Disturbance-to-State Stabilization and Quantized Control for Linear Hyperbolic Systems
We consider a system of linear hyperbolic PDEs where the state at one of the boundary points is controlled using the measurements of another boundary point. Because of the disturbances in the measurement, the problem of designing dynamic controllers is considered so that the closed-loop system is robust with respect to measurement errors. Assuming that the disturbance is a locally essentially bounded measurable function of time, we derive a disturbance-to-state estimate which provides an upper bound on the maximum norm of the state (with respect to the spatial variable) at each time in terms of -norm of the disturbance up to that time. The analysis is based on constructing a Lyapunov function for the closed-loop system, which leads to controller synthesis and the conditions on system dynamics required for stability. As an application of this stability notion, the problem of quantized control for hyperbolic PDEs is considered where the measurements sent to the controller are communicated using a quantizer of finite length. The presence of quantizer yields practical stability only, and the ultimate bounds on the norm of the state trajectory are also derived.
Partial differential equations (PDEs), or distributed parameter systems, have appeared as a tool for modeling several complex physical phenomena, and there is now a considerable literature on analysis and simulation of such systems. More recently, over the past decade, there has been a surge in control community for designing control algorithms for PDEs so that their behavior can be steered towards some desired performance level. This has led the researchers to generalize several control-theoretic questions from the finite-dimensional systems in the context of infinite-dimensional systems. In that spirit, this article formulates a robust stability notion when the measurements used for feedback control in hyperbolic PDEs are subjected to unknown disturbances. In the literature on ordinary differential equations (ODEs), the property of input-to-state stability (ISS) induces this desired robust behavior while regarding the disturbances as exogenous inputs in the closed-loop system. The Lyapunov function based techniques available for verifying ISS are thus generalized in the context of hyperbolic PDEs in this article.
Hyperbolic PDEs represent a class of such infinite dimensional systems, which have been used in modeling physical system such as shallow water equations, and also to model time-delays in engineering systems. Several results on analysis, Lyapunov stability, and feedback control design of hyperbolic systems have been published in the recent past, see the book  for an overview of results.
1.1 System Class
We consider the feedback control for the class of linear hyperbolic PDEs described by the equation
|where is the spatial variable, and is the time variable. The matrix is assumed to be diagonal and positive definite. We call the state trajectory. The initial condition is defined as|
for some function . The value of the state is controlled at the boundary through some input so that
We consider the case when only the measurement of the state at the boundary point is available for each , and this measurement is subjected to some bounded disturbance. We thus denote the output of the system by
where the disturbance may arise due to low resolution of the sensors, uncertain environmental factors, or errors in communication.
We are interested in designing a feedback control law as a function of the output measurement , that is for some operator , which stabilizes the system in some appropriate sense, and the behavior of the closed-loop system is robust with respect to the measurement disturbances. Here, we allow the possibility that may be obtained via a dynamic compensator so that is an operator with memory. In particular, it is desired that the closed-loop trajectories satisfy the following disturbance-to-state stability (DSS) estimate:
for some constants , and a class function. Here, denotes the essential supremum of for contained in , and for given and , denotes the usual Euclidean norm of . The constant is such that it depends on some norm associated with the function and possibly the initial state chosen for the dynamic compensator . The DSS property ensures that in the absence of disturbance, that is , the maximum norm of (with respect to spatial variable) decreases exponentially in time with a uniform decay rate. In the presence of nonzero disturbances, that is , the maximum value of over , at each time , is bounded by the maximum norm of the disturbance over the interval and an exponentially decaying term due to the initial condition of the system. Due to the semigroup property, the conditions we impose on the system to obtain estimate (4), also ensure that if , then also converges to zero with time, see Remark 3 and Section 4.7.
It turns out that the function (of initial state) that we compute to establish (4) is such that, even if , is not necessarily equal to zero. However, using (4), we can obtain an alternate estimate of the form
for some . This estimate guarantees attractivity of the origin in , and only practical stability.
Drawing comparisons from the literature on stability of finite-dimensional systems, it is observed that the estimates of the form (4) and (5) have been studied under the notion of input-to-state stability (ISS), pioneered in , and more generally input-to-state practical stability (ISpS) , respectively. One of the most fundamental results in the ODEs literature, which makes the ISS property extremely useful for design problems, is that the ISS estimates can be equivalently characterized in terms of Lyapunov dissipation inequalities. In our approach, we also propose a controller design which allows us to construct a Lyapunov function for the closed-loop such that the corresponding dissipation inequality is of the same form as in the finite-dimensional case. This proves to be sufficient for arriving at the estimate (4).
