Blockchain Papers

Wei‐Han Chen, Fengqi You

No abstract is available for this record.

Ori Shimony

This chapter proposes a framework for understanding the spectrum of involvement of Artificial Intelligence (AI) in Decentralized Autonomous Organizations (DAOs). This algorithmic governance model proposes combining the decentralized and deterministic nature of DAOs with the autonomous and adaptable capabilities of AI agents. It analyzes the historical development of AI and the technical affordances of DAOs for AI integration. Then, it presents the framework of AI involvement in DAOs across two dimensions (i.e., decision-making and execution) and three levels (i.e., AI-assisted, AI-dominant, and AI-only), accompanied by examples for each case. It then critically analyzes the limitations of AI-driven DAOs across infrastructural, operational, interpersonal, and societal dimensions. The chapter concludes by arguing that the development of AI-driven DAOs necessitates interdisciplinary collaboration and public dialogue to ensure their design and governance promote empowerment, inclusion, and positive social change.

Logan Nye

Zero-knowledge proofs allow verification of computations without revealing private information. However, existing systems require memory proportional to the computation size, which has historically limited use in large-scale applications and on mobile and edge devices. We solve this fundamental bottleneck by developing, to our knowledge, the first proof system with sublinear memory requirements for mainstream cryptographic constructions. Our approach processes computations in blocks using a space-efficient tree algorithm, reducing memory from linear scaling to square-root scaling--from $Θ(T)$ to $O(\sqrt{T} + \log T \log\log T)$ for computation size $T$--while maintaining the same proof generation time through a constant number of streaming passes. For widely-used linear polynomial commitment schemes (KZG/IPA), our method produces identical proofs and verification when using the same parameters and hashing only aggregate commitments into the challenge generation, preserving proof size and security. Hash-based systems also achieve square-root memory scaling though with slightly different proof structures. This advance enables zero-knowledge proofs on everyday devices and makes previously infeasible large computations verifiable, fundamentally democratizing access to privacy-preserving computation. Space-efficient zero knowledge proof systems create opportunities to reshape how trust is established in digital systems--from enabling widespread participation in decentralized networks to making verifiable scientific computing practical at unprecedented scales.

Camilla Fioravanti, Christoforos N. Hadjicostis, Gabriele Oliva

Networked Control Systems (NCS) are pivotal for sectors like industrial automation, autonomous vehicles, and smart grids. However, merging communication networks with control loops brings complexities and security vulnerabilities, necessitating strong protection and authentication measures. This paper introduces an innovative Zero-Knowledge Proof (ZKP) scheme tailored for NCSs, enabling a networked controller to prove its knowledge of the dynamical model and its ability to control a discrete-time linear time-invariant (LTI) system to a sensor, without revealing the model. This verification is done through the controller's capacity to produce suitable control signals in response to the sensor's output demands. The completeness, soundness, and zero-knowledge properties of the proposed approach are demonstrated. The scheme is subsequently extended by considering the presence of delays and output noise. Additionally, a dual scenario where the sensor proves its model knowledge to the controller is explored, enhancing the method's versatility. Effectiveness is shown through numerical simulations and a case study on distributed agreement in multi-agent systems.

Wenqian Xue, Bosen Lian, Jialu Fan, Tianyou Chai · 5 authors

In this paper, we formulate inverse reinforcement learning (IRL) as an expert-learner interaction whereby the optimal performance intent of an expert or target agent is unknown to a learner agent. The learner observes the states and controls of the expert and hence seeks to reconstruct the expert's cost function intent and thus mimics the expert's optimal response. Next, we add non-cooperative disturbances that seek to disrupt the learning and stability of the learner agent. This leads to the formulation of a new interaction we call zero-sum game IRL. We develop a framework to solve the zero-sum game IRL problem that is a modified extension of RL policy iteration (PI) to allow unknown expert performance intentions to be computed and non-cooperative disturbances to be rejected. The framework has two parts: a value function and control action update based on an extension of PI, and a cost function update based on standard inverse optimal control. Then, we eventually develop an off-policy IRL algorithm that does not require knowledge of the expert and learner agent dynamics and performs single-loop learning. Rigorous proofs and analyses are given. Finally, simulation experiments are presented to show the effectiveness of the new approach.

