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For stabilization of linear systems subject to input saturation, there exist four different approaches of low-gain design all of which are independently proposed in the literature, namely direct eigenstructure assignment, H2 and Hoo algebraic Riccati equation (ARE) based methods, and parametric Lyapunov equation based method. It is shown in Wang et al. [2010b] that for continuous-time linear systems, all these methods are rooted in and can be unified under two fundamental control theories, H2 and Hoo theory. In this paper, we extend such a result to a discrete-time setting. Both the H2 and Hoo ARE based methods are generalized to consider systems where all input channels are not necessarily subject to saturation, and explicit design methods are developed.

A recent paper (IEEE Trans. Aut. Contr. 2010; 55(9):2156–2160) considered stabilization of a class of continuous-time nonlinear sandwich systems via state feedback. This paper is a discrete-time counterpart of it. The class of nonlinear sandwich systems consists of saturation elements sandwiched between linear systems. We focus first on single-layer sandwich systems, which consist of a single saturation sandwiched between two linear systems. For such systems, we present necessary and sufficient conditions for semi-global and global stabilization problems by state feedback, and develop design methodologies to achieve the prescribed stabilization. We extend the results to single-layer sandwich systems subject to additional actuator saturation. Finally, we discuss further extension to general multi-layer sandwich systems with an arbitrary number of saturations sandwiched between linear systems, both with and without actuator saturation. The design methodologies can be viewed as extensions of classical low-gain design methodologies developed during the 1990s in the context of stabilizing linear systems subject to actuator saturation.

This paper investigates time-invariant linear systems subject to input and state constraints. We study discrete-time systems with full or partial constraints on both input and state. It has been shown earlier that the solvability conditions of stabilization problems are closely related to important concepts such as the right invertibility or non-right invertibility of the constraints, the location of constraint invariant zeros, and the order of constraint infinite zeros. In this paper, for general time-invariant linear systems with non-right invertible constraints, necessary and sufficient conditions are developed under which semi-global stabilization in the admissible set can be achieved by state feedback. Sufficient conditions are also developed for such a stabilization in the case where measurement feedback is used. Such sufficient conditions are almost necessary. Controllers for both state feedback and measurement feedback are constructed as well.

A generalized $\gamma$-level $H_\infty$ sub-optimal input decoupling (SOID) filtering problem is formulated. It is a generalization of $\gamma$-level $H_\infty$ SOID filtering problem when, besides an input with unknown statistical properties but with a finite RMS norm, there exists an additional input to the given plant or system. The additional input is a linear combination of sinusoidal signals each of which has an unknown amplitude and phase but known frequency. The analysis, design, and performance limitations of generalized $H_\infty$ optimal filters are presented.

We consider almost regulated output synchronization for heterogeneous directed networks with external disturbances where agents are non-introspective (i.e. agents have no access to their own states or outputs). A purely decentralized time-invariant protocol based on a low-and-high gain method is designed for each agent to achieve almost regulated output synchronization while reducing the impact of disturbances on the regulated output synchronization error. It is also shown that this protocol also works in the case of time-varying graphs.

For linear time invariant continuous-time systems with either unknown or white noise input, two well-known filtering problems are considered. These are the unknown input observer problem and the Kalman filtering problem. Most of the available literature on Kalman filtering considers the so-called regular filtering problem. We consider here the general singular filtering problem. We show that such a Kalman filtering problem for a given system can be transformed to the unknown input observer problem for an auxiliary system constructed from the data of the given system. Such transformations between these two filtering problems enable us to study various properties of Kalman filtering, including existence and uniqueness of Kalman filters, computation of performance indices of Kalman filtering, and performance limitations of Kalman filtering as related to the structural properties of the given system.

Discrete-time output regulation of linear systems with state and/or input constraints on magnitude is considered. Structural properties of linear plants are identified under which the so-called constrained semi-global and global output regulation problems are solvable. As in the case of continuous-time systems, an important aspect of our development is a taxonomy of constraints showing a clear relationship between the constraint types and the output regulation results. Solvability conditions are developed for semi-global and global output regulation problems. Appropriate regulators are constructed for problems that are solvable.

