Some control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive linear systemsSome control problems forpositive 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
Control systems comprise three key physical components: input, output, and state variables Input signals, whether intentional (e.g., controllers) or external (e.g., disturbances), initiate system events Outputs, on the other hand, are measured or regulated quantities (e.g., measurements, observers, controlled outputs) State variables, represented by mathematical functions or physical variables, describe system behavior under known inputs These components collectively contribute to the control and manipulation of system performance.
In many practical models, relevant states such as liquid levels in controlling tanks, concentrations of chemicals, the population size of species or the number of molecules are always nonnegative Such models are described in the state-space representation by dynamical systems, whose states and outputs driven by nonnegative inputs (including initial states) are nonnegative all the time This particular category of systems are referred to as positive systems [32] or nonnegative systems [37] (throughout this thesis we only mention as positive systems) A typical example of positive systems is com- partmental networks [79] A compartment can be viewed as a conceptual storage tank containing an amount of material which is kinetically homogeneous, where kinetically homogeneous means that any material entering the compartment is instantaneously mixed with the material of the compartment A compartmental network is a network consisting of several homogeneous compartments, describing the exchanges of nonneg- ative quantities of materials among compartments and the environment with conser- vation of mass of materials The framework of compartmental networks is also useful to establish other models which are subject to conservation laws Many other practical applications of positive systems have been found in a variety of disciplines from biology, ecology and epidemiology, chemistry, pharmacokinetics to air traffic flow networks, con- trol engineering, telecommunication and chemical-physical processes [6, 25, 51, 55, 82].
Apart from a wide range of applications, positive systems possess many elegant properties that have yet no counterpart in general linear systems [12] For instance, by the robustness and monotonicity [79] induced from the positivity, positive systems are highly evolved in designing interval observers [14, 29, 30, 83], 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 [8] or stability analysis of nonlinear time-delay systems [63, 66] Moreover, many problems known to be NP-hard, in general, turn out to be deceptively simple in the context of linear positive systems [13] Due to widespread applications and special characteristics, the systems and control theory of linear positive systems has received ever-increasing interest in the past decades (see, e.g., [12, 13, 48, 50, 72, 82] and the references therein) 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 Considerable research attention has been devoted to such problems with numerous results have been reported recently For a number of references, we refer the readers to [13, 23, 45, 46, 50, 60, 64] for various problems concerning stability analysis, controller synthesis [28, 82, 90, 95] and L1/L∞ control [11, 17, 73, 74, 76, 87].
Stability theory, central to systems and control theory, prioritizes understanding system equilibrium Its relevance extends across economic, financial, environmental, and engineering disciplines Stability ensures systems remain close to equilibrium and potentially return to it (asymptotic stability), maintaining system functionality.
In other words, the equilibrium is insensitive to small perturbation of initial conditions.
This feature is usually termed as Lyapunov stability [61] Among various concepts of stability that arise in the study of control systems, stability in the sense of Lyapunov and its variants such as asymptotic stability or exponential stability have been well- recognized as common characterizations of stability of equilibrium points with regard to the convergence of state trajectories as time tends to infinity Other stability concepts such as bounded-input bounded-output stability, input-to-state stability or finite-time stability are also significant in control engineering applications [58].
Time delays are prevalent in engineering systems, such as multi-agent systems, transmission lines, and logistic networks These delays can negatively impact system performance and stability, making their study crucial in control engineering While general linear systems theory applies to positive systems, the positivity constraints present new challenges in analysis and synthesis Traditional methods are often ineffective as positive systems are defined on convex polyhedral cones rather than linear spaces This leads to unique properties that cannot be preserved under similarity transformation, hindering the direct application of existing results from linear systems Research in positive systems aims to leverage the monotonicity induced by positivity to simplify behavioral analysis and facilitate design for achieving desired control specifications The theory of positive linear systems with delays remains an important area of investigation in control theory.