The motivation for studying the DSS property comes from the application in quantized control. When the measurement can not be passed precisely to the controller, but has to be encoded using finitely many symbols, one can see in (3) as the error between the actual value and the quantized value of the signal . The quantizers are typically designed to operate over a compact set in the output space. Within this operating region, the quantization error remains constant and hence one expects the state trajectory to converge to a ball around the origin parameterized by the size of quantization error. Hence, to obtain this practical stability, the controller must ensure that the state trajectory remains within the compact set for which the quantizer is designed.
To implement this methodology in the context of PDEs under consideration, the problem is to find a controller which ensures the DSS estimate (4) holds and that the output remains within the range of the quantizer. The DSS estimate also ensures practical stability in this setup since the eventually converges to a ball around the origin whose radius is parameterized by the sensitivity of the quantizer.
1.3 Literature Overview
In case there are no perturbations, that is, , one typically chooses such that the closed-loop boundary condition
satisfies a certain dissipative condition. This control law yields asymptotic stability of the system with respect to -norm , or -norm , depending on the dissipativity criterion imposed on . In the presence of perturbations , one has to modify the stability criteria as the asymptotic stability of the origin can no longer be established.
One finds the Lyapunov stability criteria with -norm and dissipative boundary conditions in . Lyapunov stability in -norm for nonlinear systems is treated in . Thus, the construction of Lyapunov functions in -norm for the hyperbolic PDEs with static control laws can be found in the literature. Because our controller adds dynamics to the closed-loop, the basic idea behind the construction of Lyapunov function for the closed-loop system is to use the ISS property of the hyperbolic PDE and the controller dynamics.
In the literature, one finds various instances where the ISS related tools are used for stability analysis of interconnected systems. In the paper , an integral ISS Lyapunov function is computed for a networks described by a finite-dimensional nonlinear function. Small gain theorem is crucial when interconnecting ISS systems as exploited in [9, 7].
For infinite dimensional systems, the problem of ISS has attracted attention recently but most of the existing works treat the problem with respect to uncertainties in the dynamics. See, for example , where a class of linear and bilinear systems is studied. See also  where a linearization principle is applied for a class of infinite-dimensional systems in a Banach space. When focusing on parabolic partial differential equations, some works to compute ISS Lyapunov functions have also appeared, such as [18, 17]. For time-varying hyperbolic PDEs, construction of ISS Lyapunov functions has also been addressed in .
For hyperbolic systems, when seeking robust stabilization with measurement errors, one could see that the results in  provide robust stability of in space by using static controllers and piecewise continuous solutions. However, the DSS estimate (4) requires stability in space equipped with maximum norm.
For PDEs in general, the results on stability with respect to measurement errors have not yet appeared in the literature; The only exception being the recent work reported in [12, 13] which derives ISS bounds for 1-D parabolic systems in the presence of boundary disturbances but without the use of Lyapunov-based techniques. Such questions have remained unaddressed for hyperbolic PDEs, which is the topic of this paper. Furthermore, the paper also includes a design element in the sense that the controller that achieves the DSS property is also being synthesized. On the other hand, the problem of quantized control has mostly been studied in finite-dimensional systems so far [15, 20, 22, 24], and this paper extends this problem setting to the case of PDEs. While there exist some works on quantized control of finite-dimensional systems in the presence of delays , the model of hyperbolic PDEs treated in this paper is much more general and as such no direct comparison can be drawn between the earlier approaches and the techniques developed in this paper.
To achieve the aforementioned objectives, we propose to use a dynamic controller instead of a static one, as proposed in the conference version of this article . The reason for emphasizing the use of dynamic controllers is that we are looking for a way to bound , for each , which in our knowledge is only possible if a bound on the -norm of is obtained, see Section 2 for an explanation. Existence of solutions in the space requires us to use inputs which are at least absolutely continuous. If we allow perturbations to be discontinuous, static controllers would not yield smooth enough solutions. The dynamic controller is therefore added to smoothen the discontinuity effect of the perturbations.
The addition of dynamic controller introduces a coupling of ODEs and PDEs in the closed loop which makes the analysis more challenging. Results on well-posedness of such coupled systems are proposed resulting in certain regularity of the closed-loop solutions, which is important to obtain appropriate estimates. We use Lyapunov function based analysis to synthesize the controller and guarantee DSS with respect to the perturbation . The results are then used to study the application of quantized control: We establish practical stability of the system, and derive ultimate bounds on the state trajectory in terms of the quantization error. Compared to the conference article , we provide rigorous mathematical proofs of the main results. The stability notions treated in the paper are more general, and several discussions related to connections with other stability notions are also included. Moreover, this article rigorously establishes the existence of solution for the closed-loop system in Theorem 1, which was not addressed in .
2 Refined Problem Formulation
In this section, we recall some preliminaries associated with the solution space adopted for hyperbolic PDEs in our framework. Connections between the DSS notion and the norm associated with the solution space of the PDE are made explicit. Finally, the idea of dynamic controller is proposed to guarantee solutions with appropriate regularity.
