1. Introduction
So-called hybrid dynamic systems, which essentially consist of mixed, and in general, coupled, continuous-time and either digital or discrete-time dynamics, are of an un-doubtable interest in certain engineering control problems. Such interest arises from the fact that there are certain real-world problems which retain combined continuous-time and discrete-time information, and this circumstance is reflected in the dynamics. The continuous-time information is modelled through differential equations (such as ordinary, functional or partial differential equations) while the discrete-time dynamics are modelled through difference equations. In this way, hybrid systems can sometimes be very complex to analyze, since they might involve combinations and couplings of tandems of more elementary subsystems. See, for instance, [
1,
2,
3,
4]. A major requirement in the design of control schemes is stabilization via feedback by synthesizing a stabilizing controller. Even if an open-loop system (i.e., that resulting in the absence of feedback) is stable, there is often a need to improve its stability [
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16]. A useful procedure to discuss both stability and stabilization concerns is the use of Lyapunov-type or Corduneanu-type functionals and their generalizations (for instance, Lur’e, Krasovskii, Razumikhin, Popov, etc.). See, for instance, [
1,
2,
3,
4,
5,
8,
9,
10,
11,
12,
13,
14,
15,
16] and references therein.
To fix basic ideas on hybrid systems, note that a well-known typical elementary example of such systems is that consisting of a continuous-time system in operation under a discrete-time controller. In this way, the controller does not need to keep information on the continuous-time signals for all times, but only at sampling instants. Other typical hybrid systems involve the combined use of neural nets and fuzzy logic to operate on the continuous-time and/or discrete-time dynamics, or electrical and mechanical drivelines. On the other hand, hybrid dynamic systems with coupled continuous-time and digital dynamics have been described in [
17]. Their properties of controllability, reachability and observability have been characterized in [
18,
19,
20,
21] and some of the references therein. Adaptive control methods for such systems in the case of a partial lack of knowledge of their parametrical values have been addressed in [
22,
23], while optimal “ad hoc” designs have been stated and discussed in [
24] and some of the references therein. In the above topics, it might be important to adapt the design to the multirate context, since sometimes the discretized states and/or the inputs can be subject to different sampling rates, either due to accommodating the design to the nature of such signals or improving the control performances. The finite-time stabilization of multirate networked control systems based on predictive control is discussed in [
25]. Another more general problem which can be considered in combination with different multirate designs is the eventual use of time-varying sampling rates, again to better accommodate the expected performances by adapting the sampling rates to the rates of variations in the involved signals [
26].
Dynamic systems in general, and some hybrid dynamic systems in particular, can also typically involve linear and non-linear dynamics, and they can be subject to the presence of internal delays (i.e., in the state vector) and/or external delays (i.e., in their inputs or outputs). See, for instance, [
1,
2,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16]; although, it must be pointed out that the related background literature is extensive. Typical existing real-life systems involving delays include a number of biological models, such as epidemic models, population growth or diffusion models, sunflower equation, war and peace models, economic models, etc.
This paper formulates and describes a class of linear time-varying, continuous-time systems with time-varying, continuous-time delayed dynamics. Such a class of systems is hybrid in the sense that it can consider an added contribution of delayed dynamics to its current continuous-time dynamics with respect to previously sampled values of the solution, for a certain defined sampling period. Such a dynamic contributes to the whole solution, together with both the delay-free, continuous-time dynamics and the continuous delayed dynamics. The latter is associated with a time-varying, continuously differentiable delay, which is, in general, unbounded and of a continuous-time derivative nature, being everywhere less than one. The class of hybrid systems under study might also be subject to linear output-feedback time-varying control under combined information of the output at the current time instant, the delayed one and the previous discrete-time value in a closed-loop configuration. The general solution is calculated in a closed explicit form. Special emphasis is paid to the closed-loop stabilization via linear output feedback through the appropriate design of the stabilizing control matrices. The stabilization process is investigated via Krasovskii–Lyapunov functionals.
