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Über dieses Buch

Most nonlinear systems can be modeled as linear systems with constrains on their inputs and selected outputs. Unifying two decades of research, this book is the first to establish a comprehensive foundation for a systematic analysis and design of such systems The authors address the following issues in the work:

* Internal or Lyapunov stability, external L_p stability, and simulatneous internal and external stability

* Control system feedback design laws for different types of stability

* New notions of external stability

* Additional constraints on controller architecture under decentralized internal and external stabilization

* Satisfactory performance, which also guarantees stability


Internal and External Stabilization of Linear Systems with Constraints is an excellent reference for practicing engineers, graduate students, and researchers in control systems theory and design. The book may also serve as an advanced graduate text for a course or a seminar in nonlinear control systems theory and design in applied mathematics or engineering departments. Minimal prerequisites include a first graduate course in state-space methods as well as a first course in control systems design.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
Constraints on inputs and other variables of a dynamic system are ubiquitous. Often they occur in the form of magnitude as well as rate saturation of a variable. Clearly, the capacity of every device is capped. Valves can only be operated between fully open and fully closed states, pumps and compressors have a finite throughput capacity, and tanks can only hold a certain volume. Force, torque, thrust, stroke, voltage, current, flow rate, and so on, are limited in their activation range in all physical systems. Servers can serve only so many consumers. In circuits, transistors and amplifiers are saturating components. Saturation and other physical limitations are dominant in maneuvering systems like aircrafts. Every physically conceivable actuator, sensor, or transducer has bounds on the magnitude as well as on the rate of change of its output. Thus, the saturation of a device presents a hard constraint.
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

2. Preliminaries

Abstract
In this chapter, we bring together the notations and acronyms used in this book as well as various definitions and facts related to matrices, linear spaces, linear operators, norms of deterministic as well as stochastic signals, norms of linear time- or shift-invariant systems, saturation functions, internal (Lyapunov) stability, and external stability.
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

3. A special coordinate basis (SCB) of linear multivariable systems

Abstract
What is called the special coordinate basis (SCB) of a multivariable linear time-invariant system plays a dominant role throughout this book; hence, a clear understanding of it is essential. The purpose of this chapter is to recall the SCB as well as its properties pertinent to this book. The SCB originated in [ 138, 140, 141] and was crystallized for strictly proper systems in [139] and for proper systems in [132]. Our presentation of SCB here omits all the proofs that can be found in the literature.
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

4. Constraints on inputs: actuator saturation

Abstract
This chapter is concerned with designing controllers for linear systems subject to input saturation with the purpose of achieving internal stabilization. This is a prelude to most of the subsequent chapters, and presents basic problem statements of global and semi-global internal stabilization, necessary and sufficient conditions under which such a stabilization can be achieved, as well as control design methodologies that can be utilized for an appropriate design.
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

5. Robust semi-global internal stabilization

Abstract
In the previous chapter, we discussed semi-global internal stabilization of linear systems subject to control magnitude saturation. However, all of this is obtained for an ideal saturation element σ(u). In reality, we are often faced with a saturation element which differs from the ideal saturation but still satisfies some of the basic properties as outlined in Sect. 2.6. One of the objectives of this chapter is to show in which respect the design methodologies such as low-gain and low-and-high-gain, which were described in detail in the previous chapter, still apply in case the saturation function has a different shape.
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

6. Control magnitude and rate saturation

Abstract
Chapter 4 considered global and semi-global stabilization of linear systems with actuators subject to magnitude saturation alone. In this chapter, we revisit the same internal stabilization, however, with actuators subject to both magnitude and rate saturation. Rate saturation refers to the case when actuator outputs cannot change faster than a certain value.
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