For a function , we denote its gradient by or , and for , we denote the gradient with respect to first argument by , or , and the gradient with respect to second argument by , or with the obvious interpretation that and denote the spatial and time variable, respectively. The space comprises functions for which the -th derivative, denoted , exists and . We use the shorthand for the space . The space is naturally equipped with the -norm defined as:
In literature on stability analysis of hyperbolic PDEs, we find several notions of stability depending on the norm with which the solution space is equipped. If we choose to control the -norm of the state trajectory only, the problem is that it doesn’t yield any bounds on , for a given .
2.2 Obtaining DSS using -norm
The motivation for introducing the solution space can be seen in the following proposition:
Given any function such that . It holds that, for every ,
For each , it is observed that
which gives the desired bound. ∎
Proposition 1 basically allows to get the bounds on norm of the state in terms of its norm. Then, one can work with Lyapunov functions which basically quantify norm of the state, and work with its derivative. In the literature, we see that Agmon’s inequality [14, Lemma 2.4] is also used to get a bound on norm in terms of norm as remarked in [18, Remark 4]. The inequality (7) is however different from the conventional Agmon’s inequality [14, Lemma 2.4].
The solution belongs to ;
It holds that and are bounded by the size of the disturbance plus some exponentially decreasing term in time.
To achieve these objectives, the use of static controllers of the form , will result in trajectories which are not differential with respect to spatial variable due to (possibly discontinuous) disturbances, and hence the solutions are not contained in . To remedy this problem, we propose the use of dynamic controllers for stabilization.
2.3 Using Dynamic Controller for -regular Solutions
For system class (1), (2), (3), we are interested in designing control inputs that are absolutely continuous functions of time, so that their derivative is defined Lebesgue a.e. For such inputs, we seek a solution where denotes the space of continuous functions equipped with supremum norm.
More precisely, we consider the problem of designing a dynamic controller with ODEs, which has the form
where the matrices , , and , need to be chosen appropriately. Thus, the resulting closed loop is depicted in Figure 1. Intuitively speaking, by using such a controller, the discontinuities of the output are integrated via equation (8a) which results in being absolutely continuous. The result on existence and uniqueness of solutions for the closed-loop system (1), (2), (3), (8), is formally developed in Section 3. Afterwards, in Section 4, we design the parameters of the controller (8), and derive conditions on the system and controller data which establish the DSS estimate (4).
3 Existence of Solutions
The objective of this section is to develop a result on existence and uniqueness of solutions for the closed-loop system (1), (2), (3), (8), demonstrated in Figure 1. Before presenting our result, we remark that the solutions of hyperbolic PDEs is a well-studied topic. For the intermediate results, we refer the reader to [1, Appendix A] and [5, Chapter 3]. In , the authors first present results with -regularity for the autonomous with . The results for ODE coupled with hyperbolic PDE with with regularity are also proven. However, in these works, with , which introduces certain discontinuities, the solutions with -regularity are not discussed. On the other hand, the well-posedness results are presented for systems with dynamics described by infinitesimal generators of continuous semigroups.
In this section, our contribution lies in presenting a result on well-posedness of the ODE-PDE coupled system of Figure 1. We do so by building on the results described in [1, Appendix A] and [5, Chapter 3].
To do so, we start by constructing the operator as follows:
Next, we introduce the perturbation operator as follows:
where . We now prove a result on the well-posedness of system (11). Because is possibly discontinuous, the classical solutions (where is continuous) do not exist, and one must work with the notion of a weak solution [5, Definition 3.1.6].
Definition 1 (Weak Solution).
Let . If is an infinitesimal generator of a -semigroup , then we call a weak solution to (11) when the following two conditions hold:
where , is the adjoint of the operator , and the inner product is with respect to .
The well-posedness of (11) is now obtained from the results given in [5, Chapter 3]. To invoke these results, the operators and must satisfy certain conditions. The desired properties of these operators are listed in the lemmas that follow, and their proofs are provided in the Appendix.
The operator is an infinitesimal generator of a -semigroup.
The recipe used for proving this result is inspired by the development given in [1, Appendix A, Proof of Thm. A.I], that is, we show that the operators and its adjoint are quasi-dissipative, and that is closed and densely defined. The difference in the calculations arises due to the presence of -dynamics and the domain of depends on the -dynamics. This changes the construction of the adjoint operator as well. One then invokes a generalization of Lumer-Phillips theorem reported in [5, Corollary 2.2.3] to show that the operators with such properties are infinitesimal generators of strongly continuous semigroups.
The operator is a bounded linear operator.
The linearity of the operator is obvious from its definition. For showing boundedness, one should be careful in using the right norms, because embeds a finite dimensional vector in the space .