Next, the paper deals with the derivation and analysis of sufficiency-type conditions for the closed-loop asymptotic stability, which are obtained through the definition of two Krasovskii–Lyapunov functional candidates. One of those functional candidates has a constant, leading positive-definite matrix to define the non-integral part as a quadratic function of the solution value at each time instant, while the second candidate proposes a time-varying, time-differentiable matrix function for the same purpose. There are also some extra assumptions invoked which focus on the maximum variation of the time-integral of the squared norms of the remaining matrices of delayed dynamics associated with both the continuous-time delay and with the memory on the sampled part of the hybrid system. These extra assumptions essentially rely on the fact that those time integrals vary more slowly than linearly, with any considered time interval length, in order to perform the integrals over time. The subsequent part of the manuscript is devoted to controller synthesis for the eventual achievement of closed-loop stabilization via linear output feedback, in such a way that the asymptotic stability results of the previous section are fulfilled by the feedback system. In the time-invariant, delay-free case, there are some background results available on stabilization via static linear output feedback (see, for instance, [
27,
28,
29] and some of the references therein). The synthesized controller possesses several gain time-varying matrix functions. One is designed to stabilize the delay-free dynamics, while the remaining ones have, as their objective, minimization in some appropriate sense of the contribution of the natural and the sampled delayed dynamics to the whole closed-loop dynamics. To stabilize the delay-free matrix of dynamics, the controller gain matrix function is calculated via a Kronecker product of matrices [
29,
30], associated with an algebraic system. The problem is well-posed, provided that such a system is compatible for some suitable matrix function describing the delay-free closed-loop dynamics. In case the mentioned algebraic system is not compatible, the controller gain is synthesized so as to approximate the resulting closed-loop matrix to a suitable dynamic in a best approximation context of its norm deviation, with respect to the prefixed and suitable closed-loop matrix of delay-free dynamics. This paper also discusses how to synthesize the remaining matrices, which involve natural delays, and the delayed dynamics associated with the discrete information, in such a way that the resulting matrix function of delayed dynamics has small norms in a sense of the best approximation to zero.
It can be pointed out that the previously cited literature on hybrid systems does not rely on the output-feedback stabilization of systems, which include both discrete information on the previously sampled solution values and combinations of both delay-free, continuous dynamics and delayed, continuous, time-varying dynamics. This paper also focuses on the closed-loop stabilization of the solution via linear output feedback. These concerns are the main novelty of this manuscript, and also the motivation for the study, since the class of hybrid systems under consideration is more general than those previously studied in the literature.
The paper is organized as follows.
Section 2 states and describes the linear hybrid time-varying continuous time system with combined time-varying delay-free and delayed dynamics, as well as its solution in closed explicit form in both unforced and forced cases. The forced solution also considers a particular situation where the forcing control is obtained via linear feedback of combined information on the current output, the delayed output and the previously sampled value of the output.
Section 3 deals with derivation of sufficiency-type conditions of closed-loop asymptotic stability, which are obtained through the definition of two Krasovskii–Lyapunov functionals for asymptotic stability analysis purposes. One involves a constant positive-definite matrix for the definition of the delay-free term, while the other involves a positive-definite time-varying continuous-time differentiable matrix. Controller synthesis for closed-loop asymptotic stabilization via linear output feedback is also discussed. Finally, conclusions end the paper.
Nomenclature
The following notation is used:
is a set of positive real numbers and is a set of non-negative real numbers. Similarly, the positive and non-negative integer numbers are defined by the respective sets and .
Let , then denotes that the matrix is positive-definite; denotes that it is positive-semidefinite; (respectively, ) denotes that it is negative-definite (respectively, negative-semidefinite); ; .
If and , then is the Kronecker product of the matrices and , and .
A square real or complex matrix is a stability matrix if all its eigenvalues have negative real parts.
denotes the Moore–Penrose generalized inverse, or Moore–Penrose pseudo-inverse, of . If then there exists and such that A = CD and . It satisfies the conditions , and it coincides with the inverse of if is square non-singular.
A closed-loop system, in the standard terminology, is that resulting from a state or output-feedback control law. The stability is termed to be global if the solution is bounded for all time and any given admissible function of initial conditions. It is of global asymptotic type if, in addition, it converges asymptotically to the equilibrium state.
We pay special attention in this manuscript to the synthesis of a stabilizing output linear feedback control. In the context of this manuscript, a hybrid system is one which involves mixed continuous-time and discrete-time dynamics. We consider that, in general, it also involves delayed continuous-time dynamics and discrete-time dynamics associated with a given sampling period.
