Stability analysis and control of discrete-time systems with delay

The research presented in this thesis considers the stability analysis and control of discrete-time systems with delay. The interest in this class of systems has been motivated traditionally by sampled-data systems in which a process is sampled periodically and then controlled via a computer. This setting leads to relatively cheap control solutions, but requires the discretization of signals which typically introduces time delays. Therefore, controller design for sampled-data systems is often based on a model consisting of a discrete-time system with delay. More recently the interest in discrete-time systems with delay has been motivated by networked control systems in which the connection between the process and the controller is made through a shared communication network. This communication network increases the flexibility of the control architecture but also introduces effects such as packet dropouts, uncertain time-varying delays and timing jitter. To take those effects into account, typically a discrete-time system with delay is formulated that represents the process together with the communication network, this model is then used for controller design While most researchers that work on sampled-data and networked control systems make use of discrete-time systems with delay as a modeling class, they merely use these models as a tool to analyse the properties of their original control problem. Unfortunately, a relatively small amount of research on discrete-time systems with delay addresses fundamental questions such as: What trade-off between computational complexity and conceptual generality or potential control performance is provided by the different stability analysis methods that underlie existing results? Are there other stability analysis methods possible that provide a better trade-off between these properties? In this thesis we try to address these and other related questions. Motivated by the fact that almost every system in practice is subject to constraints and Lyapunov theory is one of the few methods that can be easily adapted to deal with constraints, all results in this thesis are based on Lyapunov theory. In Chapter 2 we introduce delay difference inclusions (DDIs) as a modeling class for systems with delay and discuss their generality and advantages. Furthermore, the two standard stability analysis results for DDIs that make use of Lyapunov theory, i.e., the Krasovskii and Razumikhin approaches, are considered. The Krasovskii approach provides necessary and sufficient conditions for stability while the Razumikhin approach provides conditions that are relatively simple to verify but conservative. An important conclusion is that the Razumikhin approach makes use of conditions that involve the system state only while those corresponding to the Krasovskii approach involve trajectory segments. Therefore, only the Razumikhin approach yields information about DDI trajectories directly, such that the corresponding computations can be executed in the low-dimensional state space of the DDI dynamics. Hence, we focus on the Razumikhin approach in the remainder of the thesis. In Chapter 3 it is shown that by considering each delayed state as a subsystem, the behavior of a DDI can be described by an interconnected system. Thus, the Razumikhin approach is found to be an exact application of the small-gain theorem, which provides an explanation for the conservatism that is typically associated with this approach. Then, inspired by the relation of DDIs to interconnected systems, we propose a new Razumikhin-type stability analysis method that makes use of a stability analysis result for interconnected systems with dissipative subsystems. The proposed method is shown to provide a trade-off between the conceptual generality of the Krasovskii approach and the computationally convenience of the Razumikhin approach. Unfortunately, these novel Razumikhin-type stability analysis conditions still remain conservative. Therefore, in Chapter 4 we propose a relaxation of the Razumikhin approach that provides necessary and sufficient conditions for stability. Thus, we obtain a Razumikhin-type result that makes use of conditions that involve the system state only and are non-conservative. Interestingly, we prove that for positive linear systems these conditions equivalent to the standard Razumikhin approach and hence both are necessary and sufficient for stability. This establishes the dominance of the standard Razumikhin approach over the Krasovskii approach for positive linear discrete-time systems with delay. Next, in Chapter 5 the stability analysis of constrained DDIs is considered. To this end, we study the construction of invariant sets. In this context the Krasovskii approach leads to algorithms that are not computationally tractable while the Razumikhin approach is, due to its conservatism, not always able to provide a suitable invariant set. Based on the non-conservative Razumikhin-type conditions that were proposed in Chapter 4, a novel invariance notion is proposed. This notion, called the invariant family of sets, preserves the conceptual generality of the Krasovskii approach while, at the same time, it has a computational complexity comparable to the Razumikhin approach. The properties of invariant families of sets are analyzed and synthesis methods are presented. Then, in Chapter 6 the stabilization of constrained linear DDIs is considered. In particular, we propose two advanced control schemes that make use of online optimization. The first scheme is designed specifically to handle constraints in a non-conservative way and is based on the Razumikhin approach. The second control scheme reduces the computational complexity that is typically associated with the stabilization of constrained DDIs and is based on a set of necessary and sufficient Razumikhin-type conditions for stability. In Chapter 7 interconnected systems with delay are considered. In particular, the standard stability analysis results based on the Krasovskii as well as the Razumikhin approach are extended to interconnected systems with delay using small-gain arguments. This leads, among others, to the insight that delays on the channels that connect the various subsystems can not cause the instability of the overall interconnected system with delay if a small-gain condition holds. This result stands in sharp contrast with the typical destabilizing effect that time delays have. The aforementioned results are used to analyse the stability of a classical power systems example where the power plants are controlled only locally via a communication network, which gives rise to local delays in the power plants. A reflection on the work that has been presented in this thesis and a set of conclusions and recommendations for future work are presented in Chapter 8

