The relation between continuous time systems and discrete time systems is the main topic of this research. A continuous time system can be transformed into a discrete time system using the Cayley transform. In this process the generator of the semigroup is mapped to a difference operator, the cogenerator. Stability analysis plays a central role in the study of the relation between continuous and discrete time systems. Do stable continuous time systems correspond to stable discrete time systems? This is the main question of this dissertation. For many stable continuous time systems the corresponding discrete time systems are stable as well. In Banach spaces however, several examples are known of stable semigroups where the corresponding cogenerators have unsta- ble power sequences. In Hilbert spaces no such examples are known. It remains an open problem whether for every stable continuous time system the corre- sponding discrete time system is stable as well. This dissertation addresses the main question in three ways. First, a growth bound for the cogenerator is provided for exponentially stable semigroups in Hilbert spaces. Using Lyapunov equations it is shown that for such semigroups the power sequence of the corresponding cogenerator cannot grow faster than ln(n). Second, we extend the class of stable continuous time systems for which the corresponding discrete time systems are stable as well. For this, the notion of Bergman distance is introduced. The Bergman distance defines a metric for semigroups and a metric for power sequences. If the Bergman distance is finite, the two semigroups have the same stability behaviour. This holds for two power sequences as well. Furthermore, the Bergman distance is preserved by the Cayley transform. This enables us to extend this class. Third, the inverse of the generator is taken into account in the stability analysis. For exponentially stable semigroups on Banach spaces similarity is shown between the growth of the semigroup of the inverse and the growth of the cogenerator. For bounded semigroups on Hilbert spaces it is shown that if the semigroup generated by the inverse is bounded, the growth of the cogenerator is bounded as well.

This study deals with the application of a Model Reference Adaptive Control (MRAC) theory in discrete time to an electro-hydraulic servo system. MRAC is very useful to control a plant of which parameters are unknown or vary during operation.An electro-hydraulic servo system is generally a continuous time system. In order to control this system by micro-computer in discrete time, z-transform is used. It should be noted that the plant becomes a non-minimum phase system in discrete time for smaller sampling periods, even if it is a minimum phase system in continuous time, and the plant can no longer be controlled by MRAC in discrete time.Conditions under which the discrete plant is in non-minimum phase are being investigated. If the sampling period is less than about one half of the natural period of the electro-hydraulic system in continuous time, its discrete system using z-transform becomes a non-minimum phase system.The output velocity (or state-variable) of the plant is fed back to decrease the natural period. The modified plant is combined with MRAC in discrete time. This indicates that the problem of short sampling time is greatly reduced.

Reinforcement learning (RL) has been widely used to design feedback controllers for both discrete-time and continuous-time dynamical systems. This technique allows for the design of a class of adaptive controllers that learn optimal control solutions forward in time, and without knowing the full system dynamics. Integral reinforcement learning (IRL) and off-policy RL algorithms for continuous-time (CT) systems, and Q-learning and heuristic dynamic programing for discrete-time (DT) systems have been successfully used to learn the optimal control solutions, online in real time. The application of these methods, however, has been mostly limited to the design of optimal regulators. Nevertheless, in practice it is often required to force the states or outputs of the system to track a reference (desired) trajectory. A unified framework for both tracking and regulation problems is defined here and it is shown here how we can develop online model-free RL algorithms to solve the unified tracking and regulation control problem for both CT and DT systems.

On the discrete and continuous time infinite-dimensional algebraic Riccati equations

The standard state space solution of the finite-dimensional continuous time quadratic cost minimization problem has a straightforward extension to infinite-dimensional problems with bounded or moderately unbounded control and observation operators. However, if these operators are allowed to be sufficiently unbounded, then a strange change takes place in one of the coefficients of the algebraic Riccati equation, and the continuous time Riccati equation begins to resemble the discrete time Riccati equation. To explain why this phenomenon must occur we discuss a delay equation of difference type that can be formulated both as a discrete time system and as a continuous time system, and show that in this example the continuous time Riccati equation can be recovered from the discrete time Riccati equation. A particular feature of this example is that the Riccati operator does not map the domain of the generator into the domain of the adjoint generator, as it does in the standard case.