Exogenous disturbances are omnipresent in engineering systems due to many technical issues such as the inaccuracy of data processing, linear approximations or measurement errors [36,42] Typically, the performance of a dynamical system is char- acterized 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 For general linear systems, a popular tool to quantify the system performance is H∞ norm, which is also termed as L2-induced norm (or ℓ2-induced norm for discrete-time systems) The H∞ norm represents the maximum gain of a system which characterizes the worst-case norm of regulated outputs over all exoge- nous inputs with bounded energy [77] Another widely used measure is H2 norm, which quantifies the output variance of the system with the exogenous input [70] Un- like 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 For example, it is often desirable to analyze the maximal mass in a single compartment or the total mass of material in all the compartments of a compartmental networks While L2/ℓ2-norm stands for the energy, which is associated with quadratic of system states, the L∞-norm (ℓ∞-norm) and L1-norm (ℓ1-norm) concern with maximal quantities and the sum of quantities, respectively 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 anal- ysis and synthesis of positive systems Apart from practical considerations, by the use of L1- or L∞-induced gains, many elegant linear programming (LP) characteriza- tions 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 As shown in recent developments, some control problems can be considerably simplified owing to the positivity of dynamic systems It is noted that linear programming, 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 anal- ysis 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 con- trol or adaptive control Specific control strategies are taken according to practical situations Among those strategies, state-feedback control is widely employed because states of a system can describe dynamic behavior completely However, in many cases of control engineering, 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 sys- tems, the positivity of closed dynamics is often required which makes design problem of positive systems more complicated and challenging than general systems [72].
Although many 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 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
As mentioned above, in dealing with controller design problem, new challenges often arise since the state variables are set constraints with convex cones by which many well-established methodologies for general linear systems are not easily adapted [76].
In the literature, this problem was addressed by methods that use Lyapunov functions of quadratic or linear types to derive synthesis conditions in the form of LMIs [27, 35], iterative LMIs [9,71] or LPs [7,75,92] As mentioned in the literature, the LP approach may have a numerical advantage versus the LMI approach since the latter one often gets involve more decision variables and the existing LMI softwares cannot handle large size problems and are not numerically stable Thus, the approach based on
LP is utilized more frequently to derive analysis and synthesis conditions for positive systems [17, 76, 87, 92].
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 [7] 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 es- tablished in the literature to provide a complete solution to the static output-feedback control problem of LTI positive systems For instance, the design method of [7] is only applicable to multiple-input multiple-output (MIMO) systems with additional restric- tions on the controller gain matrix Although it overcomes the drawback of [7], theLP-based technique presented in [75] can only be applied to systems whose output or input matrix has a specific structure The decomposition method proposed in [92] only provides sufficient stabilization conditions, and thus, it cannot help to verify whether a desire static output-feedback controller (SOFC) exists In [69], based on a direct vertex algorithm approach, necessary and sufficient conditions for the state-feedback stabilization of LTI positive systems without delay were derived in terms of LP How- ever, the results of [69] are not easily extended to systems in the presence of time-delay in both state and output vectors.
Although much effort from researchers has been devoted to the stabilization prob- lem via SOFCs of LTI positive systems, a complete solution to this problem especially for positive time-delay systems is still of significance Specifically, the feasibility of existence conditions of static output-feedback stabilization is still a relevant problem for LTI positive systems with delays.
B2 L 1 -gain control of positive linear systems with multiple delays
Stability, performance analysis, and controller synthesis are integral challenges in systems and control theory These problems have received significant attention in the context of positive systems, both with and without delays Research efforts have yielded substantial contributions, providing tools and methodologies for addressing these issues effectively.
29, 46, 83, 93] For dynamical systems without positivity and, especially, for linear sys- tems with delays, the Lyapunov–Krasovskii functional (LKF) method and its variants are most widely used approaches to derive sufficient analysis and design LMI condi- tions [43,91] However, for positive systems, the monotonicity induced by the positivity is utilized as a key feature to construct co-positive Lyapunov functionals [11, 53] and especially to formulate direct comparison techniques [10, 45] This approach not only can help to derive necessary and sufficient analysis conditions [41, 69] but also is effec- tive to derive LP-based synthesis conditions [7,40,92] or simplified LMIs with diagonal Lyapunov matrix variables [12, 27, 50, 62].