We consider here both fault identification and fault signal estimation. Regarding fault identification, we seek either exact or almost fault identification. On the other hand, regarding fault signal estimation, we seek either $H_2$ optimal, $H_2$ suboptimal or Hinfinity suboptimal estimation. By appropriate combination of these two aspects, we formulate five different fault identification and fault signal estimation problems. We analyse and synthesize appropriate residual generators for fault identification and estimators which generate fault signal estimates for all these problems. Solvability conditions for all these problems are given. Also, for each of these problems, the architecture of integrating the residual generator that isolates the fault and estimators that estimate the fault signals is developed.

For discrete-time systems, an H 2-suboptimal control problem is defined and analyzed. Then, an algorithm called H 2-suboptimal state feedback gain sequence (Algorithm A1) is developed. Rather than utilizing a perturbation method, which is numerically stiff and computationally prohibitive, Algorithm A1 utilizes a direct eigenvalue assignment method to come up with a sequence of H 2-suboptimal state feedback gains. Also, although the sequence of H 2-suboptimal state feedback gains constructed by Algorithm A1 depends on a parameter ɛ, the construction procedure itself does not require explicitly the value of the parameter ɛ. Next, attention is focused on constructing a sequence of H 2-suboptimal estimator-based measurement feedback controllers. Three different estimator structures (prediction, current, and reduced-order estimators) are considered. For a given H 2-suboptimal state feedback gain, a sequence of estimator gains for any of the three estimators considered can be constructed by merely dualizing Algorithm A1. The direct method of constructing H 2-suboptimal controllers developed here has a number of advantages over the perturbation method, e.g., it has the ability to design all three types of estimator-based controllers while still maintaining throughout the design the computational simplicity of it. This paper is the discrete-time version of Ref. 1. There are some conceptual similarities as well as fundamental differences between the H 2-suboptimal control problems for continuous-time and discrete-time systems. The fundamental differences arise mainly from the fact that, in contrast to continuous-time systems, for discrete-time systems the infimum of the H 2-norm over the class of strictly proper controllers is in general different from the infimum of the H 2-norm over the class of proper controllers.

An algorithm called the 'H/sub 2/ suboptimal state feedback gain sequence' (H/sub 2/ SOSFGS) algorithm is developed. Rather than utilizing a 'perturbation method' which is numerically stiff and computationally prohibitive, the H/sub 2/ SOSFGS algorithm utilizes a direct eigenvalue assignment method to come up with a sequence of H/sub 2/ suboptimal state feedback gains. Also, although the sequence of H/sub 2/ suboptimal state feedback gains constructed by the H/sub 2/ SOSFGS algorithm depends on a parameter /spl epsi/, the construction procedure itself does not require explicitly the value of the parameter /spl epsi/. For a given H/sub 2/ suboptimal state feedback gain, a sequence of observer gains for either a full or a reduced order observer can be constructed by merely dualizing the H/sub 2/ SOSFGS algorithm.

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.

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.

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.

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.

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.

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.

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).

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 (l 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.

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..

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.

For stabilization of linear systems subject to input saturation, there exist four different approaches of low-gain design all of which are independently proposed in the literature, namely direct eigenstructure assignment, H 2 and H ∞ algebraic Riccati equation (ARE) based methods, and parametric Lyapunov equation based method. We show here all these methods are rooted in and can be unified under two fundamental control theories, H 2 and H ∞ theory. Moreover, both the H 2 and H ∞ ARE based methods are generalized to consider systems where all input channels are not subject to saturation, and explicit design methods are developed.

A simultaneous H 2 H ∞ control problem is considered. This problem seeks to minimize the H2 norm of a closed-loop transfer matrix while simultaneously satisfying a prescribed H∞ norm bound on some other closed-loop transfer matrix by utilizing dynamic state feedback controllers. Such a problem was formulated earlier by Rotea and Khargonekar (Automatica,27, 307–316, 1991) who considered only so called regular problems. Here, for a class of singular problems, necessary and sufficient conditions are established so that the posed simultaneous H 2 H ∞ problem is solvable by using state feedback controllers. The class of singular problems considered have a left invertible transfer function matrix from the control input to the controlled output which is used for the H2 norm performance measure. This class of problems subsumes the class of regular problems.

