On l1 output feedback control of positive linear systems

4 1.4.1 Definitions and lemmas on positive linear systems 4 1.4.2 L1-induced performance.. 7 2 L1-induced output-feedback controller synthesis for in-terval positive linear systems 8 2.1

HANOI PEDAGOGICAL UNIVERSITY No.2

Bachelor thesis Do Thi Van Anh

In particular, I would like to express my deep respect and gratitude

to Assoc Prof Dr Le Van Hien (Hanoi National University of cation) who has direct guidance and helps me to complete this thesis.Professionalism, seriousness and his right orientations are importantprerequisites for me to get the results in this thesis

Edu-Due to limited time, capacity, and conditions, my thesis cannotavoid errors I am looking forward to receiving valuable commentsfrom readers

Hanoi, May 5, 2019

Student

Do Thi Van Anh

Thesis Assurance

I assume that the data and the results of this thesis are true andnot identical to other topics I also assume that all the help for thisthesis has been acknowledged and that the results presented in thethesis has been identified clearly

Hanoi, May 5, 2019

Student

Do Thi Van Anh

Bachelor thesis Do Thi Van Anh

Acronyms

MIMO Multi-Input Multi-Output

SIMO Single-Input Multi-Output

R Set of real numbers

Rn Set of n-column real vectors

¯

Rn+ Set of n-dimensional nonnegative real vectors

Rn×m Set of n × m real matrices

kxk Euclidean norm of the vector x

Z ∞ 0

kx(t)k1dt

l1 space of all vector-valued functions with finite l1 norm

L1 Space of all vector-valued functions with finite L1 norm

Bachelor thesis Do Thi Van Anh

diag(A1, A2, , An) Block diagonal matrix of A1, , An on the diagonalcoli(A) The ith column of matrix A

ρ(A) max{|λi(A)|, i = 1, 2, , n}, i.e., spectral radius of matrix Aα(A) max{Reλi(A), i = 1, 2, , n}, i.e., spectral abscissa of matrix AkAk 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  B is element-wise positive

1 Introduction 1

1.1 Background 1

1.2 Literature review 2

1.3 Objective of the thesis 3

1.4 Mathematical preliminaries 4

1.4.1 Definitions and lemmas on positive linear systems 4 1.4.2 L1-induced performance 7

2 L1-induced output-feedback controller synthesis for in-terval positive linear systems 8 2.1 Performance analysis 9

2.2 Static output-feedback controller 14

3 Applications 20 3.1 Full-order output: C = I 20

3.1.1 Controller synthesis for SIMO systems 20

3.1.2 Controller synthesis for MIMO Systems 26

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3.3 An illustrative example 38

Figure 1.1: Block diagram of a control system

The input is a channel which can be plugged into a system toactivate or manipulate the process The output is a channel whichwill be measured or observed An output is controlled by varyinginput A state is a set of mathematical functions or physical, they can

be used to describe totally the future behaviour of an active system

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nonnegavtive inputs and initial conditions These systems are calledpostive systems

1.2 Literature review

In various areas of applied models, relevant states as electric charge,liquid levels in controlling tanks, population of species or the num-ber of molecules are always nonnegative Such models are typicallydescribed by systems that generate nonnegative state and output tra-jectories whenever driven by nonnegative inputs including controllersand initial states This particular category of systems, referred to

as positive systems [1] or nonnegative systems, has found tions in a variety of disciplines ranging from biology, epidemiology,chemistry, pharmacokinetics, and other models that are subject toconservation laws, to air traffic flow networks, chemical reactions orcommunication Besides that positive systems possess many elegantproperties that have yet no counterpart in general systems, such asrobustness, insensitiveness with delays or monotonicity, among manyothers [2] According to those properties, positive systems are alsoemployed into designing of interval observers and state estimations

applica-or used as comparison systems fapplica-or the analysis of complex time-delaysystems Thus, due to both practical and theoretical applications, theproblems of analysis and synthesis of positive systems have receivedever-increasing interest, which has been one of the most active researchtopics in recent years