In 1961 E. G. Albrekht presented a method for the optimal stabilization of smooth, nonlinear, finite dimensional, continuous time control systems. This method has been extended to similar systems in discrete time and to some stochastic systems in continuous and discrete time. In this paper we extend Albrekht's method to the optimal stabilization of some smooth, nonlinear, infinite dimensional, continuous time control systems whose nonlinearities are described by Fredholm integral operators.

Geometric mechanics analyzes mechanical systems in the framework of variational mechanics, while accounting for the geometry of the configuration space. From the late 1970s, developments in this area produced several schemes for geometric control of mechanical systems in continuous time and discrete time. In the mid to late 2000s, geometric mechanics was first applied to state estimation of mechanical systems, particularly systems evolving on Lie groups as configuration manifolds, like rigid body systems. Much of the existing work on geometric mechanics-based estimation has been in continuous time, using deterministic, semi-stochastic and stochastic approaches. While the body of existing literature on discrete-time estimation schemes on Lie groups is not as extensive, the literature on this topic is contemporaneous with continuous-time schemes. This work describes some recent and ongoing research on geometric mechanics-based estimation schemes in continuous and discrete time from the mid-2010s, which were developed using the Lagrange-d’Alembert principle applied to rigid body systems. This approach gives (deterministic) observer designs with strong stability and robustness properties. This work concludes with potential extensions of this approach to mechanical systems with principal fiber bundles as configuration manifolds.

Robust digital design of continuous-time nonlinear control systems using adaptive prediction and random-local-optimal NARMAX model

We provide solution techniques for the analysis of multiplexers with periodic arrival streams, which accurately account for the effects of active and idle periods and of gradual arrival. In the models considered in this paper, it is assumed that each source alternates (periodically) between active and idle periods of fixed durations. Incoming packets are transmitted on the network link and excess information is stored in the multiplexing buffer when the aggregate input rate exceeds the capacity of the link. We are interested in the probability distribution of the buffer content for a given network link speed as a function of the number of sources and their characteristics, i.e., rate and duration of idle and active periods. We derive this distribution from two models: discrete time and continuous time systems. Discrete time systems operate in a slotted fashion, with a slot defining the base unit for data generation and transmission. In particular, in each slot the link is capable of transmitting one data unit and conversely an active source generates one data unit in that time. The continuous time model of the paper falls in the category of fluid models. Compared to previous works we allow a more general model for the periodic packet arrival process of each source. In discrete time, this means that the active period of a source can now extend over several consecutive slots instead of a single slot as in previous models. In continuous time, packet arrivals are not required to be instantaneous, but rather the data generation process can now take place over the entire duration of the active period. In both cases, these generalizations allow us to account for the progressive arrival of source data as a function of both the source speed and the amount of data it generates in an active period.

Nonlinear discrete time systems with inputs in Banach spaces*

This is the first paper in a series of several papers in which we develop a state/signal linear time-invariant systems theory. In this first part we shall present the general state/signal setting in discrete time. Our following papers will deal with conservative and passive state/signal systems in discrete time, the general state/signal setting in continuous time, and conservative and passive state/signal systems in continuous time, respectively. The state/signal theory that we develop differs from the standard input/state/output theory in the sense that we do not distinguish between input signals and output signals, only between the “internal” states x and the “external” signals w. In the development of the general state/signal systems theory we take both the state space X and the signal space W to be Hilbert spaces. In later papers where we discuss conservative and passive systems we assume that the signal space W has an additional Kre1˘n space structure. The definition of a state/signal system has been designed in such a way that to any state/signal system there exists at least one decomposition of the signal space W as the direct sum W = Y ∔ U such that the evolution of the system can be described by the standard input/state/output system of equations with input space U and output space Y. (In a passive state/signal system we may take U and Y to be the positive and negative parts, respectively, of a fundamental decomposition of the Kre1˘n space W.) Thus, to each state/signal system corresponds infinitely many input/state/output systems constructed in the way described above. A state/signal system consists of a state/signal node and the set of trajectories generated by this node. A state/signal node is a triple Σ = (V ; X, W), where V is a subspace with appropriate properties of the product space X × X × W. In this first paper we extend standard input/state/output notions, such as existence and uniqueness of solutions, continuous dependence on initial data, observability, controllability, stabilizability, detectability, and minimality to the state/signal setting. Three classes of representations of state/signal systems are presented (one of which is the class of input/state/output representations), and the families of all the transfer functions of these representations are studied. We also discuss realizations of signal behaviors by state/signal systems, as well as dilations and compressions of these systems. (Duality will be discussed later in connection with passivity and conservativity.)

