Delay-Dependent Conditions for Finite Time ... - Semantic Scholar
Delay-Dependent Conditions for Finite Time Stability of Continuous Systems with Latency D.Lj. Debeljkovic*, I.M. Buzurovic**, Member, IEEE, A.M. Jovanovic*, N.J. Dimitrijevic*** *University
of Belgrade, School of Mechanical Engineering/Department of Control Engineering, Belgrade, Serbia ** Harvard Medical School, Medical Physics and Biophysics Division, Boston, MA, USA ***School of High Applied Professional Education, Vranje, Serbia [email protected], [email protected]
Abstract— In the present study, the practical and finite time stability of linear continuous system with latency has been investigated. The proposed result outlines the novel sufficient stability conditions for the systems represented by the following equation: x′′(t)=A0x(t) - A1x(t - τ). The results can be applied to the analysis of both the practical and finite time stability of the continuous systems with time delay. For the derivation of the finite time stability conditions, the Lyapunov-Krassovski functionals were used. Unlike in the previously reported results, the functionals did not have to satisfy some strict mathematical conditions, such as positivity in the whole state space and possession of the negative derivatives along the system state trajectories. The numerical examples presented in this study additionally clarified the implementation of the methodology, and the calculations of the stability conditions. Generally, it was found that the proposed sufficient conditions were less restrictive compared to the ones previously reported.
I.
INTRODUCTION
Time delay systems or systems with latency have been investigated over the past decades. Mathematical models of the physical systems contained latency for the various engineering systems, such as electrical networks, pneumatic and hydraulic mechanical systems, transmission systems, systems for chemical processes control, etc. In many systems, latency plays a crucial role and consequently, omitting the analysis on that specific parameter can possibly cause different systems' errors. Due to that fact, there is a strong motivation for developing different criteria for system stability analysis that includes the existence of time delays in the technical systems. In this study, the novel conditions for the finite time stability of continuous systems with latency has been proposed. Chronologically, two main approaches have been considered in time delay system stability analyses. One of the approaches does not include any information on time delay in the mathematical formulation of the stability conditions. The second approach incorporates the time delay into the stability conditions. The first group of conditions has been named as the delay-independent criteria, whereas the second group contains the stability dependent criteria. Mathematically speaking, the delayindependent conditions are relatively simple algebraic
equations that can significantly simplify the stability investigations. Furthermore, this study deals with finite time stability of the described systems. This specific type of stability additionally makes the analysis more complicated. The basic concept of finite time stability can be described as an interest to know the boundaries of the system trajectories. Unlike in the Lypunov-type stability, here, the system can be stable, but its dynamical characteristics, such as transient performances or similar, can lead the system toward an unacceptable dynamical behavior. Consequently, in this case, the interest is to investigate the subset of the state space where it is necessary for the system to present the desired dynamics. The subset of systems is usually predefined and depends on each specific system. This type of stability has been known as practical stability. When practical stability was analyzed over the specified time interval, it is called finite time stability. The investigations on this type of stability (sometimes named as technical stability) are important for many engineering systems; some of them were mentioned previously. So far, many practical stability definitions including finite time stability have been reported and investigated. The investigation on this topic has been applied to the continuous time systems with a constant set of trajectory bounds, as in [1]. In the following part, a brief overview of the previous results is presented. Furthermore, the proposed results in this study represent an addition to the previously reported ones. II. CHRONOLOGICAL OVERVIEW OF THE PREVIOUS RESULTS This short overview is related to the class of the continuous linear systems with latency, only to the nonLyapunov stability including the practical and finite time stability. Throughout this paper, the class of the investigated control systems is referred to as LCTDS - the linear continuous time delay systems. The finite time and practical stability conditions of the particular class of non-linear singularly perturbed multiple time delay systems were presented in [2]. Unlike the approach taken here, the similarity was noticed between the definitions presented in [1] and [2]. Various
practical stability definitions were presented in [3-6]. In [3] and [6], the results on finite time and practical stability of the systems were extended and applied to the continuous time-delay systems. The stability conditions were presented as a function of the fundamental matrix using a delay-independent approach. It means that the system latency was not involved as a parameter in the fundamental matrix. The approach based on the matrix measure presented in [4-5] and [7] clarified derivation of the finite time and practical stability conditions for linear systems with latency. In these studies, the Coppel's inequality was used to obtain algebraic delay-dependent sufficient stability conditions. The advantage of the reported approach was in the fact that there was no need to calculate a fundamental matrix of the system. The Bellman-Gronwall lemma was used for the calculation of the novel delay-dependent sufficient expressions for the practical and finite time stability of continuous systems with state delays, [7]. The presented results proved to be efficient in practical calculations. The overview of the results on the topic was presented in [8], highlighting the modified Bellman-Gronwall approach. The presented methodology was used and extended to the continuous non-autonomous continuous systems with latency in the state domain. The conditions presented were valid over the predefined finite time interval, [9].
