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2009 | Buch

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Delay Compensation for Nonlinear, Adaptive, and PDE Systems

verfasst von: Miroslav Krstic

Verlag: Birkhäuser Boston

Buchreihe : Systems & Control: Foundations & Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Some of the most common dynamic phenomena that arise in engineering practice—actuator and sensor delays—fall outside the scope of standard finite-dimensional system theory. The first attempt at infinite-dimensional feedback design in the field of control systems—the Smith predictor—has remained limited to linear finite-dimensional plants over the last five decades. Shedding light on new opportunities in predictor feedback, this book significantly broadens the set of techniques available to a mathematician or engineer working on delay systems.

The book is a collection of tools and techniques that make predictor feedback ideas applicable to nonlinear systems, systems modeled by PDEs, systems with highly uncertain or completely unknown input/output delays, and systems whose actuator or sensor dynamics are modeled by more general hyperbolic or parabolic PDEs, rather than by pure delay. Numerous examples and a detailed treatment of individual classes of problems will help the reader master the techniques.

Delay Compensation for Nonlinear, Adaptive, and PDE Systems is an excellent reference guide for graduate students, researchers, and professionals in mathematics, systems control, as well as chemical, mechanical, electrical, computer, aerospace, and civil/structural engineering. Parts of the book may be used in graduate courses on general distributed parameter systems, linear delay systems, PDEs, nonlinear control, state estimator and observers, adaptive control, robust control, or linear time-varying systems.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Time delays are ubiquitous in physical systems and engineering applications. A limited list of control applications in which delays arise includes chemical process control, combustion engines, rolling mills, control over communication networks/Internet and MPEG video transmission, telesurgery, machine tool “chatter”, road traffic systems.

Miroslav Krstic

Linear Delay-ODE Cascades

Frontmatter

Chapter 2. Basic Predictor Feedback

Abstract

In this chapter we introduce the basic idea of a PDE backstepping design for systems with input delay.We treat the input delay as a transport PDE, an elementary first–order hyperbolic PDE. Our design yields a classical control formula obtained through various other approaches–modified Smith predictor (mSP), finite spectrum assignment (FSA), and the Artstein–Kwon–Pierson “reduction“ approach. The backstepping approach is distinct because it provides a construction of an infinitedimensional transformation of the actuator state, which yields a cascade system of transformed stable actuator dynamics and stabilized plant dynamics. Our design results in the construction of an explicit Lyapunov–Krasovskii functional and an explicit exponential stability estimate.

Miroslav Krstic

Chapter 3. Predictor Observers

Abstract

In this chapter we develop a result that is an exact dual of the predictor feedback design in Chapter 2. We develop an observer that compensates for a sensor delay, namely, a delay at the plant’s output. Our design is based on observer designs for PDEs with boundary sensors [203].

The main subject of this chapter is an observer design for ODEs with sensor delay. This result is the focus of Sections 3.1, 3.2, and 3.3.

In Sections 3.4, 3.5, and 3.6 we present a side discussion that connects an observer-based predictor feedback design for systems with input delay with the classical Smith controller [201]. In this discussion we focus on structural similarities as well as the specific differences between the two designs, and present a stability analysis showing that the separation principle holds for the observer-based predictor feedback (Section 3.6).

Miroslav Krstic

Chapter 4. Inverse Optimal Redesign

Abstract

The main purpose of a Lyapunov function is the establishment of Lyapunov stability. But what else might a Lyapunov function be useful for? We explore this question in the present chapter and in Chapter 5.

As we shall see, the utility of a Lyapunov function is in quantitative studies of robustness, to additive disturbance and to parameters, as well as in achieving inverse optimality in addition to stabilization.

Miroslav Krstic

Chapter 5. Robustness to Delay Mismatch

Abstract

As we have seen in Chapter 2, the backstepping method helps construct an explicit Lyapunov–Krasovskii functional for the predictor feedback. One benefit of this construction is the possibility of inverse-optimal and disturbance attenuation designs, both presented in Chapter 4.