Lyapunov methods for time-invariant delay difference inclusions

Motivated by the fact that delay difference inclusions (DDIs) form a rich modeling class that includes, for example, uncertain time-delay systems and certain types of networked control systems, this paper provides a comprehensive collection of Lyapunov methods for DDIs. First, the Lyapunov–Krasovskii approach, which is an extension of the classical Lyapunov theory to time-delay systems, is considered. It is shown that a DDI is KL-stable if and only if it admits a Lyapunov–Krasovskii function (LKF). Second, the Lyapunov–Razumikhin method, which is a type of small-gain approach for time-delay systems, is studied. It is proved that a DDI is KL-stable if it admits a Lyapunov–Razumikhin function (LRF). Moreover, an example of a linear delay difference equation which is globally exponentially stable but does not admit an LRF is provided. Thus, it is established that the existence of an LRF is not a necessary condition for KL-stability of a DDI. Then, it is shown that the existence of an LRF is a sufficient condition for the existence of an LKF and that only under certain additional assumptions is the converse true. Furthermore, it is shown that an LRF induces a family of sets with certain contraction properties that are particular to time-delay systems. On the other hand, an LKF is shown to induce a type of contractive set similar to those induced by a classical Lyapunov function. The class of quadratic candidate functions is used to illustrate the results derived in this paper in terms of both LKFs and LRFs, respectively. Both stability analysis and stabilizing controller synthesis methods for linear DDIs are proposed

On parameterized stabilization of networked dynamical systems

The problem of stabilizing networked dynamical systems (NDS) in a scalable fashion is addressed. As a first contribution, an example is provided to demonstrate that the standard NDS stabilization methods can fail even for simple linear time-invariant systems. Then, a solution to this issue is proposed, in which the controller synthesis is decentralized via a set of parameterized local functions. The corresponding stability conditions allow for max-type construction of a Lyapunov function (LF) for the full closed-loop system, while neither of the local functions is required to be a local LF. It is shown that the provided approach is non-conservative in the sense that it is able to find a stabilizing control law for the motivating example network, whereas state-of-the-art non-centralized Lyapunov techniques fail. For input-affine NDS and quadratic parameterized local functions, the combined LF synthesis and control scheme can be formulated as a set of low-complexity semi-definite programs that are solved on-line, in a receding horizon manner

Stabilisation of linear delay difference inclusions via time-varying control Lyapunov functions

The stabilisation of linear delay difference inclusions is often complicated by computational issues and the presence of constraints. In this study, to solve this problem, a receding horizon control scheme is proposed based on the Razumikhin approach and time-varying control Lyapunov functions. By allowing the control Lyapunov function to be time varying, the computational advantages of the Razumikhin approach can be exploited and at the same time the conservatism associated with this approach is avoided. Thus, a control scheme is obtained which takes constraints into account and requires solving on-line a low-dimensional semi-definite programming problem. The effectiveness of the proposed results is illustrated via an example that also shows the computational limitations of existing control strategies

