z(k), k ≥ (4.8) Lemma 4.2.3 For any w ∈ ℓ∞ (Rnw ), the solution x(k, w) of (4.1) with zero initial condition satisfies |x(k, w)| w ℓ∞ (In − A0 − Ad )−1 Bw 1nw , k ≥ (4.9) In particular, x(k, w) also belongs to ℓ∞ (Rn ) Theorem 4.2.2 Assume that system (4.1) is positive and GES The exact value of ℓ∞ -gain of system (4.1) is obtained as Ψ where Ad = N j=1 Aj (ℓ∞ ,ℓ∞ ) = (C0 + Cd ) (In − A0 − Ad )−1 Bw + Dw ∞ , (4.10) N j=1 Cj and Cd = Theorem 4.2.3 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 ∈ Rn , η ≻ 0, that satisfies the following LP-based conditions N Aj − In η + Bw 1nw ≺ 0, (4.11a) Cj η + Dw 1nw − γ1nz ≺ (4.11b) A0 + j=1 N C0 + j=1 Theorem 4.2.4 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 ∈ Rn , ν ≻ N ν ⊤ N Aj − In A0 + + 1⊤ nz Cj ≺ 0, (4.12a) − γ1⊤ nw ≺ (4.12b) C0 + j=1 j=1 ν ⊤ Bw + 1⊤ nz Dw 4.3 Static output-feedback peak-to-peak gain control In this section, we consider the problem of ℓ∞ -gain control of the following system N Aj x(k − dj (k)) + Bu(k) + Bw w(k), x(k + 1) = A0 x(k) + k ≥ 0, j=1 N (4.13) Cj x(k − hj (k)) + Du(k) + Dw w(k), z(k) = C0 x(k) + j=1 y(k) = Ex(k) + F w(k), where y(k) ∈ Rno is the measurement output, u(k) ∈ Rnu is the control input We now focus on the existence of an SOFC of the form u(k) = −Ky(k), k ≥ 0, (4.14) that makes the closed-loop system N c Aj x(k − dj (k)) + Bw w(k), x(k + 1) = Ac x(k) + j=1 (4.15) N c Cj x(k − hj (k)) + Dw w(k), z(k) = Cc x(k) + k ≥ 0, j=1 21 c = positive, stable and has prescribed ℓ∞ -gain performance, where Ac = A0 − BKE, Bw c = D − DKF For a given γ > 0, system (4.15) is Bw − BKF , Cc = C0 − DKE and Dw w positive, stable and has ℓ∞ -gain performance at level γ if and only if A0 Bw C Dw − B KH D (4.16) and one of the following two LP-based conditions is feasible η , ≺γ 1nz 1nw B KH D Ξ− B KH D Ξ− ⊤ ν , ≺γ 1nw 1nz A0 + Ad − In Bw C0 + Cd Dw where H = E F and Ξ = η ≻ 0, ν ≻ 0, (4.17) (4.18) 4.3.1 Matrix transformation approach Using certain type of matrix transformations is a very common approach in the existing literature to obtain a desired controller gain K Revealed by (4.18), we define the transformation K ⊤ B ⊤ D⊤ ν = Z ∈ Rno 1nz (4.19) then, conditions (4.16) and (4.18) hold if and only if A0 Bw C Dw B ⊤ ν + D ⊤ 1nz − Ξ⊤ B Z ⊤H D 0, (4.20a) ν − H⊤ Z ≺ γ 1nw 1nz (4.20b) Proposition 4.3.1 For a given γ > 0, there exists an SOFC in the form of (4.14) that makes the closed-loop system (4.15) positive, stable and has ℓ∞ -gain performance at level γ if the coupled LP-based condition (4.20) is feasible for vectors ν ∈ Rn , ν ≻ 0, and Z ∈ Rno The controller gain is obtained as Z ⊤ (4.21) K= ⊤ B ν + D ⊤ 1nz 4.3.2 Vertex optimization approach Let G = H= only if B = (gi ) D · · · n+nw , A0 Bw = (mij ) We decompose C Dw ∈ Rno , and K = k ⊤ , k ∈ Rno , then condition (4.16) holds if and 0, G = 0, and M = j k⊤ j ≤ σj∗ i=1,2, ,n+nz mij : gi > gi (4.22) According to (4.22), we define a polyhedron P as follow P = k ∈ Rno k ⊤ H σ∗ , (4.23) ∗ With the common assumption that H is a full-row rank where σ ∗ = σ1∗ σ2∗ · · · σn+n w matrix, (4.23) defines a nonempty convex polyhedron Let S be the set of vertices of P For 22 a fixed vector χ ≻ 0, the function ̺χ : Rno → R+ , k → ̺χ (k) = k ⊤ Hχ, is continuous Thus, ̺χ (k) attains its maximum on the compact ∆ = P ∩ {k ∈ Rno : ̺χ (k) ≥ 0} Lemma 4.3.1 There exists a vertex kv ∈ S such that ̺χ (kv ) = maxk∈P ̺χ (k) Theorem 4.3.1 For a given γ > 0, there exists an SOFC in the form of (4.14) that makes the closed-loop system (4.15) positive, stable and has ℓ∞ -gain performance at level γ if and only if there exists a vertex kv∗ ∈ S of the polyhedron P such that the following LP-based condition is feasible η , (4.24) ≺γ Ξ − Gkv∗⊤ H 1nz 1nw where Ξ = A0 + Ad − In Bw C0 + Cd Dw and G = B D As a special case, for single-input single-output systems, an optimal controller gain is explicitly obtained as mij ∗ : gi j > K = kop = gi j and a desired SOFC (4.14) exists if and only if there exists a vector ≺ η ∈ Rn satisfying ∗ Ξ − kop GH η ≺γ 1nz 1nw (4.25) Conclusion of Chapter In this chapter, the problem of peak-to-peak gain control has been studied for a class of discrete-time positive linear systems with multiple time-varying