Real vector space Complex vector space Identity matrix
( )
F = fij ∈ R F
T
ℑ = t0 , ( t0 + T ) ∈ R + .
(3)
The choice of function f assures the continuous dependence of the solutions denoted as x ( t , t0 , x 0 ) with respect to time t and the initial conditions. Value T has either a positive real value or it is equal to + ∞ . Taking this approach, both the practical stability and the finite time stability can be analyzed at the same time. Generally speaking, it is not necessary for condition (4) to be fulfilled: f ( t , 0, 0 ) ≡ 0 .
(4)
Consequently, the origin of autonomous system (1) does not need to be in the equilibrium state. The state space of system (1) has been denoted as R n ,
( ⋅)
is
the
Euclidean
norm.
Function
V : ℑ× R → R is defined to be a potential aggregate
function. In that case V ( t , x ( t ) ) is bounded only if trajectory norm
V& ( t , x ( t ) ) =
F >0 F ≥0
Positive definite matrix Positive semi definite matrix
λ (F)
Eigenvalue of matrix F
x (t )
is bounded as well. An Eulerian
∂V ( t , x ( t ) ) ∂t
T
+ grad V ( t , x ( t ) ) f (
).
(5)
Additionally, the time invariant sets have been defined as follows: S ( ) is an open bounded set; S β is a set of all allowable trajectories (states) of system (1) for ∀t ∈ ℑ . Set S α has been defined to fulfill condition S α ⊂ S β , and it denotes the set of all initial states. Sets S α a S β are
F Euclidean matrix norm of F = λmax ( AT A ) A general mathematical description of the control system was presented using a non-linear set of differential equations with state latency, as follows:
connected and initially defined for system (1). λ ( denotes the eigenvalues of a matrix
( ).
)
λ max and
λ min are the maximum and minimum eigenvalues of the system, respectively.
,
(1)
IV. PREVIOUS RESULTS
where x ( t ) ∈ R n is a state vector, u ( t ) ∈ R m is a control vector, φ ∈ C = C ([ −τ , 0] , R n ) is a functional containing
the admissible initial states, C = C ([ −τ , 0] , R n ) is the Banach's space of the continuous functions.
It is assumed that function f is smooth in order to guarantee existence and uniqueness of the solutions of the system over a finite time interval
trajectory of system (1) has been denoted as
Transpose of matrix F
x (t ) = φ (t ) , −τ ≤ t ≤ 0
(2)
derivation of aggregate function V ( t , x ( t ) ) along the
Real matrix
x& ( t ) = f ( t , x ( t ) , x ( t − τ ) , u ( t ) ) , t ≥ 0
) : ℑ× R n × R n × R m → R n .
n
The following notations were used in this study. Additionally, some preliminaries were given to connect the main results with the reported ones.
n×n
f(
and
III. PRELIMINARIES
R C I
interval [ −τ , 0] to R n with uniformly converging topology. Vector function f fulfills:
It maps
The following LCTDS system has been analyzed throughout this study: x& ( t ) = A 0 x ( t ) + A1x ( t − τ ) .
(6.a)
The initial conditions for these systems are known, and they are represented as in (6.b):
x ( t ) = φx ( t ) ,
−τ ≤ t ≤ 0 .
(6.b)
Matrices A 0 and A1 are constant with adequate dimensions. Several stability definitions of importance have been presented in the sequel. A.
Stability definitions
Definition 1. System (6) is stable with respect to {α , β , − τ , T , x } , α ≤ β if and only if for any x 0 < α implies x ( t ) < β
trajectory x ( t ) condition
where
(⋅)
fundamental matrix of the system without latency [3], [6]. In the case when τ = 0 , and consequently A 1 = 0 , the analyzed system has been reduced to the regular linear control systems. It was shown in [10] that the presented results are valid for the regular systems. Theorem 2. Autonomous system (6.a) with initial function (6.b) is finite time stable with respect to {S α , S β , τ , T } if condition (8) is satisfied:
∀t ∈ [ −∆, T ] , ∆ = τ max , [2].
e
Definition 2. Autonomous system (6) is contractively stable with respect to {α , β , γ , − τ , T , x } , γ < α < β ,
if and only if for any trajectory x ( t ) condition x0 < α implies: (i) stability with respect to {α , β , − τ , T , x } , (ii) there exists
t ∗ ∈ ] 0, T
is the Euclidean norm and Φ 0 ( t ) is the
[ such
that
where
x (t )
< β , t ∈ ℑ , where ζ ( t ) is a scalar function with
the property 0 < ζ ( t ) ≤ α , – τ ≤ t ≤ 0, where α is a real
∀t ∈ ℑ ,
,
(8)
denotes the Euclidean norm, [5].