The second major benefit of the Lyapunov construction is that one can prove robustness of exponential stability of the predictor feedback to a small mismatch in the actuator delay, in both the positive and negative directions.

Miroslav Krstic

Chapter 6. Time-Varying Delay

Abstract

Before we complete Part I of the book, on predictor feedback for linear systems, and before we move on to delay-adaptive control in Part II, we examine a somewhat related problem of linear (time-invariant) systems with time-varying input or output delays.

Predictor-based control for time-varying delays has a rather intuitive extension from the case of constant delay. The key is to calculate the state prediction over a nonconstant window, starting with the current state as an initial condition, and using past controls over a window of time of nonconstant length.

Miroslav Krstic

Adaptive Control

Frontmatter

Chapter 7. Delay-Adaptive Full-State Predictor Feedback

Abstract

Adaptive control in the presence of actuator delays is a challenging problem. Over the last 20 years, several control designs have been developed that address this problem. However, the existing results deal only with the problem where the plant has unknown parameters but the delay value is known. The remaining theoretical frontier, and a problem of great practical relevance, is the case where the actuator delay value is unknown and highly uncertain (of completely unknown value). This problem is open in general even in the case where no parametric uncertainty exists in the ODE plant.

Miroslav Krstic

Chapter 8. Delay-Adaptive Predictor with Estimation of Actuator State

Abstract

In Chapter 7 we solved the problem of adaptive stabilization in the presence of a long and unknown actuator delay, under the assumption that the actuator delay state is available for measurement. In this chapter we dispose of this assumption.

The result that we obtain in this chapter is not global, as the problem where the actuator state is not measurable and the delay value is unknown at the same time is not solvable globally, since the problem is not linearly parametrized.

Miroslav Krstic

Chapter 9. Trajectory Tracking Under Unknown Delay and ODE Parameters

Abstract

In Chapter 7 we presented an adaptive control design for an ODE system with a possibly large actuator delay of unknown length.We achieved global stability under full-state feedback.

In this chapter we generalize the design from Chapter 7 in two major ways: We extend it to ODEs with unknown parameters and extend it from equilibrium regulation to trajectory tracking. These issues were not pursued immediately in Chapter 7 for pedagogical reasons, to prevent the tool from achieving global adaptivity in the infinite-dimensional (delay) context from being buried under standard but nevertheless complicated details of ODE adaptive control.

Miroslav Krstic

Nonlinear Systems

Frontmatter

Chapter 10. Nonlinear Predictor Feedback

Abstract

The difference between the two categories of systems is that those in category 1 will lend themselves to global stabilization by predictor feedback in the presence of an arbitrarily long delay, whereas systems in category 2 may suffer finite escape before the control “kicks in” at t=D and for this reason are not globally stabilizable (the achievable region of attraction will depend on D).

The system category 2 is clear–it includes many systems that have high-growth nonlinearities and where the control must act aggressively to prevent explosive instability. Such systems have been studied using the nonlinear “backstepping” design [112] for strict-feedback systems and other approaches.

Miroslav Krstic

Chapter 11. Forward-Complete Systems

Abstract

As we have seen in Chapter 10, the predictor feedback has no chance of achieving global stability for systems that are prone to finite escape in an open loop, which is due to the absence of control during the actuator’s “dead time”. For this reason, in this chapter we focus on the class of forward-complete systems, which are guaranteed to have solutions that remain bounded (despite a possible exponential instability) for all finite time, as long as the input remains finite. This is not a small class of systems. It includes all linear systems–both stable and unstable. It also includes various nonlinear systems that have linearly bounded nonlinearities, such as systems in mechanics that contain trigonometric nonlinearities, as a result of rotational motions.

Miroslav Krstic

Chapter 12. Strict-Feedforward Systems

Abstract

The similarity in name between forward-complete and strict-feedforward systems is a pure coincidence. For forward-complete systems, “forward” refers to the direction of time. Such systems have finite solutions for all finite positive time. With feed forward systems, the word “forward” refers to the absence of feedback in the structure of the system. The system consists of a particular cascade of scalar systems.