On parameterized Lyapunov and control Lyapunov functions for discrete-time systems

his paper deals with the existence and synthesis of parameterized-(control) Lyapunov functions (p-(C)LFs) for discrete-time nonlinear systems that are possibly subject to constraints. A p-LF is obtained by associating a finite set of parameters to a standard LF. A set-valued map, which generates admissible sets of parameters, is defined such that the corresponding p-LF enjoys standard Lyapunov properties. It is demonstrated that the so-obtained p-LFs offer non-conservative stability analysis conditions, even when Lyapunov functions with a particular structure, such as quadratic forms, are considered. Furthermore, possible methods for synthesizing p-CLFs for discrete-time nonlinear systems are discussed. These methods make use of the receding horizon principle and require solving on-line a low-complexity convex optimization problem

On parameterized Lyapunov and control Lyapunov functions for discrete-time systems

This paper deals with the existence and synthesis of parameterized-(control) Lyapunov functions (p-(C)LFs) for discrete-time nonlinear systems that are possibly subject to constraints. A p-LF is obtained by associating a finite set of parameters to a standard LF. A set-valued map, which generates admissible sets of parameters, is defined such that the corresponding p-LF enjoys standard Lyapunov properties. It is demonstrated that the so-obtained p-LFs offer non-conservative stability analysis conditions, even when Lyapunov functions with a particular structure, such as quadratic forms, are considered. Furthermore, possible methods for synthesizing p-CLFs are discussed. These methods require solving on-line a low-complexity convex optimization problem

On the construction of D-invariant sets for linear polytopic delay difference inclusions

The construction of model predictive control schemes for linear polytopic delay difference inclusions (lpDDIs) is a complex task, especially for lpDDIs with large or time-varying delays. The main challenge is to formulate a control scheme that is computationally tractable. A ¿rst step towards such a control scheme is the construction of a set with particular invariance properties, called D-invariance, which allows to formulate the controller for a relatively low-dimensional system and guarantees a type of delay-independent invariance. Therefore, necessary and suf¿cient conditions for the existence of a D-invariant set are presented in this paper. Furthermore, synthesis algorithms for both polyhedral and ellipsoidal D-invariant sets are also derived. The applicability of the proposed results is illustrated via an example

On parameterized dissipation inequalities and receding horizon robust control

This paper considers the standard input-to-state stability (ISS) inequality for discrete-time nonlinear systems, which involves a candidate Lyapunov function (LF) and a supply function that dictates the ISS gain of the system. To reduce conservatism, a set of parameters is assigned to both the LF and the supply function. A set-valued map, which generates admissible sets of parameters for each state and input, is defined such that the corresponding parameterized LF and supply function enjoy the standard ISS inequality. It is demonstrated that the so-obtained parameterized ISS inequality offers non-conservative analysis conditions, even when LFs and supply functions with a particular structure, such as quadratic forms, are considered. For bounded inputs, it is then shown how parameterized ISS inequalities can be used to synthesize a closed-loop system with an optimized envelope of trajectories. An implementation method based on receding horizon optimization is proposed, along with a recursive feasibility and complexity analysis. The advances provided by the proposed synthesis methodology are illustrated for a continuous stirred tank reactor

Further results on stabilization of linear systems with time-varying input delay

This paper deals with stabilization of continuous-time linear systems affected by uncertain time-varying input delay. Existing solutions usually augment the state vector with the previous input and make use of standard robust stability analysis and synthesis techniques. Instead of discarding knowledge of the previous input, we employ a relaxed version of control Lyapunov functions (CLFs) that can cope with the additive term formed by the previous input. By solving an optimization problem on-line, in a receding horizon manner, we allow the CLF to be locally non-monotone,while taking into account state and input constraints.We then propose a special constraint that governs the non-monotonicity of the CLF such that attractivity is still attained. Moreover, we show that for CLFs defined using the infinity norm the developed method can be implemented as a single linear program, which can be solved explicitly via multiparametric programming. The developed theory is validated on a benchmark example: control of a DC-motor affected by time-varying input del