The most important notations and acronyms used throughout in this Thesis are listed below in this section.
Mathematical notations
Z the set of integer numbers
N the set of nonnegative integers
C the set of complex numbers
R the set of real numbers
R+ the set of non-negative real numbers
Rd the space of d-dimensional real column vectors Rd1×d2 the space of all d1×d2 real matrices
xi ith element of vectorx
xT transpose of vectorx
Ld∞ the set of bounded functions R+→Rd
˙
x= dxdt time derivative of x
AT transpose of matrix A
diag{d1, . . . , dd} d×ddiagonal matrix with the elementsd1, . . . , dd in the main diagonal I the unit matrix of appropriate size I =diag{1, . . . ,1}
0 the zero matrix of appropriate size
k · k maximum normkxk:= max1≤i≤d|xi|of a vector x∈Rd
k · kτ the maximum norm of a continuous function x: [−τ,0]→Rddefined by kxkτ := max−τ≤t≤0kx(t)k
k · k∞ supremum norm of a function r∈Ld∞ defined bykrk∞:= supt≥0kr(t)k C(X, Y) the set of continuous functions mapping from X toY
S [−τ,0],Rd
the set of finite sequences {ψ:{−τ,−τ+ 1, . . . ,0} →Rd}, where τ ∈N kψkτ the norm of a sequenceψ∈S([−τ,0],Rd) is defined by
kψkτ := max−τ≤n≤0kψ(n)k.
Chapter1. Introduction 6 Acronyms
BIBO bounded input bounded output
EPCAs equations with piecewise constant arguments IVP initial value problem
LTI linear time invariant
ODEs ordinary differential equations VDEs Volterra difference equations VIEs Volterra integral equations
Chapter 2
Theoretical Background
In this chapter we review the most important concepts and known results which are used or referred to later in the thesis. Research works on control theory, feedback control, delay differential equations, delay difference equations, nonlinear VIEs, nonlinear VDEs, stability and boundedness on the solutions are reviewed.
2.1 Basic notions in systems and control
Control means to regulate, direct, command, or govern. A system is a collection, set, or arrange-ment of elearrange-ments (subsystems). A control system is an interconnection of components forming a system configuration that will provide a desired system response. Hence, a control system is an arrangement of physical components connected or related in such a manner as to command, regulate, direct, or govern itself or another system. A control system provides an output as shown in Figure 2.1.
Figure 2.1: Description of a control system
Nonlinear systems
Any system which does not obey superposition principle is said to be a nonlinear system. Physical systems are in general nonlinear and analysis of such systems is complicated.
7
Chapter2. Theoretical Background 8 A general class of systems can be represented by the following state space model
˙
x(t) =f(x(t), u(t)), t≥t0
y(t) =h(x(t)),
(2.1.1)
wherex(t0) =x0andf :Rd×Rd1 →Rd,h:Rd→Rd2are nonlinear functions for positive integers d,d1,d2. The system or state model (2.1.1) is called time-invariant because the functionsf and h do not depend explicitly on time t; there are more general time-varying systems where the functions do depend on time. The model consists of two functions: the functionf gives the rate of change of the state vector as a function of state x and input u, and the output function h gives the measured values as a function of state x. We can model many physical systems in this form by carefully choosing the state variables and the analysis of nonlinear systems needs more sophisticated methods (see [59,67,69,108,112]).
Linear time invariant (LTI) systems
For linear systems, the state model (2.1.1) takes the special form
˙
x(t) =Ax(t) +Bu(t) y(t) =Cx(t),
(2.1.2)
where x ∈ Rd, u ∈ Rd1, y ∈ Rd2, A ∈ Rd×d, B ∈ Rd×d1, C ∈ Rd2×d and d, d1, d2 are positive integers. If the time is regarded to be a continuous variablet∈R+ in the above system (2.1.2), then the representation is a continuous time. The other possibility is to choose the time set to be discrete which results in a discrete time system model (see [8,67,68,124]).
LTI systems form the basis for engineering design in many situations. They have the advantage that there is a rich and well-established theory for analysis and design of this class of systems.
The importance of LTI system models is twofold: a lot of nonlinear ODEs can be transformed to an LTI one with an appropriate control input function, giving rise to the application of linear controllers on the linearized model.