While forward-complete systems yield global stability when predictor feedback is applied to them, the strict-feedforward systems have the additional property that, despite being nonlinear, they can be solved explicitly. Consequently, the predictor state can be defined explicitly. Related to this, the direct infinite-dimensional backstepping transformation can be explicitly constructed.

Miroslav Krstic

Chapter 13. Linearizable Strict-Feedforward Systems

Abstract

Most strict-feedforward systems are not feedback linearizable; however, a small class of strict-feedforward systems is linearizable, and, in fact, it is linearizable by coordinate change alone, without the use of feedback.

In this chapter we review the conditions for the linearizability of strict-feedforward systems, present a control algorithm that results in explicit formulas for control laws, present formulas for predictor feedbacks that compensate for actuator delays (which happen to be nonlinear in the ODE state but linear in the distributed actuator state), derive formulas for closed-loop solutions in the presence of actuator delay, and, finally, present a few examples of third-order linearizable strictfeedforward systems.

Miroslav Krstic

PDE-ODE Cascades

Frontmatter

Chapter 14. ODEs with General Transport-Like Actuator Dynamics

Abstract

In this chapter we start the development of feedback laws that compensate actuator (or sensor) dynamics of a more complex type than the pure delay. Having dealt with the pure delay, i.e., the transport PDE in Chapter 2, in this chapter we expand our scope to general first-order hyperbolic PDEs in one dimension.

We first focus on first-order hyperbolic PDEs alone, without a cascade with an ODE. First-order hyperbolic PDEs serve as a model for such physical phenomena as traffic flows, chemical reactors, and heat exchangers.We design controllers using the backstepping method–with the integral transformation and boundary feedback, the unstable PDE is converted into a “delay line” system that converges to zero in finite time.

Miroslav Krstic

Chapter 15. ODEs with Heat PDE Actuator Dynamics

Abstract

In this chapter we use the backstepping approach from Chapter 2 to expand the scope of predictor feedback and build a much broader paradigm for the design of control laws for systems with infinite-dimensional actuator dynamics, as well as for observer design for systems with infinite-dimensional sensor dynamics.

In this chapter we address the problems of compensating for the actuator and sensor dynamics dominated by diffusion, i.e., modeled by the heat equation. Purely convective/first-order hyperbolic PDE dynamics (i.e., transport equation or, simply, delay) and diffusive/parabolic PDE dynamics (i.e., heat equation) introduce different problems with respect to controllability and stabilization. On the elementary level, the convective dynamics have a constant-magnitude response at all frequencies but are limited by a finite speed of propagation. The diffusive dynamics, when control enters through one boundary of a 1D domain and exits (to feed the ODE) through the other, are not limited in the speed of propagation but introduce an infinite relative degree, with the associated significant roll-off of the magnitude response at high frequencies.

Miroslav Krstic

Chapter 16. ODEs with Wave PDE Actuator Dynamics

Abstract

In Chapter 15 we provided a first extension of the predictor feedback concept to systems whose actuator dynamics are infinite-dimensional and more complex than a simple pure delay. We provided a design for actuator dynamics governed by the heat PDE.

In this chapter we consider actuator dynamics governed by the wave PDE, a rather more challenging problem than that in Chapter 15. To imagine the physical meaning of having a wave PDE in the actuation path, one can think of having to stabilize a system to whose input one has access through a string. The challenges of overcoming string/wave dynamics in the actuation path include their infinite dimension, the finite (limited) propagation speed of the control signal (large control doesn“t help), and the fact that all of their (infinitely many) eigenvalues are on the imaginary axis.

Miroslav Krstic

Chapter 17. Observers for ODEs Involving PDE Sensor and Actuator Dynamics

Abstract

This chapter parallels the development in Chapter 3 but for the more challenging cases where the sensor dynamics are not of a pure delay type but instead aremodeled by heat or wave PDEs (see Fig. 17.1). The chapter consists of two distinct halves, the first half dealing with the heat PDE case in Sections 17.1, 17.2, and 17.3, and the second half dealing with the heat PDE case in Sections 17.4, 17.5, and 17.6.