7. State and input constraints: Semi-global and global stabilization in admissible set

Abstract
Chapter 4 considers internal stabilization of linear systems subject to control magnitude constraints, while Chap. 6 considers the same however with both control magnitude and rate constraints. Although such constraints on control variables occur prominently, magnitude as well as rate constraints on state variables are also of a major concern in many plants. Nearly every application imposes constraints on state as well as control variables. We observe that dynamic models of physical systems are often nonlinear. Linear approximations of such nonlinear systems are obviously valid only in certain constraint regions of state and control spaces. In process control, state and control constraints arise from economic necessity of operating the plants near the boundaries of feasible regions. In connection with safety issues, state and control constraints are a major concern in many plants. In certain possibly hazardous systems, such as a nuclear power plant, safety limits on some variables are often imposed. The violations of such predetermined safety measures may cause system malfunction or even damage. This implies that magnitude constraints or bounds on states must be taken as integral parts of any control system design.
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

8. Solvability conditions and design for semi-global and global stabilization in the admissible set

Abstract
In Chap. 7, we formulated two important problems, (1) the semi-global stabilization problem in the admissible set and (2) the global stabilization problem in the admissible set. Moreover, based on the structural properties of the mapping from the control input to the constrained output, a taxonomy of constraints was developed there. In view of such a taxonomy, this chapter concentrates on semi-global as well as global stabilization problems in the admissible set. The nature and solvability of these stabilization problems as well as appropriate design of controllers differ profoundly for the two different cases of right and non-right-invertible constraints. Because of this, we consider here these two cases separately. In particular, we consider the case of right-invertible and non-right-invertible constraints, respectively, in Sects. 8.2 and 8.3 for continuous-time systems. Similarly, we consider the same in Sects. 8.4 and 8.5 for discrete-time systems. This chapter is primarily based on our work by Saberi et al. ( Automatica 38(4):639654, 2002; International Journal on Robust & Non-linear Control 14(5):435461, 2004; International Journal on Robust & Nonlinear Control 14(13–14):1087-1103, 2004).
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

9. Semi-global stabilization in the recoverable region: properties and computation of recoverable regions

Abstract
As in Chap. 8, we consider in this chapter constraints on state as well as input variables. As discussed in detail in Chap. 8, if the given system has at least one of the constraint invariant zeros in the open right-half plane (continuous time) or outside the unit disc (discrete time), that is, if it has non-minimum-phase constraints, then neither semi-global nor global stabilization in the admissible set is possible. Thus, whenever we have non-minimum-phase constraints, the semi-global stabilization is possible only in a certain proper subset of the admissible set. This gives rise to the notion of a recoverable region (set), sometimes also called the domain of null controllability or null controllable region. Generally speaking, for a system with constraints, an initial state is said to be recoverable if it can be driven to zero by some control without violating the constraints on the state and input. We can appropriately term the set of all recoverable initial conditions as the recoverable region. The recoverable region is thus indeed the maximum achievable domain of attraction in stabilizing linear systems subject to non-minimum-phase constraints. The goal of stabilization is to design a feedback, say u = f(x), such that the constraints are not violated and moreover the region of attraction of the equilibrium point of the closed-loop system is equal to the recoverable region or an arbitrarily large subset contained within the recoverable region. Such a stabilization is termed as the semi-global stabilization in the recoverable region, and this is what we pursue in this chapter.
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

10. Sandwich systems: state feedback

Abstract
We studied internal stabilization of linear systems subject to actuator magnitude saturation in Chap. 4, and the same in Chap. 6, however, when the actuator is subject to both magnitude and rate saturation. The block diagram of Fig. 10.1 depicts the setup. Although such actuator saturation occurs ubiquitously, in this chapter we consider a broader class of nonlinear systems than that depicted in Fig. 10.1. As pointed out in Chap. 1, an important and common paradigm of nonlinear systems is that they are indeed linear systems in which nonlinear elements are sandwiched or embedded as shown in Fig. 10.2. A model of a common nonlinear element is a static nonlinearity followed by a linear system or vice-versa. In either case, the block diagram of Fig. 10.2 depicts a commonly prevailing situation besides linear systems subject to merely actuator saturation.
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

11. Simultaneous external and internal stabilization

Abstract
So far, we developed various design methodologies that attain internal stabilization in different contexts for linear systems with constraints and in particular for linear systems subject to actuator saturation. We recall internal stability protects against a single impulse-like disturbance, whereas input–output stability or otherwise called external stability or L p ( p ) stability protects against external inputs or noise disturbances.
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