2. The Hybrid Continuous-Time/Discrete-Time Differential System Subject to a Time-Varying Delay
Consider the following dynamic control system subject to, in general, a time-varying delay:
under a bounded piecewise continuous function of initial conditions
, where
is the sampling period,
,
,
and
are, respectively, the state solution on
and the output and input vector functions with
and
;
with
and
;
. The matrix functions of dynamics
,
and
, and the control
and output
matrix functions, are piecewise, continuous and bounded. The control vector is piecewise and constant with eventual finite jumps at the sampling instants
;
(the set of non-negative integer numbers) and is the input (or control) vector
; with
;
(the set of non-negative real numbers), and
is the time-varying delay subject to
;
and
finite. The above system is continuous-discrete hybrid in the sense that the state evolves forced by its current value at time
with a memory effect on its last preceding sampled value at the sampling instant
under a periodic sampling of period
and the control operating jointly at both instants
and
. The major interest of the subsequent investigation is the output-feedback controls of the form:
where
,
and
are the controller gain matrices to be synthesized and
. The replacement of the output vector by the state vector in (3) leads to the most restrictive state output-feedback control type. Through the paper, we will refer to (1) and (2) as the open-loop system, since the control via feedback is not yet selected. Its unforced solution is that corresponding to just the initial conditions, that is, when
. The forced solutions correspond to nonzero controls. Note that the controlled system (1) and (2) as well as the closed-loop configuration (1)–(3) resulting via feedback control are parameterized, in general, by time-varying matrices. The closed-loop system is the combination of (1) to (3), that is, that resulting after replacing the control law (3) in (1). The solution of (1) is characterized in the subsequent theorem.
Theorem 1. The solution of the unforced system (1), for any bounded piecewise continuous function of initial conditions, is unique and given by:where the evolution matrix functionis subject tofor,(the-the identity matrix);, and it satisfies:where the dot symbol denotes the time derivative with respect to the first argument t. The whole solution of (1), including the unforced and the forced contributions, is:with.
Proof. The uniqueness of the solution is obvious since the matrix functions which parameterize (1) are bounded, piecewise, and continuous, and the expression (4), subject to (5), is the solution of the unforced (1), as it can be directly verified as follows. One obtains by replacing (5) into the time-derivative of (4) with the subsequent use of the claimed solution (4):
with
for
, thus (7) coincides with the unforced differential system (1) so that the unforced solution is (4) and the evolution matrix function
subject to
for
,
satisfies (5). As a result, the whole solution of (1) is (6). □
Remark 1. Ifcommutes withfor allthen the evolution matrix function of (1) which is the solution to (5) is:for. In particular, ifis constant, thenfor. □ An interesting property of the evolution matrix through time is given in the subsequent result, which is useful to characterize analytically and eventually compute the solution:
Proposition 1. Consider arbitrary time instants. Then, the evolution matrix function satisfies: Proof. The first and the right-hand-side expressions of (10) have to be identical for any given function of initial conditions so that (9) holds. □
Let us define by
the strip of the solution of
the interval
for the given function of initial conditions
, with
being
. In accordance with (4), define the interval-to-point evolution operator
as follows:
for any
t ≥
t0 ≥ 0, where
is the space of the unforced solutions of (1), for any given function of initial conditions
with
for
, so that, for any
,
so that the evolution operator satisfies for
:
It can be noticed that the interval-to-point evolution operator is related to the evolution matrix function via the identities (12), and, under the additional assumption that the delay function is non-increasing discussed in the subsequent result, it is also related to an interval-to-interval evolution operator.
Proposition 2. If is non-increasing, then the following properties hold:
- i
andfor any.
- ii
Define the interval-to-interval evolution operatoras follows for any:
so that for any t0, t1(≥t0), t2(≥t1) ∈ R0+, one has:andis a strongly continuous one-parameter semigroup. Proof: follows directly since is non-increasing. Now assume, on the contrary to the second property, that for some . Then, which contradicts that is non-increasing. Thus, for any so that Property (i) is proved. Note that, since is non-increasing, then (13) is well-posed, since for any and (14) follows from (12), (13). Now, note that is the identity operator on for any , (see (14)), and so that the interval-to-interval evolution operator is continuous in the strong operator topology. Property (ii) has been proved. □
Note that Proposition 2 also holds in particular if the delay is constant.