delays A characterization of ℓ∞ -gain subject to diverse time-varying delays has been formulated and necessary and sufficient tractable LP-based conditions for the existence of a static output-feedback controller have been presented 23 CONCLUDING REMAKRS Main contributions Proposed a novel and systematic approach based on an optimization procedure for the problem of static output-feedback control of LTI positive systems with time-delay in the state and output vectors By utilizing the proposed method, necessary and sufficient stabilization conditions are derived in the form of tractable LP conditions, which can be effectively verified using various optimization algorithms Established a characterization of L1 -induced norm and derived necessary and sufficient L1 -induced performance conditions for a class of positive linear systems with multiple delays Based on some vertex optimization techniques, a complete solution to the stabilization problem under L1 -gain control scheme has been formulated through tractable LP conditions Derived a new characterization of ℓ∞ -gain for discrete-time positive systems with heterogeneous interval delays Then, based on an optimization procedure, a complete solution to the problem of static output-feedback control under ℓ∞ -gain scheme has been presented Future works: Potential further extensions • Extend the results obtained in Chapter to positive linear systems with general timevarying delays including discrete and/or distributed delays • An analogous version of L∞ established in Chapter for continuous-time positive systems with mixed time-varying delays is still left open Tackling this problem seems to be not a simple task, which needs further investigation and development • Overall, the research of this thesis is mainly focused on the problems of stability analysis and control under L1 and ℓ∞ /L∞ schemes of linear time-invariant systems Extending the obtained results in this thesis to various complex systems such as positive switched systems, impulsive systems or periodic systems proves to have significant practical and scientific meaning This will be considered in potential future works 24 LIST OF PUBLICATIONS [P1] Le Van Hien and Mai Thi Hong (2019), An optimization approach to static outputfeedback control of LTI positive systems with delayed measurements, Journal of the Franklin Institute, vol 356, pp 5087-5103 (SCIE, Q1) [P2] Mai Thi Hong and Le Van Hien (2021), Solvability of L1-induced controller synthesis for positive systems with multiple delays, International Journal of Control, Automation and Systems DOI: 10.1007/s12555-020-0510-x (SCIE, Q2) [P3] Mai Thi Hong, Le Van Hien and Trinh Thi Minh Hang (2021), Satic output-feedback peakto-peak gain control of discrete-time positive linear systems with diverse interval delays (Revision submitted) The results of this dissertation have been presented at • The weekly seminar on Differential and Integral Equation, Division of Mathematical Analysis, Faculty of Mathematics and Informatics, Hanoi National University of Education • Seminar of the Division of Mathematics, Faculty of Informatics Technology, National University of Civil Engineering • PhD Annual Conferences, Faculty of Mathematics and Informatics, Hanoi National University of Education, 2019, 2020 • Workshop Dynamical Systems and Related Topics, Vietnam Institute for Advanced Study in Mathematics (VIASM), Hanoi, December 23-25, 2019 • Workshop Selected Problems in Differential Equations and Control, Vietnam Institute for Advanced Study in Mathematics (VIASM), Tuan Chau, Ha Long, November 5-7, 2020 ... time-delay systems is essential in the field of control engineering, which has attracted significant research attention in the past two decades A very rich literature with a large number of important... extensively studied, the induced L∞ and ℓ∞ theory has received less attention despite the existing results have demonstrated good attenuation for persistent peakbounded disturbances Typically, L∞... Nguyen Minh Tuan VNU University of Education This dissertation is presented to the examining committee at Hanoi National University of Education, 136 Xuan Thuy Road, Hanoi, Vietnam At the time of