12. Simultaneous external and internal stabilization: input-additive case

Abstract
For linear systems subject to saturation, Chap. 11 formulates five simultaneous global external and global internal stabilization problems and two simultaneous semi-global external and semi-global internal stabilization problems. As discussed there, two distinct cases arise, the first case corresponds to the situation when the control inputs and the disturbance signals are additive, while in the second case, such signals are nonadditive. In this chapter, under the condition that the given system is asymptotically null controllable with bounded control (ANCBC), we present control strategies that solve all the five global problems and the two semi-global problems for both continuous- and discrete-time systems. For state feedback, we will solve these problems completely. However, for the measurement feedback case, only partial results are available..
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

13. Simultaneous external and internal stabilization: non-input-additive case

Abstract
For the case when the external signals appear additive to the control input, Chap.​ 12 develops control strategies for several simultaneous stabilization problems in both global and semi-global setting for both continuous- and discrete-time systems. In this chapter, we tackle the same problems, however, for the case of non-input-additive external signals. Unlike in the input-additive case where our results are more or less complete, we present here only some limited results.
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

14. The double integrator with linear control laws subject to saturation

Abstract
As discussed in previous chapters, for linear systems subject to actuator saturation, simultaneous external and internal stabilization is possible only for systems which are asymptotically null controllable with bounded control (ANCBC). Even then, in general, nonlinear feedback control laws are required. Of particular interest is the use of linear feedback control laws. Clearly, as shown in Chaps. 12 and 13, for open-loop neutrally stable systems, the global asymptotic stability and external L p  ( p ) stability for p ∈ [1, ) with arbitrary initial conditions can be achieved by linear feedback control laws. We continue in this chapter the theme of pursuing systems other than open-loop neutrally stable ones for which such a simultaneous stabilization is feasible. We focus here on a canonical class of ANCBC systems, namely, a double integrator which is ubiquitously used. In fact, the double-integrator system is commonly seen in control applications including low-friction, free rigid-body motion, such as single-axis spacecraft rotation and rotary crane motion (see [117], and the references therein). All these issues are the motivating factors why we choose in this chapter to reexamine the notions of external and internal stabilization via linear feedback control laws for a double integrator with linear feedback control laws subject to saturation.
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

15. Simultaneous internal and external stabilization in the presence of a class of non-input-additive sustained disturbances: continuous time

Abstract
We continue here with the theme of simultaneous internal and external stabilization in the presence of non-input-additive disturbances. As discussed in Chap. 13, for such non-input-additive disturbances, L p stabilization with finite gain is impossible, but L p stabilization without finite gain is always attainable via a nonlinear dynamic low-gain feedback law for all p ∈ [1, ) (i.e., for all disturbances whose “energy” vanishes asymptotically). In the case of open-loop neutrally stable system, this can be done via a linear state feedback law.
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

16. Simultaneous internal and external stabilization in the presence of a class of non-input-additive sustained disturbances: discrete time

Abstract
For discrete-time general critically unstable linear systems subject to actuator saturation, this chapter is a counterpart of Chap. 15 which pertains to continuous-time systems. That is, our goal here for discrete-time systems is to identify a set of non-input-additive sustained disturbances for which a feedback control law can be determined such that: 1.In the absence of disturbances, the origin of the closed-loop system is globally asymptotically stable. 2.If the disturbances belong to the given set, the states of the closed-loop system are bounded for any arbitrarily specified initial conditions.
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

17. External and internal stabilization under the presence of stochastic disturbances

Abstract
So far, all the results discussed in this book are in deterministic setting. A new frontier for the next phase of research is in stochastic setting. That is, to consider disturbances which are modeled as colored noise which in turn can be modeled as white noise followed by a linear system. Then, the goal is to investigate simultaneous external and internal stabilization of linear systems subject to constraints when the disturbances are modeled stochastically.
Ali Saberi, Anton A. Stoorvogel, Peddapullaiah Sannuti

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