{α , β , τ , T , µ ( A ) ≠ 0} if condition (9) is satisfied, [6]: 2
e
( )
of Belgrade, School of Mechanical Engineering/Department of Control Engineering, Belgrade, Serbia ** Harvard Medical School, Medical Physics and Biophysics Division, Boston, MA, USA ***School of High Applied Professional Education, Vranje, Serbia [email protected], [email protected]
Abstract— In the present study, the practical and finite time stability of linear continuous system with latency has been investigated. The proposed result outlines the novel sufficient stability conditions for the systems represented by the following equation: x′′(t)=A0x(t) - A1x(t - τ). The results can be applied to the analysis of both the practical and finite time stability of the continuous systems with time delay. For the derivation of the finite time stability conditions, the Lyapunov-Krassovski functionals were used. Unlike in the previously reported results, the functionals did not have to satisfy some strict mathematical conditions, such as positivity in the whole state space and possession of the negative derivatives along the system state trajectories. The numerical examples presented in this study additionally clarified the implementation of the methodology, and the calculations of the stability conditions. Generally, it was found that the proposed sufficient conditions were less restrictive compared to the ones previously reported.
I.
INTRODUCTION
Time delay systems or systems with latency have been investigated over the past decades. Mathematical models of the physical systems contained latency for the various engineering systems, such as electrical networks, pneumatic and hydraulic mechanical systems, transmission systems, systems for chemical processes control, etc. In many systems, latency plays a crucial role and consequently, omitting the analysis on that specific parameter can possibly cause different systems' errors. Due to that fact, there is a strong motivation for developing different criteria for system stability analysis that includes the existence of time delays in the technical systems. In this study, the novel conditions for the finite time stability of continuous systems with latency has been proposed. Chronologically, two main approaches have been considered in time delay system stability analyses. One of the approaches does not include any information on time delay in the mathematical formulation of the stability conditions. The second approach incorporates the time delay into the stability conditions. The first group of conditions has been named as the delay-independent criteria, whereas the second group contains the stability dependent criteria. Mathematically speaking, the delayindependent conditions are relatively simple algebraic
equations that can significantly simplify the stability investigations. Furthermore, this study deals with finite time stability of the described systems. This specific type of stability additionally makes the analysis more complicated. The basic concept of finite time stability can be described as an interest to know the boundaries of the system trajectories. Unlike in the Lypunov-type stability, here, the system can be stable, but its dynamical characteristics, such as transient performances or similar, can lead the system toward an unacceptable dynamical behavior. Consequently, in this case, the interest is to investigate the subset of the state space where it is necessary for the system to present the desired dynamics. The subset of systems is usually predefined and depends on each specific system. This type of stability has been known as practical stability. When practical stability was analyzed over the specified time interval, it is called finite time stability. The investigations on this type of stability (sometimes named as technical stability) are important for many engineering systems; some of them were mentioned previously. So far, many practical stability definitions including finite time stability have been reported and investigated. The investigation on this topic has been applied to the continuous time systems with a constant set of trajectory bounds, as in [1]. In the following part, a brief overview of the previous results is presented. Furthermore, the proposed results in this study represent an addition to the previously reported ones. II. CHRONOLOGICAL OVERVIEW OF THE PREVIOUS RESULTS This short overview is related to the class of the continuous linear systems with latency, only to the nonLyapunov stability including the practical and finite time stability. Throughout this paper, the class of the investigated control systems is referred to as LCTDS - the linear continuous time delay systems. The finite time and practical stability conditions of the particular class of non-linear singularly perturbed multiple time delay systems were presented in [2]. Unlike the approach taken here, the similarity was noticed between the definitions presented in [1] and [2]. Various
practical stability definitions were presented in [3-6]. In [3] and [6], the results on finite time and practical stability of the systems were extended and applied to the continuous time-delay systems. The stability conditions were presented as a function of the fundamental matrix using a delay-independent approach. It means that the system latency was not involved as a parameter in the fundamental matrix. The approach based on the matrix measure presented in [4-5] and [7] clarified derivation of the finite time and practical stability conditions for linear systems with latency. In these studies, the Coppel's inequality was used to obtain algebraic delay-dependent sufficient stability conditions. The advantage of the reported approach was in the fact that there was no need to calculate a fundamental matrix of the system. The Bellman-Gronwall lemma was used for the calculation of the novel delay-dependent sufficient expressions for the practical and finite time stability of continuous systems with state delays, [7]. The presented results proved to be efficient in practical calculations. The overview of the results on the topic was presented in [8], highlighting the modified Bellman-Gronwall approach. The presented methodology was used and extended to the continuous non-autonomous continuous systems with latency in the state domain. The conditions presented were valid over the predefined finite time interval, [9].