The problems of performance analysis and synthesis under L1-gain and ℓ1-gain control schemes have drawn significant research attention in the past few years [16,
17, 57] Typically, in the existing literatures, stability and performance analysis condi- tions are derived in terms of linear or semidefinite programmings using certain types of co-positive Lyapunov functions [11, 87, 96] The controller synthesis problem un- der 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 [12] For example, in [19], 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 con- ditions for the existence of stabilizing controllers even for LTI systems without delay.
In [76], by utilizing a characterization ofL1-gain performance derived in [11], necessary and sufficient stabilization conditions subject to an equality constraint were derived for LTI systems For positive systems with delays, the use of solution representations is no longer suitable for obtaining an L1-gain characterization In addition, a direct ap- proach 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 underL1-gain scheme, the proposed methods, for example, in [11,12,93,97], are still not effective for the feasibility verification problem In other words, with the established methodologies, ones cannot assure whether a desired L1-gain controller exists 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
It is well recognized that exogenous disturbances are omnipresent in engineering systems due to many technical issues such as the inaccuracy of data processing, linear approximations or measurement errors [36] Thus, dealing with dynamical systems with uncertainties, certain schemes in robust control theory are important and effective tools Typically, in robust control, two paradigms are widely used for modeling plant and signal uncertainty associated with signal energy (L2- or ℓ2-norm for continuous- or discrete-time domain) and signal peak value, which give rise to H∞ theory and L1 or L∞ induced theory, respectively [1] The problems of analysis and control under the H∞ performance index, which is to minimize the worst energy-to-energy (L2-L2) gain, have been fully investigated in the literature [15, 67, 68, 80] Robust L2-L∞ or
ℓ 2 -ℓ ∞ stability and controller design, which is to minimize the worst energy-to-peak gain from the noise input to filtering error output, have also been developed for various models of dynamical systems [5, 47, 89].
The H∞ and l2-ℓ∞ performance indices assume energy-bounded disturbance inputs In practice, external disturbances in engineering systems are often persistent and amplitude-bounded, requiring specifications on disturbance energy Thus, existing control schemes are inapplicable Instead, the peak-to-peak gain, which measures the worst-case amplification from disturbance to output, provides a more suitable performance index.
ℓ∞-induced) control problem Roughly speaking,ℓ∞-induced optimal design is to mini- mize the maximum peak-to-peak gain of a closed-loop system that is driven by bounded amplitude disturbances Thus, the ℓ∞-gain minimization is an useful and effective approach to the problem of examining the responses of dynamic systems corrupted by persistently bounded disturbances [4] During the past two decades, the induced
ℓ∞ theory has attracted considerable attention due to its theoretical significance and wide applications It is because that this control scheme is a natural counterpart to the existing H∞ stability theory and can deal with persistently bounded distur- bances, which are more frequently concerned in control engineering [24] We refer the readers to recent works dealing with various control and filtering problems for one-dimensional [26, 31, 38, 52, 54, 84] and two-dimensional systems [2, 3] A common approach in the aforementioned works is the use of Lyapunov or LKFs to derive suffi- cient LMI conditions that guarantee an upper bound ofL∞/ℓ∞-induced gain It should be noted here that the existing method cannot help to derive tractable conditions for optimal peak-to-peak gain as it usually produces much conservativeness in the estab- lished conditions, not only in terms of feasibility but also in terms of computational burden.
For positive systems, according to the nature of positive state trajectories, the applications of linear co-positive Lyapunov functions [53] often lead to analysis and syn- thesis 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
Research topics
This thesis is concerned with some problems in the systems and control of pos- itive linear systems with delays The research includes the methodology development and establishment of new results, which are utilized to derive necessary and sufficient conditions for the existence of desired controllers Specifically, the following topics will be studied and presented in this thesis.