Recently, the robust output regulation problem for continuous-time linear systems with both input and communication time-delays was studied. This paper will further present the results on the robust output regulation problem for discrete-time linear systems with input and communication delays. The motivation of this paper comes from two aspects. First, it is known that the solvability of the output regulation problem for linear systems is dictated by two matrix equations. While, for delay-free systems, these two matrix equations are same for both continuous-time systems and discrete-time systems, they are different for continuous-time time-delay systems and discrete-time time-delay systems. Second, the stabilization methods for continuous-time timedelay systems and discrete-time time-delay systems are also somehow different. Thus, an independent treatment of the robust output regulation problem for discrete-time time-delay systems will be useful and necessary.

This study presents an effective approach to stabilizing a continuous‐time (CT) nonlinear system using dithers and a discrete‐time (DT) fuzzy controller. A CT nonlinear system is first discretized to a DT nonlinear system. Then, a Neural‐Network (NN) system is established to approximate a DT nonlinear system. Next, a Linear Difference Inclusion state‐space representation is established for the dynamics of the NN system. Subsequently, a Takagi‐Sugeno DT fuzzy controller is designed to stabilize this NN system. If the DT fuzzy controller cannot stabilize the NN system, a dither, as an auxiliary of the controller, is simultaneously introduced to stabilize the closed‐loop CT nonlinear system by using the Simplex optimization and the linear matrix inequality method. This dither can be injected into the original CT nonlinear system by the proposed injecting procedure, and this NN system is established to approximate this dithered system. When the discretized frequency or sampling frequency of the CT system is sufficiently high, the DT system can maintain the dynamic of the CT system. We can design the sampling frequency, so the trajectory of the DT system and the relaxed CT system can be made as close as desired.

In this paper, sliding mode observer design principles based on the equivalent control approach are discussed for a linear time invariant system both in continuous and discrete time. For the continuous case, the observer is designed using a recursive procedure; however, the observer is eventually expressed as a replica of the original system with an additional auxiliary input with a certain nested structure. A direct discrete time counterpart of the sliding mode realization of a reduced order asymptotic observer using the discrete time equivalent control is also developed. Simulation of a linearized truck-trailer during a manoeuvre illustrates the approach. Results show the effectiveness and the finite time convergence characteristics of the proposed discrete and continuous time sliding mode observers.

In this chapter several anti windup control strategies for SISO and MIMO systems are proposed to diminish or eliminate the unwanted effects produced by this phenomena, when it occurs in PI or PID controllers. Windup is a phenomena found in PI and PID controllers due to the increase in the integral action when the input of the system is saturated according to the actuator limits. As it is known, the actuators have physical limits, for this reason, the input of the controller must be saturated in order to avoid damages. When a PI or PID controller saturates, the integral part of the controller increases its magnitude producing performance deterioration or even instability. In this chapter several anti windup controllers are proposed to eliminate the effects yielded by this phenomena. The first part of the chapter is devoted to explain classical anti windup architectures implemented in SISO and MIMO systems. Then in the second part of the chapter, the development of an anti windup controller for SISO systems is shown based on the approximation of the saturation model. The derivation of PID SISO (single input single output) anti windup controllers for continuous and discrete time systems is implemented adding an anti windup compensator in the feedback loop, so the unwanted effects are eliminated and the system performance is improved. Some illustrative examples are shown to test and compare the performance of the proposed techniques. In the third part of this chapter, the derivation of a suitable anti windup PID control architecture is shown for MIMO (multiple input multiple output) continuous and discrete time systems. These strategies consist in finding the controller parameters by static output feedback (SOF) solving the necessary linear matrix inequalities (LMI’s) by an appropriate anti windup control scheme. In order to obtain the control gains and parameters, the saturation is modeled with describing functions for the continuous time case and a suitable model to deal with this nonlinearity in the discrete time case. Finally a discussion and conclusions sections are shown in this chapter to analyze the advantages and other characteristics of the proposed control algorithms explained in this work.