Real vector space Complex vector space Identity matrix
( )
F = fij ∈ R F
T
ℑ = t0 , ( t0 + T ) ∈ R + .
(3)
The choice of function f assures the continuous dependence of the solutions denoted as x ( t , t0 , x 0 ) with respect to time t and the initial conditions. Value T has either a positive real value or it is equal to + ∞ . Taking this approach, both the practical stability and the finite time stability can be analyzed at the same time. Generally speaking, it is not necessary for condition (4) to be fulfilled: f ( t , 0, 0 ) ≡ 0 .
(4)
Consequently, the origin of autonomous system (1) does not need to be in the equilibrium state. The state space of system (1) has been denoted as R n ,
( ⋅)
is
the
Euclidean
norm.
Function
V : ℑ× R → R is defined to be a potential aggregate
function. In that case V ( t , x ( t ) ) is bounded only if trajectory norm
V& ( t , x ( t ) ) =
F >0 F ≥0
Positive definite matrix Positive semi definite matrix
λ (F)
Eigenvalue of matrix F
x (t )
is bounded as well. An Eulerian
∂V ( t , x ( t ) ) ∂t
T
+ grad V ( t , x ( t ) ) f (
).
(5)
Additionally, the time invariant sets have been defined as follows: S ( ) is an open bounded set; S β is a set of all allowable trajectories (states) of system (1) for ∀t ∈ ℑ . Set S α has been defined to fulfill condition S α ⊂ S β , and it denotes the set of all initial states. Sets S α a S β are
F Euclidean matrix norm of F = λmax ( AT A ) A general mathematical description of the control system was presented using a non-linear set of differential equations with state latency, as follows:
connected and initially defined for system (1). λ ( denotes the eigenvalues of a matrix
( ).
)
λ max and
λ min are the maximum and minimum eigenvalues of the system, respectively.
,
(1)
IV. PREVIOUS RESULTS
where x ( t ) ∈ R n is a state vector, u ( t ) ∈ R m is a control vector, φ ∈ C = C ([ −τ , 0] , R n ) is a functional containing
the admissible initial states, C = C ([ −τ , 0] , R n ) is the Banach's space of the continuous functions.
It is assumed that function f is smooth in order to guarantee existence and uniqueness of the solutions of the system over a finite time interval
trajectory of system (1) has been denoted as
Transpose of matrix F
x (t ) = φ (t ) , −τ ≤ t ≤ 0
(2)
derivation of aggregate function V ( t , x ( t ) ) along the
Real matrix
x& ( t ) = f ( t , x ( t ) , x ( t − τ ) , u ( t ) ) , t ≥ 0
) : ℑ× R n × R n × R m → R n .
n
The following notations were used in this study. Additionally, some preliminaries were given to connect the main results with the reported ones.
n×n
f(
and
III. PRELIMINARIES
R C I
interval [ −τ , 0] to R n with uniformly converging topology. Vector function f fulfills:
It maps
The following LCTDS system has been analyzed throughout this study: x& ( t ) = A 0 x ( t ) + A1x ( t − τ ) .
(6.a)
The initial conditions for these systems are known, and they are represented as in (6.b):
x ( t ) = φx ( t ) ,
−τ ≤ t ≤ 0 .
(6.b)
Matrices A 0 and A1 are constant with adequate dimensions. Several stability definitions of importance have been presented in the sequel. A.
Stability definitions
Definition 1. System (6) is stable with respect to {α , β , − τ , T , x } , α ≤ β if and only if for any x 0 < α implies x ( t ) < β
trajectory x ( t ) condition
where
(⋅)
fundamental matrix of the system without latency [3], [6]. In the case when τ = 0 , and consequently A 1 = 0 , the analyzed system has been reduced to the regular linear control systems. It was shown in [10] that the presented results are valid for the regular systems. Theorem 2. Autonomous system (6.a) with initial function (6.b) is finite time stable with respect to {S α , S β , τ , T } if condition (8) is satisfied:
∀t ∈ [ −∆, T ] , ∆ = τ max , [2].
e
Definition 2. Autonomous system (6) is contractively stable with respect to {α , β , γ , − τ , T , x } , γ < α < β ,
if and only if for any trajectory x ( t ) condition x0 < α implies: (i) stability with respect to {α , β , − τ , T , x } , (ii) there exists
t ∗ ∈ ] 0, T
is the Euclidean norm and Φ 0 ( t ) is the
[ such
that
where
x (t )
< β , t ∈ ℑ , where ζ ( t ) is a scalar function with
the property 0 < ζ ( t ) ≤ α , – τ ≤ t ≤ 0, where α is a real
∀t ∈ ℑ ,
,
(8)
denotes the Euclidean norm, [5].
{α , β , τ , T , µ ( A ) ≠ 0} if condition (9) is satisfied, [6]: 2
e
( )