Yi Yu, Shuai Liu, Xingwei Zhou, Wenshan Hu

Networked control systems (NCSs) are widely used in practical applications because of their flexibility in deployment. However, due to the dependence on the communication network, NCSs could be vulnerable to malicious cyberattacks. To address this problem, a novel blockchain technology-assisted networked predictive secure control approach is presented for the first time in this article. First, the introduction of blockchain technology brings a significant boost to the inherent resilience of the NCS in an active manner without relying on any prior knowledge of the system or potential attacks. However, blockchain technology would induce time delays unfavorable to the NCS, which could result in the low real-time performance of the control system. Subsequently, a networked Kalman filter-based predictive control is specially designed to compensate for the low real-time property of blockchain technology. A detailed analysis of the security and stability of the closed-loop NCS with the developed networked predictive secure controller is also presented, while sufficient conditions for the closed-loop system to be simultaneously stable and safe in a probabilistic sense are given. Finally, to verify the performance of the proposed approach in terms of practicality, an experimental prototype of a photovoltaic (PV)-based power generation system subjected to random cyber-attacks is built for voltage regulation.

Johannes Fottner, Dana Clauer, Fabian Hormes, Michael Freitag · 13 authors

The paper at hand presents a definition of autonomous intralogistics systems and a classification of intralogistics systems with regard to their degree of autonomy. Intralogistics -; a complex interplay of different logistics functions - covers the organization, control, execution and optimization of internal material and information flows. Over the past two decades, numerous authors have observed and proclaimed an increase in complexity in manufacturing and supply chain operations. A key approach to face this challenge is a paradigm shift from centralized, hierarchical organization structures towards, networked and autonomous systems. Autonomous intralogistics systems enable self-contained, decentralized planning, execution, control, and optimization of internal material and information flows through cooperation and interaction with other systems and with humans.Based on the definition of autonomous intralogistics systems, the authors propose a two-dimensional classification framework covering different automation stages for different intralogistics task levels. The developed classification framework is applied to various industry use cases to evaluate and discuss the state of the art regarding the implementation of autonomous intralogistics systems. Finally, the paper provides an outlook on future research and poses key research questions.