On the other hand, the L1-gain control is an essential problem inthe control theory of continuous-time positive systems It provides

an interpretation of a mass balance with linear supply rates and ear co-positive storage functions, which naturally arise in dissipativetheory of positive systems [3] Following this pioneering work, theperformance analysis and synthesis problems under L1-gain schemeshave attracted considerable research attention with numerous inter-esting results have been reported in the literature (see, [4]-[5] and thecited references therein) Based on certain types of co-positive Lya-punov functions, performance conditions are derived in terms of linear

lin-or semidefinite programs The controller synthesis problem of positivesystems under L1-gain scheme is often more challenging and manywell-established methodologies developed for general linear systemsare not adaptable for positive systems

1.3 Objective of the thesis

Our main objective in this graduation thesis is to study the problems

of performance analysis and controller synthesis for L1-gain control ofpositive linear systems based on recent works [4] and [5, Chapter 3]

We are also looking forward to suitable extensions of the results of[4], for example, to a hot research topic of positive systems with timedelays

Bachelor thesis Do Thi Van Anh

1.4 Mathematical preliminaries

1.4.1 Definitions and lemmas on positive linear systems

In this section, some basic definitions about positive linear systems arefirst introduced Then, the positivity and stability characterization ofpositive systems are presented

Definition 1.4.1 (Discrete-time positive linear systems)

Consider a discrete-time linear system

(1.1)

where x(k) ∈ Rn, w(k) ∈ Rm and y(k) ∈ Rq are the system state, inputand output, respectively A, Bw, C and Dw are real constant matriceswith appropriate dimensions System (1.1) is called a discerete-timepositive linear system (DPLS) if for any x0 ≥ 0 and input w(k) ≥ 0,

it holds that x(k) ≥ 0, y(k) ≥ 0 for k ≥ 0

Definition 1.4.2 (Continuous-time positive linear systems)

Consider a continuous-time linear system

(1.2)

This system is called a continuous-time positive linear system (CPLS)

if for any x0 ≥ 0 and input w(t) ≥ 0, it holds that x(t) ≥ 0, y(t) ≥ 0for t > 0

Definition 1.4.3 For a matrix A ∈ Rm×n, A ≥≥ 0 (respectively,

A  0) means that all elements of A are nonnegative, that is, aij ≥ 0for all i, j (respectively aij > 0), where aij denotes the element located

at the ith row and the jth column

Definition 1.4.4 A matrix A ∈ Rn×n is said to be a Metzler matrix

if all its off diagonal elements are non-negative (aij ≥ 0 for all i 6= j).Lemma 1.4.5 (Positivity [1])

1 System (1.1) is a DPLS if and only if

A ≥≥ 0, Bw ≥≥ 0, C ≥≥ 0, Dw ≥≥ 0

2 System (1.2) is a CPLS if and only if

A is Metzler , Bw ≥≥ 0, C ≥≥ 0, Dw ≥≥ 0

Lemma 1.4.6 (Stability [1])

1 The DPLS (1.1) is asymptotically stable if and only if there exists

a vector p  0 satisfying the following condition

p>(A − I)  0

Bachelor thesis Do Thi Van Anh

2 The CPLS (1.2) is asymptotically stable if and only if there exists

a vector p  0 such that

p>A  0

In the following, we introduce some other important definitions andlemmas which will be used in the sequel

Definition 1.4.7 A vector x = [x1, , xn]> ∈ Rn is said to be sparse

if its l0-norm is small compared to the dimension of the vector, where

Lemma 1.4.8 For any two nonnegative matrices A1, A2 ∈ Rn×n, if

A1 ≥≥ A2 then ρ(A1) ≥ ρ(A2)

