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.OME CONTROL PROBLEMS FOR POSITIVE LINEAR SYSTEMS.

MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION ——————–o0o——————— MAI THI HONG SOME CONTROL PROBLEMS FOR POSITIVE LINEAR SYSTEMS DISSERTATION OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HA NOI-2021 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 A dissertation submitted to Hanoi National University of Education for fulfilled requirements of the degree of Doctor of Philosophy in Mathematics Under the guidance of Associate Professor LE Van Hien DECLARATION I am the creator of this dissertation, which has been conducted at the Faculty of Mathematics and Informatics, Hanoi National University of Education, under the guidance and direction of Associate Professor Le Van Hien I hereby affirm that the results presented in this dissertation are truly provided and have not been included in any other dissertations or theses submitted to any other universities or institutions for a degree or diploma “I certify that I am the PhD student named below and that the information provided is correct” Full name: Mai Thi Hong Signed: Date: ACKNOWLEDGMENT First and foremost, I would like to express my sincere thanks to my supervisor, Associate Professor Le Van Hien, for his enlightening guidance, insightful ideas, and endless support during my candidacy at Hanoi National University of Education His rigorous research ethics, diligent work attitude, and wholehearted dedication to his students have been an inspiration to me and will influence me forever I am grateful to Associcate Professor Tran Dinh Ke and other members of the weekly seminar at the Division of Mathematical Analysis, Faculty of Mathematics and Informatics, Hanoi National University of Education, for their discussions and valuable comments on my research results I am also grateful to my colleagues at the Division of Mathematics, Faculty of Information Technology, National University of Civil Engineering, for their help and support during the time of my PhD research and study I am forever grateful to my parents for endless love and unconditional support they have been giving me Last but not least, I am indebted to my beloved husband, Mr Trung Kien, my beautiful daughters, Hoang Mai, Gia Linh, and lovely son, Minh Giang, who always stay beside me None of this would have been possible without their continuous and unconditional love, kindness and comfort through my journey The author TABLE OF CONTENTS Page Declaration Acknowledgment List of symbols and acronyms INTRODUCTION A Background B C Literature review 13 B1 Static output-feedback control of positive linear systems 13 B2 L1-gain control of positive linear systems with multiple delays 14 B3 Peak-to-peak gain control of discrete-time positive linear systems 15 B4 Objectives .17 Research topics 18 C1 Static output-feedback control of positive linear systems with timevarying delay 18 C2 L1-gain control of positive linear systems with multiple delays 19 C3 Peak-to-peak gain control of discrete-time positive linear systems with diverse interval delays 20 D Outline of main contributions 21 E Thesis structure 23 PRELIMINARIES 24 1.1 Nonnegative and Metzler matrices 24 1.2 Lyapunov stability 26 1.2.1 Stability concepts 26 1.2.2 Stability and stabilization of LTI systems .28 