Feedback systems
In most cases the control aim is reached by using feedback, i.e. the output signal is fed back to the input through a subsystem called controller (see Figure2.3).
We can investigate systems without and with controllers.
Open Loop Systems: A system in which the output has no effect on the control action is known as an open loop system.
Closed Loop Systems: A system which maintains a prescribed relationship between the con-trolled variable and the reference input, and uses the difference between them as a signal to activate the control, is known as a feedback control system. The output or the controlled
Chapter2. Theoretical Background 9
Figure 2.2: Block diagram of an output feedback control system
variable is measured and compared with the reference input. Thus the controlled variable is continuously fed back and compared with the input signal.
Feedback is a central notion in control theory, it is the property of a closed-loop system, which allows the output to be compared with the input to the system such that the appro-priate control action may be formed as some function of the input and output. In some cases we can measure the whole state, i.e., all components of the state. Then the structure of the controller can be a state feedback, which means, that the control input is determined as a function of the states
u(t) =k(x(t)), k:Rd→Rd1.
The linear state feedback law has the formu(t) =−Kx(t), whereK ∈Rd1×dis state gain matrix. In the case of affine state feedback Lawu(t) =−Kx(t) +r(t), whereK ∈Rd1×dis state gain matrix andr(t)∈Rd1 is the reference input (see Figure2.2).
Figure 2.3: Block diagram of state feedback control system
Output feedbacks use only output information to generate input, i.e., u(t) = ¯k(y(t)), ¯k:Rd2 →Rd1.
Chapter2. Theoretical Background 10 BIBO stability
Stability is a basic system property. There are various but related notions in systems and control theory to characterize the stability of a system. Internal or asymptotic stability is when the disturbance acts as an impulse and then the system behavior is analyzed when time goes to infinity, while external or BIBO stability describes the behavior of the system if it is subject to bounded but permanent disturbances.
There are different notions of BIBO stability. First consider the following definition.
Definition 2.1.1. [11, 59] A system is external or BIBO stable if for any bounded input it responds with a bounded output:
ku(t)k ≤M1 <∞=⇒ ky(t)k ≤M2 <∞, t∈[0,∞).
The following definition is asymptotic stability of LTI systems (see e.g., [11,59,93,117]).
Definition 2.1.2. The LTI system (2.1.2) is internally (asymptotically) stable if the solution x(t) of the equation
˙
x(t) =Ax(t), x(t0) =x0 6= 0, t≥t0 fulfills
t→∞lim x(t) = 0.
The following theorem from [11, 59] shows the relationship between BIBO and asymptotic stability in the case of LTI systems.
Theorem 2.1.3. Asymptotic stability implies BIBO stability for LTI systems.
Conversely, Perron’s theorem [98] yields that if the solutions of ˙x = Ax+u, x(0) = 0 are bounded for all bounded u = u(t), then all eigenvalues of A have negative real parts, i.e., the LTI system (2.1.2) is asymptotically stable. Note that this result was generalized for LTI systems with delay in [35,57].
An other version of the BIBO stability was introduced by Wu and Mizukami in [116].
Definition 2.1.4. The system (2.1.1) with state feedback law u(t) = −Kx(t) +r(t) is BIBO stable if there exist some positive constants θ1 and θ2 satisfying
ky(t)k ≤θ1krk∞+θ2, t≥0 for every reference inputr ∈Ld∞.
Chapter2. Theoretical Background 11 This is a stronger requirement compared to Definition2.1.1, since the constants θ1 and θ2 are independent of the reference input r. In the remaining part of the Thesis the notion of BIBO stability is used in the sense of Definition2.1.4.
In recent years, many researchers have focused their interest on the analysis of BIBO stability (see [34,59,73,113,118]) and BIBO stabilization (see [84–86,119,120]). We note that in most of the above papers the notion of BIBO stability is used in the sense of Definition2.1.4.
In Chapter 4 and Chapter 6 we investigate the BIBO stability of nonlinear continuous and discrete systems with delays respectively. To proof the BIBO stability we need some known results on delay differential equations, delay difference equations, VIEs, VDEs, boundedness of the solutions and stability.