Minseok Kim, Jaehoon Park

—In this paper, a sampled-data model predictive tracking control method is presented for mobile robots which is modeled as constrained continuous-time linear parameter varying (LPV) systems. The presented sampled-data predictive controller is designed by linear matrix inequality approach. Based on the input delay approach, a controller design condition is derived by constructing a new Lyapunov function. Finally, a numerical example is given to demonstrate the effectiveness of the presented method. Keywords—Model predictive control, sampled-data control, linear parameter varying systems, LPV I. INTRODUCTION OBILE robots nowadays move autonomously by recognizing external environment and determining the situation through the remote control. With the development of network communication, implementation employing wireless & wired network is widespread [1]. Though control through network is advantageous in maintenance, installation, flexibility and cost, it has to be carefully designed in reality. It may cause instability and performance degradation without considering network induced delay or data packet losses. Therefore, the design of control scheme should consider with aspects and performances of whole systems. Model predictive control (MPC) scheme is very useful since it provides good tracking performance and the MPC tuning parameters are explicitly related to the key characteristics safety, comfort, and fuel economy. But if the model is not accurate, the control technique does not guarantee the stability and performance [2]. Also, an important issue in the implementation of MPC algorithm is the discretization. A continuous-time model is much more natural and accurate in terms of describing the behavior of a system, Also, in network control systems, choosing proper sampling interval is very important for designing suitable controllers. It is clear that a longer sampling period will lead to lower communication channel occupation, few actuation of the controller, and less signal transmission. Thus, it is very important to consider the stabilizing control design problem under a bigger sampling period [5]. For sampled-data systems, the input delay approach has been widely used [4], which is based on the representation of the sampled-data system as a continuous-time system Fig. 1 Mobile robot in X-Y coordination with a delayed control input. Then, the Lyapunov Krasovskii functional (LKF) method can be used to establish the stability conditions. Recently, based on the input delay approach, the sampled-data control problem of dynamical systems with time-varying delay has been investigated in [3], [4]. In this paper, we consider a continuous-time LPV model to handle mobile robot systems and present a model predictive control method for the systems with sampled-data. To the best of authors' knowledge, there are no approaches considering sampled-data MPC for mobile robots. The presented synthesis condition is formulated by construction of a suitable Lyapunov-Krasovskii's functional and control inputs are obtained by minimizing the upper bound of the cost function satisfying the cost monotonicity. Finally, we demonstrate the effectiveness of the proposed approach via numerical simulation. II. DESCRIPTION OF MOBILE ROBOT The dynamics of mobile robot with a rigid body and wheels can be described as follows [1] , (1) where [x,y,θ] denotes the position and orientation of the center with respect to a global frame, v is the translational velocity, and w is the angular velocity. For the given mobile robot, the reference trajectory is set to , (2) where xr, yr, θr are references in Cartesian coordination, vr is the reference translational velocity, and ωr is the reference angular velocity. Considering local coordinate frame, define From (1)-(3), the error dynamics is obtained as In general, systems represented by nonlinear systems can be transformed into Linear Parameter Varying (LPV) systems X˙ (t) = A(¯v(t),ω¯(t),vr(t))X(t) + BU(t), where A(·) is system matrices containing a time varying parameter vector v¯(t),ω¯(t),vr(t), X = [xe,ye,θe] − [¯xe,y¯e,θ¯e], and U = [v − v,ω¯ − ω¯]. By computing Jacobian matrix, the system matrices are given as . For a given sampling rates, the matrix A(¯v(t),ω¯(t),vr(t)) is subject to a polytope set Ω. (5) where Ω = {A1,A2,...,AL} is the convex hull. In the typical system architecture, control signals are conveyed through network communication. In network environments, the control signals pass through zero-order-hold (ZOH) which generate functions with a sequence of hold times 0 ≤ t0 < t1 < ··· < tk ··· < lim tk = +∞. Taking k→∞ consideration of ZOH, the control input is U(t) = KX(tk), t ∈ [tk,tk+1). (6) where K is the control gain matrix. Without loss of generality, it is assumed that the sampled time interval is bounded by h(t) ≤ hM where h(t) = tk+1 − tk, and hM is the maximum sampled delay. Using sampled signals, the systems are expressed as delayed LPV systems X˙ (t) = AiX(t) + BU(t − h(t)). (7) Lemma 1. [5] For given matrices Λ1,Λ2,Ψ, and a scalar 0 ≤ Lemma 2. [6] For given matrices H,N,R > 0 and a continuously differentiable function x(t) in [a,b] ∈ Rn, the following inequality is ensured. (10) (11) , where is any vector,, and x(s) . −b−a a III. MAIN RESULTS The main purpose of this paper is to design a proper sampled-data model predictive controller. Model Predictive Control is used to approximately obtain optimal trajectories. Therefore, choosing the following performance index is reasonable: (12) where Q, R are coefficients. For the given performance index, if the following condition is satisfied . (13) where · denotes 2-norm, then the upper bound of the performance index can be derived instead of directly minimizing performance index. By integrating (13) from i = 1 to i = ∞, one can notice the upper bound of the performance index is less than the Lyapunov function. Before presenting main results, we employed the following representations for simplicity. The matrices ei = R4n×n for i = 1,2,...,4 are matrices composed of nth zero elements with ith identity matrix. (For example, e1 = [I 0 0 0] and e3 = [0 0 I 0]). . With predefined Lemmas and notations, we present design methodology of model predictive control for delayed LPV systems by deriving a set of linear matrix inequality conditions. Theorem 1. For a given parameter hM and a vector X(tk), if U¯ U¯ there exist positive matrices G, 0,V >¯ 0, Y , Z¯1,Z¯2, satisfying the following LMI conditions, the control input at time instant tk guarantees the performance index (12) with γ . (14) (15) (16) (17) (18) where , with then, the state feedback gains are given as K = Y G−1. Proof. Choosing the following Lyapunov-Krasovskii functional (LKF) for t ∈ [tk,tk+1) yields V (xt) = V1(t) + V2(t) + V3(t) (19) where , Differentiate the LKF From Lemma 2, the following holds (23) where Z1,Z2 are auxiliary variables. Taking into account system dynamics (7), (24) Summing up from (20) to (24) leads to V˙ + XT(t)QX(t) + UT(t)RU(t) ≤ ζ(tk)Σ¯ζ(tk) (25) where Pre-and post-multiplying with a matrix γ1/2 × diag{G,G,G,G}, the followings are satisfied with Lemma 1. , (26) Σ1 + hMΣ3 < 0 (27) time (sec) Fig. 2 error response of the system in Example 1 where U¯ = GUG, V¯ = GV G, Z¯1 = GZ1G, Z¯2 = GZ2G, and K = Y G−1. Using Schur complement, The equations in (25) and (26) are equivalent to those of (16) and (17). For every sampling instance, V2 and V3 vanish. Then, the upper bound of LKF is expressed in terms of V1. XT(tk)GP¯1GX(tk) ≤ γ, (28) where γ denotes the bound of optimal performance index. The effect of input saturation is considered similar to the method in [7]. This ends the proof. IV. NUMERICAL EXAMPLE Example 1 This example considered the dynamical equations of the system represented from error dynamics. X˙ (t) = AiX(t) + BU(t − h(t)) (29) where ⎡ −0 ωr − 0.05 0 ⎤⎦ A1 =ω 0.05 0 vr(t) , 0 0 0 ⎡ 0 ωr + 0.05 0 ⎤ A2 =ω + 0.05 0 vr(t) , 0 0 0 ⎦ ⎡−1 0 ⎤ B = 0 0 . ⎣ 0 −1⎦ The model parameters are calculated with a sampling time 0.1s. The sampling time h(t) is less than 0.1 s. Along the reference trajectory, the input is constrained to −0.1 ≤ u(1) ≤ 0.1 and −0.05 ≤ u(2) ≤ 0.05. The corresponding controller gain matrix is Fig. 2 shows the simulation result which is obtained with the above controller gain, taking Q = I, R = I,α = 0.1. V. CONCLUSION The sampled-data MPC method for mobile robot systems have been investigated by considering constrained polytopic LPV model. Based on the input delay model, sufficient conditions for the sampled-data MPC controller design are obtained by constructing a new Lyapunov functional. The effectiveness of the presented method has been verified by illustrating numerical simulation. REFERENCES W. Lucia, F. Tedesco. "A networked-based receding horizon scheme for constrained LPV systems," European Journal of Control, vol. 25, pp. 69-75, 2015. S. Lee, Ju H. Park, D. Ji, S. Won, "Robust model predictive control for LPV systems using relaxation matrices," IET. Control Theory Appl., vol. 1, no. 6, pp. 1567-1573, 2007. A. Seuret, F. Gouaisbaut, Wirtinger-based integral inequality: application to time-delay systems, Automatica, vol. 49, no. 8, pp. 2860-2866, 2013. S. Lee, O. Kwon, Quantised MPC for LPV systems by using new LyapunovKrasovskii functional, IET. Control Theory Appl., vol. 11, no. 3, pp. 439-445, 2017. D. Yue, E. Tian, Y. Zhang, and C. Peng, "Delay-distribution-dependent stability and stabilization of T-S fuzzy systems with probabilistic interval delay," IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 39, no. 2, pp. 503–516, 2009. C.K. Zhang, Y. He, L. Jiang, W. Lin, M. Wu, "Delay-dependent stability analysis of neural networks with time-varying delay: A generalized free-weighting-matrix," Applied Mathematics and Computation, vol. 294, no. 1, pp. 102-120, 2017. E. Fridman and M. Dambrine, "Control under quantization, saturation and delay: An LMI approach," Automatica, vol. 45, no. 10, pp. 2258–2