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≥0, y(t) = Cx(t) +C d x(t−δt), x(t) =φ(t), t ∈[−δ∗,0],
The system under consideration is described by the following state-space model:x(t) = A x(t) + A d x(t-δ t ) + Bu(t)y(t) = C x(t) + C d x(t-δ t )where x(t) ∈ R n , u(t) ∈ R m and y(t) ∈ R p are the state vector, control input and measured output vectors, respectively A, A d ∈ R n×n , B ∈ R n×m , C, C d ∈ R p×n are known real matrices δ t is an unknown time-varying delay confined in interval [0,δ∗], where δ∗ is a prescribed constant.
An SOFC is designed in the form u(t) =−Ky(t) = −KCx(t)−KC d x(t−δ t ), (2) where K ∈ R m×p is the controller gain, which will be determined By incorporating the SOFC (2), the closed-loop system of (1) is presented as ˙ x(t) = (A−BKC)
The objective is to derive testable necessary and sufficient conditions for the exis- tence of an SOFC in the form of (2) that make 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. This topic will be studied and presented in Chapter 2.
C2 L 1 -gain control of positive linear systems with multiple delays
In Chapter 3, we study the problem ofL1-gain control for a class of LTI systems with multiple delays of the following form ˙ x(t) =A0x(t) +
(4) 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 A0, A k , Bw, C0, C k and Dw, k ∈ 1, m, are given real matrices with appropriate dimensions h k and τ k are known scalars involving time delays, φ ∈ C([−d,0],R n ) is the initial condition initializing the system state on interval [−d,0].
System (4) withw= 0 isglobally exponentially stable (GES) if there exist positive constants α and β such that any solution x(t,φ) of (4) satisfies
∥x(t,φ)∥ ≤β∥φ∥ C e −αt , t≥0, where ∥φ∥ C = sup −d≤t≤0 ∥φ(t)∥ In addition, system (4) is said to be L1-stable if it holds simultaneously that (i) for w = 0, system (4) is GES; and (ii) for any nonzero disturbance w ∈ L1(R + ,R n w ), the corresponding solution of (4) satisfies x(t,φ) ∈
0 ∥x(t,φ)∥ 1 dt < ∞ In the first part, we will show that, for positive system (4), GES and L1-stability are indeed equivalent Assume that system (4) is stable (in the sense of GES) The input-output operator is defined as Σ:L1(R + ,R n w ) −→ L 1 (R + ,R n z ), w /→ z, and L1-gain of system (4) under zero initial conditionis
∥z∥ L 1 (5) The objective of Chapter 3 is to
Formulate a characterization ofL1-gain ∥Σ∥ (L 1 ,L 1 ) of system (4)
Derive conditions for L1-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 L1-gain perfor- mance index.
By an alternative approach using Laplace transformation, a characterization of
Using the L1-induced norm of the input-output operator, necessary and sufficient conditions are established for achieving a prescribed level of L1-induced performance By applying optimization techniques, a comprehensive solution to the stabilization problem is derived under an L1-gain control scheme, with the solution conditions formulated as tractable 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 x(k+ 1) =A0x(k) +
(6) 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 A0,Aj,Bw, C0,Cj and
D w , j ∈1, N, are known real matrices of appropriate dimensions d j (k) and h j (k) are unknown time-varying delays which satisfy d j ≤ dj(k)≤dj, h j ≤hj(k)≤hj, (7) where d j ,dj,h j and hj are positive integers involving lower bounds and upper bounds of delays, d= max 1≤j≤N {d j , h j } φ(k) is the sequence of initial states.