We consider the problem of state-feedback stabilization for a class of sandwich systems, consisting of two linear systems connected in cascade via a saturation. In particular, we present design methodologies for constructing semiglobally and globally stabilizing controllers for such systems when the input is itself subject to saturation. The design is carried out under a set of assumptions that are proven to be both necessary and sufficient. The presented design methodologies are extensions of classical low-gain design methodologies developed for stabilizing linear systems subject to input saturation. The methodologies can be further extended to multilayer sandwich systems, consisting of an arbitrary number of cascaded linear systems with saturations sandwiched between them.

In this paper, we consider the problems of semi- global and global internal stabilization of a class of sandwich systems consisting of two linear systems with a saturation element in between. We develop here low-and-high gain and scheduled low-and-high gain state feedback design methodolo- gies to solve the posed stabilization problems.

Several sets (e.g. lists of integers, polynomials, etc.) which remain invariant under a group of transformations including static output feedback control of a linear dynamic system are identified. A nested feedback look decomposition of any given system is given which not only identifies the new invariants but also gives an algorithmic procedure of computing them. As is known, structural invariants such as the ones introduced here, play important roles in many theoretical studies of control theory. To show one such case, here an upper bound to the minimum order of a dynamic output feedback compensator required to stabilize a given system is calculated in terms of the newly identified invariants. This paper can be considered as an extension of the work of Morse who identified structural invariants under static-state feedback, where, as this paper identifies, there are structural invariants under static output feedback.

An almost disturbance decoupling technique guaranteeing internal stability of a linear system while using either state feedback or only measurement feedback is considered. Under a set of necessary and sufficient conditions, a state feedback controller is explicitly constructed. Similarly, a full or reduced order observer-based measurement feedback controller is explicitly constructed under a set of necessary and sufficient conditions. All the developed controllers guarantee the asymptotic stability of the closed-loop system in the absence of disturbances. Furthermore, they can reduce H ∞ norm of the transfer function from the external disturbance to the controlled output to any arbitrarily specified value. Unlike other methods which merely give the solvability conditions, our method explictly constructs them. Also our controllers are parametrized in terms of tuning parameters, by the tuning of which the disturbance reflected in the output can be attenuated to any desired degree of accuracy

Explicit expressions for two different cascade factorizations of any detectable left invertible nonminimum phase systems are given. The first one is a well known minimum phase/all-pass factorization by which all nonminimum phase zeros of a transfer function G(s) are collected into an all-pass factor V(s), and G(s) is written G/sub m(s)V/$ where G/sub m/s is considered as a minimum phase image of G(s). The second one is a new cascade factorization by which G(s) is rewritten as G/sub M/(s)U(s) where U(s) collects all 'awkward' zeros including all nonminimum phase zeros of G(s). Both G/sub m/(s) and G/sub M/(s) retain the given infinite zero structure of G(s). Further properties of G/sub m/(s), G/sub M/(s), and U(s) are discussed. These factorizations are useful in several applications including loop transfer recovery. >

In this paper, we consider the problems of semi-global and global internal stabilization of a class of sandwich systems consisting of two linear systems with a saturation element in between. We provide necessary and sufficient conditions for solvability of these problems by state feedback, and develop controllers for semi-global and global stabilization.

Global asymptotic stabilization of nominally linear uncertain systems is considered where uncertain elements in the plant are modelled as cone bounded non-linearities. At first, a linear time-invariant state feedback law ensuring global asymptotic closed-loop stability is obtained. The algorithm which calculates such a feedback law is simple and straightforward. It does not involve repeated solutions of a parameter-dependent algebraic Riccati or any such non-linear equations. Any state feedback law thus developed can then be implemented via an observer specially designed to preserve the global asymptotic stability of the closed-loop system.

In an earlier work, a procedure for squaring down proper systems by static and dynamic compensators has been developed. The compensators designed there have the following properties: (a) they are asymptotically stable; (b) the additional finite zeros induced by them are assignable to the open left half complex plane; and (c) they preserve the fundamental properties such as stabilizability, detectability, infinite zero structure and minimum phase nature of the original system. The earlier design procedure is extended to non-strictly proper systems while retaining all the above-mentioned properties.