Luis Luna, Erick Asiain, R. Garrido

This paper describes preliminary results on a Proportional plus Adaptive Disturbance Observer (P+ADOB) controller applied to velocity regulation tasks in a servo system. Adaptation law is obtained to estimate the servo system input gain, which is subsequently employed in the design of a Disturbance Observer. Compared with previous approaches, this feature relaxes the assumption on exact knowledge on the input gain, and only upper and lower bounds on this term are assumed known. A stability proof assuming constant disturbances allows concluding that the estimate of the input gain is bounded, and the velocity tracking error converges to zero. Real-time experiments illustrate the performance of the proposed controller.

Zahra Zahedi, Mohammad Mehdi Arefi, Alireza Khayatian, Hamidreza Modares

In this paper, a solution for Nash equilibrium seeking problem for N-players static non-cooperative games with non-quadratic payoff functions is proposed. The proposed solution is a non-model based approach, in the sense that the players do not need any knowledge about the agent's model or other players' actions, and can attain the Nash equilibrium using only measurements of payoff values. To overcome the shortcoming of existing non-model based algorithms, for which the Nash equilibrium stays within a small neighborhood and oscillates, the proposed approach adjusts the classical extremum seeking algorithms so that the amplitude of excitation sinusoidal signal converges to zero locally and exponentially. Therefore, with removing steady-state oscillation, not only the deleterious effects of steady-state oscillation is eliminated but also Nash equilibrium is achieved faster. The details of proof and the analysis for stability and convergence are provided. Finally, the efficiency and effectiveness of the algorithm are illustrated with a numerical example and simulation.

Georgios Darivianakis, Angelos Georghiou, John Lygeros

Distributed model predictive control (MPC) has been proven a successful method in regulating the operation of large-scale networks of constrained dynamical systems. This paper is concerned with cooperative distributed MPC in which the decision actions of the systems are usually derived by the solution of a system-wide optimization problem. However, formulating and solving such large-scale optimization problems is often a hard task which requires extensive information communication among the individual systems and fails to address privacy concerns in the network. Hence, the main challenge is to design decision policies with a prescribed structure so that the resulting system-wide optimization problem to admit a loosely coupled structure and be amendable to distributed computation algorithms. In this paper, we propose a decentralized problem synthesis scheme which only requires each system to communicate sets which bound its states evolution to neighboring systems. The proposed method alleviates concerns on privacy since this limited communication scheme does not reveal the exact characteristics of the dynamics within each system. In addition, it enables a distributed computation of the solution, making our method highly scalable. We demonstrate in a number of numerical studies, inspired by engineering and finance, the efficacy of the proposed approach which leads to solutions that closely approximate those obtained by the centralized formulation only at a fraction of the computational effort.

Nhan T. Nguyen

No abstract is available for this record.

Sayan Basu Roy, Shubhendu Bhasin, Indra Narayan Kar

Convergence of controller parameters in standard model reference adaptive control (MRAC) requires the system states to be persistently exciting (PE), a restrictive condition to be verified online. A recent data-driven approach, concurrent learning, uses information-rich past data concurrently with the standard parameter update laws to guarantee parameter convergence without the need of the PE condition. This method guarantees exponential convergence of both the tracking and the controller parameter estimation errors to zero, whereas, the classical MRAC merely ensures asymptotic convergence of tracking error to zero. However, the method requires knowledge of the state derivative, at least at the time instances when the state values are stored in memory. The method further assumes knowledge of the control allocation matrix. This paper addresses these limitations by using a memory-based finite-time system identifier in conjunction with a data-driven approach, leading to convergence of both the tracking and the controller parameter estimation errors without the PE condition and knowledge of the system matrices and the state derivative. A Lyapunov based stability proof is included to justify the validity of the proposed data-driven approach. Simulation results demonstrate the efficacy of the suggested method.