System (6) with w= 0 is GES if there exist constantsα∈(0,1),β >0 such that any solution x(k,φ) of (8) satisfies
∥x(k,φ)∥ ∞ ≤ β∥φ∥ ∞ α k , k ∈N 0 , where ∥φ∥ ∞ = max l∈ Z [−d,0]∥φ(l)∥ ∞ In Chapter 4, we first derive necessary and sufficient conditions by which positive system (6) with w = 0 is GES It will also be shown that under the derived conditions system (6) is also ℓ∞-stable in the sense that
!∞ k=0∥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 given γ > 0, system (6) is said to have ℓ∞-gain performance of level γ if ∥Ψ∥ (ℓ ∞ ,ℓ ∞ ) < γ By novel comparison techniques involving steady states of upper and lower scaled systems with peak values of exogenous disturbances, we first establish a characterization for ℓ∞-induced norm of the input-output operator The obtained
Utilizing ℓ∞-gain characterization, necessary and sufficient conditions for ℓ∞-induced performance with a prescribed level are derived The resulting performance analysis enables the synthesis of SOFCs that minimize worst-case amplification from disturbances to regulated outputs subject to peak-to-peak gain This complete solution leverages a vertex optimization technique to address the synthesis problem.
Outline of main contributions
This thesis is concerned with the problems of stabilization, L1-gain and ℓ∞-gain control via static output feedback and state feedback of positive linear systems with delays Main results presented in this thesis can be summarized as follows.
The stabilization of LTI positive systems with time-varying delay in the state and output vectors via static output feedback control is investigated A novel approach based on optimization procedures is proposed to derive necessary and sufficient LP-based conditions for the existence of desired controllers.
2 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.
3 The problem of static output-feedback optimal peak-to-peak gain control is ad- dressed for discrete-time positive linear systems with heterogeneous interval de- lays 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.
The aforementioned results have been published in 03 papers in international journals (ISI, Q1-Q2) and have been presented at
The weekly seminar onDifferential 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.
WorkshopSelected 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-
STATIC OUTPUT-FEEDBACK CONTROL OF POSITIVE LINEAR SYSTEMS WITH TIME-VARYING DELAY
In this chapter, we consider 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 Based on the derived stability conditions, necessary and sufficient stabilization conditions are formulated in terms of matrix inequalities This general setting is then transformed into suitable vertex optimization problems by which necessary and sufficient conditions for the existence of a desired SOFC are obtained The proposed synthesis conditions are presented in the form of linear programming 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.
Consider the following LTI system with delay ˙ x(t) = Ax(t) +A d x(t−δt) +Bu(t), t≥0, y(t) = Cx(t) +C d x(t−δ t ), x(t) =φ(t), t ∈[−δ ∗ ,0],
(2.1) wherex(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,Ad ∈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 andy(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
For system (2.1), an SOFC is designed in the form u(t) =−Ky(t) = −KCx(t)−KC d x(t−δ t ), (2.2) where K ∈ R m×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)
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).
A closed-loop system (2.3) is positive, with x(t) remaining non-negative for all t ≥ 0 and φ ≥ 0, if A c is a Metzler matrix and A dc is nonnegative This condition ensures the stability of the system, as defined by Proposition 2.1.1.
Since system (2.3) is subject to an unknown time-varying delayδt, similar to [60], we consider the following scaled system ˙ˆ x(t) =Acx(t) +ˆ A dc x(tˆ −δ∗), t ≥0, ˆ x(t) = ˆφ(t), t∈[−δ∗,0],
(2.4) where ˆφ(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 x(t)ˆ of (2.4) satisfies
(c) There exists a vector ν ∈R n , ν ≻0, such that ν ⊤ (Ac+Adc)≺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 ν ⊤ (A c +A dc )≽0.
Let ˆx(t) be the solution of (2.4) with constant initial function ˆφ = ν We define a co-positive Lyapunov functional as v(ˆx(t)) =ν ⊤ x(t) +ˆ
Then, we have d dtv(ˆx(t)) =ν ⊤ x(t) +˙ˆ ν ⊤ A dc (ˆx(t)−x(tˆ −δ∗))
=ν ⊤ (Ac+A dc )ˆx(t)≥0, as ˆx(t)≽0 for all t≥0 Therefore, v(ˆx(t))≥v(ˆx(0)) =ν ⊤
≥ν ⊤ ν=∥ν∥ 2 >0, which yields a contradiction as v(ˆx(t)) → 0 when t → ∞ Thus, ν ⊤ (Ac+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): Letη = (η i )∈R n be a positive vector satisfying (2.5) Then, we have
,a c ij +a c dij - η j