In Section 17.1 we develop a dual of our actuator dynamics compensator in Chapter 15 and design an infinite-dimensional observer that compensates the diffusion dynamics of the sensor. In Section 17.3 we combine an ODE observer with the full-state feedback compensator of the heat PDE actuator dynamics in Chapter 15 and establish a form of a separation principle, where the observer-based compensator is stabilizing for the overall systems consisting of the ODE plant, ODE observer, heat PDE actuator dynamics, and heat PDE observer. The heat PDE observer is a simple copy of the system since the heat PDE dynamics are exponentially stable, so the observer error for that part of the system is exponentially convergent.

Miroslav Krstic

Delay-PDE and PDE-PDE Cascades

Frontmatter

Chapter 18. Unstable Reaction-Diffusion PDE with Input Delay

Abstract

In this chapter and in Chapter 20 we introduce the problems of stabilization of PDEPDE cascades. First, we deal with PDEs with input delays (the reaction-diffusion PDE in this chapter, and the antistable wave PDE in Chapter 20). Then we deal with cascades of unstable heat and wave PDEs (in either order) in Chapter 20.

Stability analysis for cascades of stable PDEs from different classes, when interconnected through a boundary, virtually explodes in complexity despite the seemingly simple structure where one PDE is autonomous and exponentially stable and feeds into the other PDE. The difficulty arises for two reasons. One is that the connectivity through the boundary gives rise to an unbounded input operator in the interconnection. The second reason is that the two subsystems are from different PDE classes, with different numbers of derivatives in space or time (or both). This requires delicate combinations of norms in the Lyapunov functions for the overall systems.

Miroslav Krstic

Chapter 19. Antistable Wave PDE with Input Delay

Abstract

In this chapter we continue with the designs for delay-PDE cascades as in Chapter 18. Here we deal with an antistable wave PDE, which has all of its infinitely many eigenvalues in the right half-plane (all located on a vertical line).

The wave PDE problem with input delay is much more complex than the delayheat cascade in Chapter 18. The primary reason is the second-order-in-time character of the wave equation, though the “antistability” of the plant also creates a challenge.

Due to the extra complexity of the wave PDE, in this chapter we forego the derivation of explicit closed-loop solutions such as those that we derived in Section 18.8. However, we do derive the explicit expressions for the control gains and present a stability analysis.

We present the design for an antistable wave PDE with input delay in Section 19.1 and explain its origins in the baseline delay for an antistable wave PDE without delay in Section 19.2. The explicit solutions for the controller’s gain kernels are derived in Section 19.3. The stability analysis is presented in two steps, first for the target system in Section 19.4, and then for the system in the original variables in Section 19.5.

Miroslav Krstic

Chapter 20. Other PDE-PDE Cascades

Abstract

In this chapter we deal with cascades of parabolic and second-order hyperbolic PDEs. These are example problems. The parabolic-hyperbolic cascade is represented by a heat equation at the input of an antistable wave equation. The hyperbolic-parabolic cascade is represented by a wave equation at the input of an unstable reaction-diffusion equation.

The topic of PDE-PDE cascades is in its infancy. Its comprehensive coverage is beyond the scope of this book. Unlike the previous chapters in this book, our presentation in this chapter is fairly informal. We do derive the feedback laws and make statements of closed-loop eigenvalues; however, we forego a detailed Lyapunov stability analysis and the associated estimates for the transformations between the plant and the target system.

Miroslav Krstic

Backmatter

Titel: Delay Compensation for Nonlinear, Adaptive, and PDE Systems
verfasst von: Miroslav Krstic
Verlag: Birkhäuser Boston
Electronic ISBN: 978-0-8176-4877-0
Print ISBN: 978-0-8176-4876-3
DOI: https://doi.org/10.1007/978-0-8176-4877-0