Kelley E. Hashemi

Adaptive control techniques are often avoided in aerospace systems due to stringent plant structural requirements and validation difficulties. This dissertation seeks to broaden the range of aerospace engineering applications that can utilize an adaptive controller through the development of an extended model reference adaptive control (MRAC) design. First, a partitioned control framework is presented that permits the combined use of an adaptive control law and a nonadaptive control law. The partitioned framework is used to shift full control authority away from the adaptive portion of the system. Next, two MRAC variations that can accommodate the nonminimum phase zeros often seen in aerospace applications are discussed for use as the adaptive system. The parallel feedforward compensator approach proposes inclusion of a user--defied fictitious model in parallel with the plant that is designed to make the plant appear nonminimum phase. The surrogate tracking error approach modifies the typical MRAC structure to handle nonminimum phase plants by requiring knowledge of its nonminimum phase zeros. A tracking error convergence proof is provided for this continuous-time MRAC variant. The partitioned design using the surrogate tracking error approach is applied to the control tasks of an experimental, flexible wing aircraft. A simulation is used to demonstrate much improved flight path angle command tracking when compared to use of the aircraft's existing nonadaptive control law, even in the presence of large--scale modeling error. A second simulation is used to show the design applied to flexible motion control of the same aircraft model and exhibits similarly improved performance.

Yang Liu, Xiao He, Zidong Wang, Zhou Donghua

In this paper, the fault detection and diagnosis problems are considered for a class of discrete nonlinear systems with decentralized event-triggered measurement transmissions. Each sensor determines, according to certain triggering rules, whether to transmit the present measurement to remote filters based on only locally available information. A set of filters is designed where each filter aims to jointly estimate the system states and a specific possible fault. Upper bounds of the estimation error covariances are obtained in the simultaneous presence of the linearization errors and decentralized event-triggered transmissions, and then the filter gains are calculated to minimize such bounds. The filters are designed in a recursive way and thus the algorithm is applicable for online implementation. When a fault is detected, the filter with the least residual is regarded as the one corresponding to the actual fault and its output can be seen as the states and fault estimation. The effectiveness of the proposed method is illustrated by a simulation example.

Weijie Mai, C. Y. Chung

In order to increase the utilization of renewable energy sources and to reduce the need of generator-provided ancillary services and inefficient peaking generation, buildings are progressively transforming into smarter electricity consumers and active participants in power system operations. In this work, an R-C thermodynamic model of building that strongly relates to the power consumption of heating ventilation air-conditioning system (HVAC) is firstly introduced. To offer the aggregated flexible HVAC power to the grid as regulating power, we propose a building-aggregator-grid contract framework and formulate a robust model predictive control (MPC) algorithm which both maximizes the profit of aggregator and minimizes the payment of each participating building to optimally declare power flexibility. Lastly, simulation results on a group of real and hypothetical buildings validate the feasibility of the proposed method.

Nathalie Verdière, Lilianne Denis-Vidal, Ghislaine Joly-Blanchard

This paper is devoted to the identification of a pollution source in a river. A simple mathematical model of such a problem is given by a one-dimensional linear advection-dispersion-reaction equation with a right hand side spatially supported in a point (the source) and a time variant intensity, both unknown. The identifiability of the distributed system was established for two points of observations one upstream, the other downstream from the source provided the pollutant flow rate is zero on an interval [T, T + deltaT] (T > 0). But the distributed system has to be discretized in order to do a numerical estimation of the unknown parameters. It is why this paper is devoted to the identifiability of the differential system obtained by using a semi-discretization scheme in space. The proof of the identifiability does not require the restrictive assumption about the pollutant flow rate but the knowledge of the initial condition and one observation located upstream from the source. Moreover, from this study, a numerical procedure is deduced for estimating the unknown parameters. It does not necessitate a priori knowledge about the parameters and the unknown function is not expanded on a basis of special functions. Both aspects play an important role in the real applications

Sagar Sen, Sagar Sen, Hans Agarwal, Sagar Sen · 5 authors

No abstract is available for this record.