Lemma 1.4.9 For a Metzler matrix A, −A−1 ≥≥ 0 if and only ifα(A) < 0

Lemma 1.4.10 For two Metzler matrices A1, A2 ∈ Rn×n, if A1 ≥≥

A2 then α(A1) ≥ α(A2) Moreover, if α(A1) < 0 then −A−11 ≥≥

−A−12

Lemma 1.4.11 For any matrices 0 ≤≤ M∗ ≤≤ M and 0 ≤≤ N∗ ≤≤

N of compatible dimensions, we have

0 ≤≤ M∗N∗ ≤≤ M N

ky(t)k1dt < γ

Z ∞ 0

kw(t)k1dt,

where γ > 0 is given scalar

Chapter 2

controller synthesis for interval

positive linear systems

This chapter is concerned with the design of L1-induced output-feedbackcontroller for CPLSs with interval uncertainties A necessary and suf-ficient condition is derived in terms of matrix inequality ensuring sta-bility and L1-induced performance of the system Based on this, con-ditions for the existence of robust static output-feedback controllersare established and an iterative convex optimization approach is de-veloped to solve the proposed conditions

where x(t) ∈ Rn, w(t) ∈ Rm and y(t) ∈ Rr are the system state, inputand output, respectively

Assume that the positive linear system (2.1) is stable Then, its

L1-induced norm is defined as

where γ > 0 is a given scalar

First, we give the following result by which the value of L1-inducednorm of system (2.1) can be computed directly

Theorem 2.1.1 For a stable positive linear system given in (2.1),

Bachelor thesis Do Thi Van Anh

the exact value of the L1-induced norm is given by

G(t − τ )w(τ )dτ (2.6)From [2], we have

Let ci, bwj, dwij denote the ith row vector, the jth column vector andthe (i, j)-element of matrices C, Bw and Dw, respectively Equation(2.8) can be written as

¯

Gij =

Z ∞ 0

cieAtbwj + dwijδ(t)
dt = −ciA−1bwj + dwij (2.9)

which yields (2.4)

Then, our objective is to develop a novel characterization underwhich system (2.1) is asymptotically stable and satisfies the perfor-mance in (2.3).

Theorem 2.1.2 The positive linear system in (2.1) is asymptoticallystable and satisfies kykL1 < γkwkL1 if and only if there exists a vector

if x(t) ≡ 0, we have y(t) = Dww(t) and from (2.11), kykL1 < γkwkL1

γkw(t)k1dt

Bachelor thesis Do Thi Van Anh

ε1>x(t)dt + V (T ) < 0,and therefore

Z T 0

Since the system is asymptotically stable, when T → ∞ we have

which yields

kykL1 < γkwkL1.Necessity: We assume that system (2.1) is asymptotically stableand satisfies kykL1 < γkwkL1 By Theorem 2.1, the following inequal-

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2.2 Static output-feedback controller

In this section, we consider the robust stabilization problem via staticoutput feedback for positive systems with uncertainties Based onthe analysis results, a static output-feedback controller (SOFC) is de-signed Here, we consider the positive interval system

(2.15)

where the system matrices A, Bw, Cz, Dzw, C are unknown, but belong

to the following interval uncertainty domain:

A ∈ [A, A], Bw ∈ [Bw, Bw], Cz ∈ [Cz, Cz], Dzw ∈ [Dzw, Dzw], C ∈ [C, C]

(2.16)

If the system (2.15) is positive and asymptotically stable for all missible uncertainty domain in (2.16), it is called positive and robustlystable The following theorem provides a performance characteristicfor positive SI systems over the entire uncertain domain in (2.16)

ad-Theorem 2.2.1 Assume the system SI is positive Then, it is bustly stable with kzkL1 < γkwkL1 for any A in[A, A], Bw ∈ [Bw, Bw], Cz ∈[Cz, Cz], Dzw ∈ [Dzw, Dzw] under x(0) = 0 if and only if there exists avector p ≥≥ 0 satisfying

which, based on the results in Theorem 2.2, implies that the system SI

is robustly stable and satisfies kzkL1 < γkwkL1 over all interval doubtdomain under x(0) = 0

Necessity: We assume that SI is robustly stable and satisfies kzkL1 <γkwkL1 under x(0) = 0 By Theorem 2.1.2 we get