1.3 Positive LTI systems 30 1.3.1 Stability analysis and controller design 31 1.3.2 L1-induced performance 32 1.3.3 ℓ∞-induced performance 33 1.4 KKM Lemma 33 STATIC OUTPUT-FEEDBACK CONTROL OF POSITIVE LINEAR SYS- TEMS WITH TIME-VARYING DELAY 35 2.1 Problem formulation 35 2.2 Stability analysis .36 2.3 Controller synthesis 40 2.3.1 Single-input single-output systems 43 2.3.2 Single-input multiple-output systems 44 2.3.3 Multiple-input single-output systems 47 2.3.4 Multiple-input multiple-output systems .48 2.4 Numerical examples .50 2.5 Conclusion of Chapter 54 ON L1-GAIN CONTROL OF POSITIVE LINEAR SYSTEMS WITH MULTIPLE DELAYS .55 3.1 Problem statement 55 3.2 Stability analysis .57 3.3 L1-induced performance .61 3.4 L1-gain control 65 3.5 Illustrative examples 71 3.6 Concluding remarks .74 PEAK-TO-PEAK GAIN CONTROL OF DISCRETE-TIME POSITIVE LINEAR SYSTEMS WITH DIVERSE INTERVAL DELAYS .76 4.1 Problem formulation 76 4.2 Stability analysis 78 4.3 Peak-to-peak gain characterization 80 4.4 Static output-feedback peak-to-peak gain control 87 4.4.1 Matrix transformation approach 89 4.4.2 Vertex optimization approach 89 4.5 Illustrative examples 91 4.6 Conclusion of Chapter 95 Concluding remarks 96 List of publications 98 References 98 LIST OF SYMBOLS AND ACRONYMS Rn the n-dimensional Euclidean space max-norm maxi=1,2, ,n |xi| of a vector x = (xi) ∈ Rn ∥x∥∞ ∥x∥1 , , 1)n ⊤ ∈ Rx ≼ y n 1-norm Σn i= |xi| of a vector x = (xi) ∈ R1n the column vector (1, component-wise comparison between vectors x and y More precisely, for x = (xi) ∈ Rn and y = (yi) ∈ Rn, x ≼ y if xi ≤ yi for i = 1, , , n x≺ y if xi < yi for i = 1, , , n x≽ y if xi ≥ yi for i = 1, , , n (or y ≼ x) x≻ y if xi > yi for i = 1, , , n (or y ≺ x) n positive orthant of Rn R+ n , i.e., the set {x ∈ Rn : x ≽ 0} the absolute of a vector x = (xi) ∈ | + x | = ( | x i | ) ∈ R RRn n×m the set of n × m real matrices m×n |A| = (|aij|) ∈ R+ the absolute of a matrix A = (aij) ∈ Rdiag{A, B} the diagonal matrix formulated by stacking A and BA ⊤ m×n the transpose matrix of a matrix A A the inverse matrix of a matrix A A≽0 nonnegative matrix A = (aij) ∈ Rm×n (aij ≥ for all i, j) A≻ positive matrix A (i.e aij > for all i, j) A>0 positive-definite matrix A (i.e x⊤ Ax > 0, ∀x ∈ Rn , x ̸= 0) −1 Sn+ the set of symmetric positive-definite matrices in RIn identity matrix the set of Metzler matrices in R∥A∥∞ the max-norm of a matrix A, in RMn i.e., for A = (aij) ∈ Rn×n n×n n×n m×n , ∥A∥∞ = ∥|A|1n∥∞ = max1≤i≤m |aij| Σn j= ∥A∥1 the 1-norm of a matrix A, i.e., for A 1= (aij) ∈ Rm×n, ∥A∥1 = max1≤j≤n rank(A) Σm rank of a matrix A i=|aij| σ(A) spectrum (the set of eigenvalues) of a matrix A ∈ Rρ(A) max{|λ| : λ ∈ σ(A)}, spectral radius of A µ(A) max{Reλ : λ ∈ σ(A)}, spectral abscissa of A N0 the set of natural numbers N the set of positive integers Z the set of integers Z[a, b] the set {p ∈ Z : a ≤ p ≤ b} ∥f (t)∥1 Σn i= n |fi (t)|, 1-norm of a vector f (t) ∈ Rn×n ∥f ∥L1 R∥x(k)∥∞ ∫∞ ∥f (t)∥1 dt, L1 -norm of a function f : R+ → maxi=1,2, ,n |xi(k)|, max-norm of a vector x(k) ∈ R∥f ∥ℓ∞ (k)∥∞ , ℓ∞ -norm of a function f : Z+ → RL1 (R+ , Rn n n n the set {f : R+ → Rn : ∥f ∥L1 < ∞} ) ℓ∞ (Rn ) ∥Σ∥(L1,L1) the set {f : Z+ → Rn : ∥f ∥ℓ∞ < ∞} L1-induced norm of the operator Σ ∥Ψ∥(ℓ∞,ℓ∞) ℓ∞-induced norm of the operator Ψ C = C([a, b], Rn) ∥φ∥C the set of Rn-valued continuous