On l1 output feedback control of positive linear systems

HANOI PEDAGOGICAL UNIVERSITY No.2 DEPARTMENT OF MATHEMATICS ——————–o0o——————— DO THI VAN ANH On L1 output - feedback control of positive linear systems BACHELOR THESIS Major: Applied Mathematics Hanoi, May 2019 HANOI PEDAGOGICAL UNIVERSITY No.2 DEPARTMENT OF MATHEMATICS ——————–o0o——————— DO THI VAN ANH On L1 output - feedback control of positive linear systems BACHELOR THESIS Major: Applied Mathematics Supervisor: Assoc Prof Dr LE VAN HIEN Hanoi, May 2019 Bachelor thesis Do Thi Van Anh Thesis Acknowledgement I would like to express my gratitudes to the teachers of Hanoi Pedagogical University No.2, especially the teachers in the Department of Mathematics The lecturers have imparted valuable knowledge and facilitated for me to complete the course and the thesis In particular, I would like to express my deep respect and gratitude to Assoc Prof Dr Le Van Hien (Hanoi National University of Education) who has direct guidance and helps me to complete this thesis Professionalism, seriousness and his right orientations are important prerequisites for me to get the results in this thesis Due to limited time, capacity, and conditions, my thesis cannot avoid errors I am looking forward to receiving valuable comments from readers Hanoi, May 5, 2019 Student Do Thi Van Anh i Bachelor thesis Do Thi Van Anh Thesis Assurance I assume that the data and the results of this thesis are true and not identical to other topics I also assume that all the help for this thesis has been acknowledged and that the results presented in the thesis has been identified clearly Hanoi, May 5, 2019 Student Do Thi Van Anh ii Bachelor thesis Do Thi Van Anh Acronyms MIMO Multi-Input Multi-Output SIMO Single-Input Multi-Output iii Bachelor thesis Do Thi Van Anh Symbols R Set of real numbers Rn Set of n-column real vectors ¯n R + Set of n-dimensional nonnegative real vectors Rn×m Set of n × m real matrices x Euclidean norm of the vector x n x(t) |xi (t)| i=1 ∞ x x(k) l1 k=0 ∞ x x(t) dt L1 l1 space of all vector-valued functions with finite l1 norm L1 Space of all vector-valued functions with finite L1 norm ∈ Belong to Defined as End of proof I Identity matrix AT Transpose of the matrix A A−1 Inverse of the matrix A iv Bachelor thesis Do Thi Van Anh diag(A1 , A2 , , An ) Block diagonal matrix of A1 , , An on the diagonal coli (A) The ith column of matrix A ρ(A) max{|λi (A)|, i = 1, 2, , n}, i.e., spectral radius of matrix A α(A) max{Reλi (A), i = 1, 2, , n}, i.e., spectral abscissa of matrix A A Spectral norm of the matrixA A>B A − B is positive definite A≥B A − B is positive semi-definite A ≥≥ B A − B is element-wise nonnegavtive A is element-wise positive B v Contents Introduction 1.1 Background 1.2 Literature review 1.3 Objective of the thesis 1.4 Mathematical preliminaries 1.4.1 Definitions and lemmas on positive linear systems 1.4.2 L1 -induced performance L1 -induced output-feedback controller synthesis for interval positive linear systems 2.1 Performance analysis 2.2 Static output-feedback controller 14 Applications 3.1 3.2 20 Full-order output: C = I 20 3.1.1 Controller synthesis for SIMO systems 20 3.1.2 Controller synthesis for MIMO Systems 26 3.1.3 Sparse Controller Synthesis 28 Dynamic output-feedback controller design 33 vi Bachelor thesis 3.3 Do Thi Van Anh An illustrative example References 38 42 vii Chapter Introduction 1.1 Background A control system is a system, which provides desired responses by controlling outputs The following figure shows a simple block diagram of a control system Inputs Plant Outputs Figure 1.1: Block diagram of a control system The input is a channel which can be plugged into a system to activate or manipulate the process The output is a channel which will be measured or observed An output is controlled by varying input A state is a set of mathematical functions or physical, they can be used to describe totally the future behaviour of an active system if the inputs are known In fact, there is a type of systems whose state variables and outputs are always positive or nonnegavtive for Bachelor thesis Do Thi Van Anh the vector, where l0 -norm of x is defined as n x |sign(xi )| = i=1 The problem of L1 -induced sparse controller design (L1SCD) is formulated as follows Problem L1SCD: Given a positive system S:   x(t) ˙ = Ax(t) + Bw w(t), (3.13)  y(t) = Cx(t) + Du(t)D w(t), w where A ∈ [A, A], B ∈ [B, B], Bw ∈ [B w , B w ]C ∈ [C, C], D ∈ [D, D] and Dw ∈ [Dw , Dw ] Find find a state-feedback controller u(t) = Kx(t) such that The closed-loop system   x(t) ˙ = (A + BK) x(t) + Bw w(t), (3.14)  y(t) = (C + DK) x(t) + D w(t), w is positive and robustly stable The l0 -norm of the controller gain K is minimized subject to the L1 -induced performance, that is, K 29 Bachelor thesis Do Thi Van Anh subject to y L1