1>Cz + p>A  0,

p>Bw + 1>Dzw − γ1>  0which implies (2.17) and (2.18) hold The prove is completed

In the following, the problem of L1-induced static output-feedbackcontroller design (L1SOFCD) is formulated

Problem L1SOFCD: Given system SI positive Find an SOFCu(t) = Ky(t) such that the closed-loop system

SC :



˙x(t) = (A + BKC)x(t) + Bww(t),

(2.20)

Bachelor thesis Do Thi Van Anh

is positive, robustly stable and satisfies the L1-induced performance

Cz + DzKC ≥≥ Cz + DzKC ≥≥ 0 (2.25)

which shows the closed-loop system (2.20) is positive

It follows from (2.20), (2.24), by Theorem 2.2.1, that the loop system (2.20) is strongly stable and satisfies kzkL1 < γkwkL1

closed-equalities in (2.25) do not necessarily hold, which leads to difficulty inthe synthesis of a desired controller for interval positive system (2.15).The nonpositiveness of K facilitates the problem of controller synthe-sis It should be noted that in special cases when B, C, Dz are knownconstant matrices, the constraint K ≤≤ 0 can be removed As for de-signing a sign-indefinite K, it has not been resolved and needs furtherstudy.

Note that the Lyapunov vector p is coupled with the controllermatrix K in (2.23), which can not be easily solved However, fixingmatrix K, (2.23) turns out to be linear with respect to the remaingvariables Thus, we can fix K and solve from (2.21) to (2.24) bylinear programming Therefore, the following iterative algorithm can

be proposed to solve the conditions

(2.26)

with

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Step 2 Fix Ki ≤≤ 0, solve the following optimization problem for

pi ≥≥ 0 and γi > 0 Minimize γi subject to the following constraints

Step 3 For fixed pi, solve the following optimization problem for

Ki ≤≤ 0 Minimize γi subject to the following constraints

Step 4 Set i = i + 1 and Ki = Ki−1, then go to Step 2

Remark 2.2.4 Given system SI positve and suppose that K1 ≤≤ 0,

SC is positive and robustly stable for any A ∈ [A, A], B ∈ [B, B], Cz ∈[C , C ], D ∈ [D , D ], C ∈ [C, C] if and only if there exist a diagonal

matrix P > 0, a scalar  > 0, and matrices L ≤≤ 0, X ≤≤ 0 such that

Remark 2.2.5 The parameter γ is optimized iteratively Note that

γi+1∗ ≤ γi∗ since the parameters obtained in Step 4 will be used as theinitial conditions to get a smaller γ Thus, the convergence of theiteration process is naturally guaranteed

Chapter 3

Applications

This chapter deals with sepecial cases of the design problem considered

in the preceding chapter

3.1 Full-order output: C = I

When C = I, the general problem of the static output-feedback troller design is reduced to a problem of state feedback controller syn-thesis Here, an analytic method is established for the synthesis of

con-a SFC for SIMO intervcon-al positive systems Extensions to the MIMOcase is also presented On the other hand, an L1-gain sparse controller

is designed for positive systems

3.1.1 Controller synthesis for SIMO systems

The following theorem provides conditions for the existence of a SFC.Theorem 3.1.1 For positive system SI with C = I, system SC with

an SFC u(t) = Kx(t) is positive, robustly stable and satisfies kzkL <

γkwkL1 under x(0) = 0 if and only if there exists a matrix K such that

A + BK is Metzler, (3.1)

Cz + DzK ≥≥ 0, (3.2)α(A + BK) < 0, (3.3)

If the elements of A + BK are small enough, (3.3) is more likely

to be satisfied and from (3.5), the L1-gain norm of system (2.20) with

C = I will also be small when the elements of A+BK and Cz+D+zKare small enough

Indeed, for SIMO systems, the off-diagonal element minimizationproblem of A + BK and all the elements of Cz + DzK can be ana-lytically solved To show this, we rewrite B and Dz as vectors given