functions defined on [a, b] uniform norm supa≤t≤b ∥φ(t)∥ GAS global asymptotic stability GES global exponential stability LMIs linear matrix inequalities BMIs bilinear matrix inequalities LTI linear time-invariant LP linear programming LKF Lyapunov-Krasovskii functional SFC state-feedback controller SOFC static output-feedback controller SISO single-input single-output SIMO single-input multiple-output MISO multiple-input single-output MIMO multiple-input multiple-output ✷ completeness of a proof supk∈Z+ ∥f to equilibria x∗ = [1.0194 0.4694]⊤ and x∗ = [0.6796 0.3129]⊤ , respectively This shows the validity of theoretical results 1.2 x+(k) 1 0.8 x (k) x- (k) 1 0.4 0.2 0 50 100 k 150 200 (a) x1(k) 0.6 + x (k) 0.5 0.4 x (k) 0.3 - x (k) 0.2 0.1 0 50 100 k (b) x2(k) 150 200 Figure 4.1: State trajectories of system (4.1) with random delay ≤ d(k) ≤ and disturbance ≤ w(k) ≤ 1.5 Example 4.5.2 Consider the following positive system x(k + 1) = A0x(k)+ A1x(k − d(k)) + Bww(k), z(k) = C0x(k)+ Dww(k), (4.46) ⎡ ⎤ , 0.16 and d(k) is a time-varying delay satisfying ≤ d(k) ≤ By resolving iteratively the LP-based conditions in (4.28) with respect to η ≻ using linprog toolbox in Matlab, it is found that condition (4.28) is feasible for γ ≥ γmin ¾ 1.9691 To validate this result, some norm-state trajectories of ∥z(k)∥∞ with respect to ∥w∥ℓ∞ = and zero initial condition are conducted It can be seen from Fig.4.2 that sup∥w∥ℓ =1 ∥z(k)∥∞ = γmin This shows the 0.35 wher e ⎡ 15⎢ 0.05⎤ ⎥ ⎡ ⎢ 0.2 0.1 ⎤ ⎥ ∞ merit of the result of Theorem 4.3.2 A = 0.2 0.15 0.1 0.2 ⎢ ⎣ 5⎥ ⎦ , A 1.9691 1.5 ∥z( k) ∥∞ w(k) = random ≤ w(k) ≤ random 0.9 ≤ w(k) ≤ random 0.5 ≤ w(k) ≤ 0.5 = ⎢ ⎣ 5⎥ k 600 800 1000 Example 4.5.3 Consider the following positive system 0.03 0.2 08 B , 02 D 5 w 0 = 15 200 400 Figure 4.2: State trajectories of ∥z(k)∥∞ with respect to ∥w∥ℓ∞ = ⎦ , ⎢ ⎥ x(k + 1) = A0x(k)+ Bu(k)+ Bww(k), 0.25 z(k) = C0x(k)+ C1x(k − h(k)) + Dww(k), 0.15 y(k) = Ex(k)+ Fw(k), Σ ⎣ ⎦ Σ (4.47) Σ Σ easy to verify that the unforced system of (4.47) (i.e with u(k) = where 0) is unstable Simulation results of ∥z(k)∥∞ with respect to ∥w∥ℓ∞ ⎡ 0⎢ 0.2 0.15⎤ ⎡1.0⎤ ⎡ 0.1 ⎤ = 1, ≤ h(k) ≤ and zero initial condition given in Fig.4.3 show that ∥z(k)∥∞ is unbounded ×10 ⎢ 0.2 0.5 0.35 0⎥ B ⎦ ∥z( k) ∥∞ , C 0.2 = 0.25 0.26 , 0.1 1.5 10 20 k 40 30 50 0.5 ≤ w(k) ≤ Σ 0.7 ≤ w(k) ≤ 0.9 ≤ w(k) ≤ 0.5 Σ 0 10 20 30 40 50 k Figure 4.3: Open-loop trajectories of ∥z(k)∥∞ with ≤ h(k) ≤ and ∥w∥ℓ∞ = , 0.4 0.18 0.15 0.11 D = 1.0 ≤ wk) ≤ 11 C = 0.1 0.1 0.05 Apply Theorem 4.4.1, it is found that the set of vertices S contains a unique w o 0.1 Σp 0.7 w( k) Σ element = 0.2 k mingik mij j ̸=0 gj kj = 0.2 which is the optimal controller gain Moreover, condition (4.43) (and also (4.45)) is feasible if and only if γ > ,E= γmin = 1.5316 By Theorem 4.4.1, the closed-loop system of Σ (4.47) is positive, stable and has ℓ∞-gain 0 ,F = 0.1 Σ Some closed-loop state trajectories of ∥z(k)∥ performance at level γ > γmin ∞ with respect to ∥w∥ℓ∞ = 1, ≤ h(k) ≤ and zero initial condition are presented in Fig.4.4 Again, the simulation results show the effectiveness of the obtained theoretical results I t i s 1.8 w(k) = 1.5316 ∥z( k) ∥∞ w(k) = 0.5 0.5 0 200 400 k 600 800 1000 Figure 4.4: Closed-loop trajectories of ∥z(k)∥∞ with 0.5 ≤ w(k) ≤ and ≤ h(k) ≤ 4.6 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 