The operator is an infinitesimal generator of a -semigroup.
The proof then follows by invoking [5, Theorem 3.2.1]. In fact, the resulting semigroup has a lower triangular structure due to special form of and .
The so-called compatibility conditions on the initial condition , that are required for -regularity are imposed by requiring that belong to . Such a condition is essential and hence the choice of depends upon . It is noted that in , the authors propose two compatibility conditions for the initial state because they seek solution . We only need solutions where is -regular, so only one such condition appears in our analysis.
4 Closed Loop and Stability Analysis
As a solution to the problem formulated in Section 2, we now provide more structure for the controller dynamics, and study the stability of the closed-loop system. The conditions on the system parameters that guarantee stability are then provided by constructing a Lyapunov-function.
4.1 Control Architecture and Closed Loop
The controller that we choose for our purposes is described by the following equations:
For the system in the closed loop, the dynamics of the state trajectory are given by
For what follows, we are also interested in analyzing the dynamics of which are derived as follows:
To obtain the boundary condition, from (13c), we have
Substituting for each , we get
Note that the equation (15) would be well defined for with the obvious interpretation that , and . The same interpretation holds for (13c). In the sequel, when carrying out calculations in stability analysis, it will be assumed that , and by the density argument, the same conclusion would hold for , as done in .
4.2 Stability Result
The second main contribution of the paper is to present conditions on the controller dynamics (12) which results in stability of system (1) and robustness with respect to the measurement disturbances . To state the result, we introduce some notation. Let denote the set of diagonal positive definite matrices. For scalars and , let ; let , and for ; and finally, let . We denote by the symmetric matrix
in which are some positive constants, and denotes the transposed matrix block. In the following statement, we denote the induced-Euclidean norm of a matrix by .
Assume that there exist scalars , a matrix , the gain matrix , and the positive constants in the definition of such that
for some scalar . Then, the closed-loop system satisfies the DSS estimate (4) with
Remark 5 (DSS implies ISpS).
For given in (17), we obtain
where we used the obvious relation that . Substituting this bound on in (4), our DSS estimate leads to
This is a more conventional input-to-state practical stability notion in with respect to disturbance . In particular, with , we have practical stability of in in the following sense: For every , there exists such that the following implication holds
It must be noted that, if the initial condition of the closed-loop system is chosen such that and , then . Indeed, since implies that is continuous, and implies that almost everywhere on , we must have .
In the statement of Theorem 2, condition (16a) requires which also appears in the more general context of nonlinear systems  when analyzing stability with respect to -norm. However, the condition (16b) is introduced in our work to compensate for the lack of proportional gain in the feedback law. It definitely restricts the class of systems that can be treated with our approach and relaxing this condition or obtaining different criteria is a topic of further investigation.
At this moment, we do not have a precise characterization of the parameters of system (1) for which (16) admits a solution. As a particular instance, assume that (16a) holds with . In that case, the matrix simplifies greatly as . Using the Schur complement, one can immediately find the constants that result in being positive definite, and hence satisfying (16b). By applying the continuity argument for solutions of matrix inequalities with respect to parameter variations, the solution to (16b) will also hold for , but sufficiently small.
The remainder of this section is devoted to the proof of Theorem 2 and is divided into several steps for the ease of reading. In Section 4.3, we construct a function . By computing the derivative of this function in Section 4.4, an upper bound on along the solutions of (12)-(13) is obtained under condition (16) which yields
4.3 Construction of the Lyapunov Function
The primary idea is to introduce a Lyapunov function and analyze its derivative with respect to time. As a candidate, we choose given by
where is defined as,
where is a diagonal positive definite matrix that will be specified later. Similarly, is given by
where is a diagonal positive definite matrix that will be specified later, and finally is given by
where is a symmetric positive definite matrix that will be specified later.
It is evident that there exist constants , such that, for all , and ,
4.4 Lyapunov Dissipation Inequality
We now derive the bound on that was used in Section 4.3 to obtain the desired ISS estimate. This is done by analyzing the time derivative of each of the three functions in the definition of the Lyapunov function.
Analyzing : Using an integration by parts and recalling that is a diagonal positive definite matrix, the time derivative of yields
where in which denotes the minimal eigenvalue of .
We now impose the boundary conditions by substituting the value of control given in (13c) to get
which results in
Pick a diagonal positive definite matrix such that (16a) holds. We check that
Let for some . With (22), we thus get
We will see in the sequel that the dynamic controller is chosen so that the last term vanishes in the analysis of .
Analyzing : Repeating the same calculations as in the case of , we get
Let , then using (16a) again
and we choose to obtain
It is observed that because and are diagonal matrices, so that
The terms involving appear in because the -dynamics are driven by the output which includes disturbances. Cancelation of certain terms yields