The following result is closely related to Theorem 1, except for that the hybrid system considers the contribution of the dynamics of the last preceding sampling instant to the current continuous one instead of the delay between them both.
Corollary 1. Consider the differential system:
The unforced solution for any bounded, piecewise, continuous function of initial conditionsis unique, and given bywhere the evolution matrix functionis subject tofor,;, and it satisfies:and the whole solution of (1) is:withand;. □
The proof of Corollary 1 is similar to that of Theorem 1 by noting that an auxiliary delay for allows us to write and , which leads to (17) being identical to (5) for such a delay. Note that the hybrid continuous/discrete differential system (15) has a finite memory contribution of the state and control at the sampling instants on each next inter-sample time interval, which is incorporated into the continuous-time dynamics.
Remark 2. The unforced and the total solutions (16) and (18) of (1) can also be written equivalently as follows, by taking initial conditions on the interval: The closed-loop differential system (1) is obtained by replacing the feedback control (3) into (1), taking into account (2), to yield:
where
The solution of (21) and (22) is found directly by replacing the evolution matrix function of Theorem 1 by that associated with (21), subject to (22), which leads to the subsequent result:
Theorem 2. The solution of the closed-loop differential system (21) and (22) for any given bounded, piecewise, continuous function of initial conditions, is unique, and given by:with;, where the evolution matrix functionis subject tofor,;, and it satisfies: Remark 3. A parallel conclusion to that of Remark 1 for the closed-loop system is that, if A(t) commutes withfor all, then the evolution matrix function of (23), and solution of (21) subject to (22), isfor.
The following result addresses the fact that the global Lyapunov stability and asymptotic stability for any bounded function of initial conditions of the unforced differential systems (1) and (15), and that of the closed-loop hybrid system (21) and (22), obtained via the feedback control law (3), depend directly on the boundedness and vanishing conditions of their respective evolution matrix functions.
Theorem 3. The following properties hold:
- i
The unforced system (1) is globally stable in the Lyapunov´s sense, if, and only if, the evolution matrix function, being the solution to (5), and its given constraints, is bounded for anyand, with, and any given bounded functions of initial conditions. The unforced system (1) follows Lyapunov´s global asymptotic stablity, if and only if, in addition,as.
- ii
The unforced system (15) follows Lyapunov´s global stability, if and only if the evolution matrix function, being the solution to (17), and its given constraints, is bounded for anyand, with, and any given bounded functions of initial conditionsfor all. The unforced system (15) follows Lyapunov´s asymptotic global stability if, in addition,asand.
- iii
The closed-loop system (21) and (22), obtained from (1) under the control law (3), is globally Lyapunov´s stable if and only if the evolution matrix function, being the solution to (17), and its given constraints, is bounded for anyand, with, and any given bounded functions of initial conditionsfor all. The closed-loop system is globally Lyapunov´s asymptotically stable if, in addition,asand.
- iv
The time-derivative matrix functions given by (5), (17) and (24) of the respective evolution operators of (1), (15), (21) and (22) are bounded for all time if such respective operators are bounded for all time, andfor (1) and (15) andfor (21) and (22) are, in addition, uniformly continuous. Furthermore,for (1) and (15) andfor (21) and (22), asandif their respective evolution operators converge to zero asymptotically, asandprovided that.
Proof. Property (i). Note that, in order for (4) to be bounded, for all time for any given for any given , the evolution operator being the solution to (5) has to satisfy ; , . The converse is also true in the sense that if such a norm is bounded then is bounded for all time for any given finite . Thus, ; , is a necessary and sufficient condition for the global Lyapunov´s stability of the unforced differential system (1). This condition, together with as , guarantees, in addition, that as , and vice versa, so that the unforced differential system (1) is globally Lyapunov´s asymptotically stable, i.e., asymptotically stable for any bounded initial conditions. Property (i) has been proved. Properties (ii)–(iii) are proved in a similar way via equations (15) to (17), (21), (22), (23) and (24), respectively. Property (iv) follows directly from the above properties in view of expressions (5), (15) and (24), since the parameterizing matrix functions of the differential systems (1), (15), (21) and (22) are bounded for all time. The uniform continuity of the respective evolution operators follows from the continuity of their time-derivative operators. □