LP-based conditions which guarantee a ℓ∞-induced performance corresponding to prescribed level have been derived On the basis of the obtained analysis results, necessary and sufficient tractable LP-based conditions for the existence of a static output-feedback controller are presented The desired controller makes the incorporated closed-loop system be positive, stable and satisfy the prescribed peak-to-peak gain performance index Numerical examples with simulations have been given to illustrate the effectiveness of the obtained results CONCLUDING REMAKRS Main contributions In this thesis, the problems of stabilization, L1-gain control and ℓ∞-gain control via static output-feedback and state-feedback have been studied for some classes of positive linear systems with delays Main contributions of this thesis can be specified as follows Proposed a novel and systematic approach based on an optimization procedure for the problem of static output-feedback control of LTI positive systems with timedelay in the state and output vectors By utilizing the proposed method, necessary and sufficient conditions for stabilization are derived in the form of tractable LP conditions, which can be effectively verified using various optimization algorithms Established a characterization of exact value of L1-induced norm of the inputoutput operator for a class of positive linear systems with multiple delays The obtained L1-induced norm characterization has been utilized to derive necessary and sufficient conditions subject to L1-induced performance with prescribed level Then, 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 The obtained results in this thesis also leave much room for further development, which can be considered in future works Some research topics can be studied as As discussed in the concluding section, the results concerning performance analysis of L1-gain obtained in Chapter are only applicable for positive LTI systems with possibly unknown but constant delays Whether the obtained results are extended to positive systems with general time-varying delays proves to be an interesting problem It seems that the method proposed in this chapter cannot be simply extended to such systems The lack of systematic approaches, which can be utilized to deal with L1-gain control of positive linear systems with general time-varying delays including discrete and/or distributed delays, clearly requires significant technical development 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 time-varying 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 LIST OF PUBLICATIONS [P1] Le Van Hien and Mai Thi Hong (2019), An optimization approach to static output- feedback 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), Static outputfeedback peak-to-peak gain control of discrete-time positive linear systems with diverse interval delays (Revision submitted) REFERENCES [1] J Abedor, K Nagpal and K Poolla (1996), A linear matrix inequality approach to peak-to-peak gain minimization, Int J Robust Nonlinear Control, 6, pp 899– 927 [2] C.K Ahn, P Shi and L Wu (2016), l∞-gain performance analysis for twodimensional Roesser systems with persistent bounded disturbance and saturation nonlinearity, Info Sci., 333, pp 126–139 [3] C.K Ahn, P Shi and M.V 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