HL Trentelman, AA Anton Stoorvogel, Malo Hautus, Larry Dewell

9R15. Control Theory for Linear Systems. - HL Trentelman (Dept of Math, Univ of Groningen, PO Box 800, Groningen, 9700 AV, Netherlands), AA Stoorvogel, M Hautus (Dept of Math and Comput Sci, Eindhoven Univ of Tech, PO Box 513, Eindhoven, 5600 MB, Netherlands). Springer-Verlag London Ltd, Surrey, UK. 2001. 389 pp. ISBN 1-85233-316-2. $84.95.Reviewed by L Dewell (Adv Tech Center, Lockheed Martin Advanced Tech Center, 3251 Hanover St, Palo Alto CA 94304).In the preface of this book, the authors stake out a strikingly broad boundary: “the theory of feedback control design for linear, finite-dimensional, time-invariant state space systems with inputs and outputs.” It is a testament to the recent growth of this field that had such a title appeared in the early 1970s, it would scarcely require 100 pages, let alone the nearly 400 pages of this book. Indeed, such a wide scope of a book today may either elicit suspicions of the authors of either being overly cursory in their treatment or overly parochial in their interpretation of the subject. In this book, however, the authors treat the subject with a coherence and an organization that made the large scope seem perfectly natural and pleasantly illuminating. The book appears to be adapted primarily to a reader interested in the theory of control for finite-dimensional linear systems, as opposed to applied engineers or scientists. The very restriction of the subject to such systems, however, allows for the applied control systems analyst to gain insights into recent developments in the field. The focus is certainly towards the development of a tight theoretical basis for the results, rather than applied examples, case studies or exercises. The authors assume that the reader is proficient in the linear algebra tools to analyze state space linear systems. In the opinion of this reviewer, it is probably best that the reader approach the book with an existing knowledge of the major results of the past few decades on the control of state space systems—most particularly linear quadratic regulators (LQR) and the linear quadratic Gaussian (LQG) problem. This book can quite readily stand on its own as an introduction to the recent results of H-infinity control theory from this foundation. The authors approach their subject from the point of view of geometric control theory, which requires a particular nomenclature and foundation with which the reader may not initially be familiar. The authors spend the first two chapters building such a foundation. In the next two chapters, the authors deal with the problem of disturbance decoupling and the notions of controlled and conditioned invariant subspaces. These are very fascinating chapters which alone make the book a worthwhile read. The following three chapters delve deeply into the structural properties of linear control systems, including system zeros and system invertibility. The next five chapters “deliver the goods” that the authors advertised in their preface—tracking and regulation, LQ optimal control, stochastic optimal control, and H-infinity control in finite- and infinite-time, with full-state and with measurement feedback. Each chapter was superbly organized into an introduction, a presentation of ideas, and a set of concluding notes. The notes were particularly pleasant to read, as they give the reader a clear connection between the material and the work of others over the last several decades. The index of the book is excellent (it includes a symbol index, which is frequently overlooked), and all results are painstakingly supported by proofs. If there is one area where the book seems inconsistent, it is in the paucity of numerical examples or case studies to illustrate the material of the book. This was doubtless a conscious decision of the authors, but the decision resulted in the work having a certain opacity and theoretical detachment that was distracting at times. The authors motivate the book in Chapter 1 by discussing, in detail, a practical problem of stationkeeping control of a geosynchronous satellite. After such a strong practical introduction, it would seem appropriate to apply the theory to a practical problem, perhaps even to the satellite problem suggested by the authors. Moreover, the authors stress in Chapters 4 and 6 that a disturbance decoupling controller can actually be numerically computed by algorithms they provide. An actual demonstration of such a controller synthesis would have been very powerful. Control Theory for Linear Systems is a fine contribution to the literature of control theory of linear systems. It offers a powerful organization and structure to the very broad subject of the control of finite-dimension linear systems and brings the torrent of activity of this field over the last several decades to a tight focus and perspective that has been lacking heretofore. This book is highly recommended for purchase by libraries and researchers in the field.

Roméo Ortega, Gustavo Sanchez Galindo

A complete proof is given of global convergence to zero of the prediction error and asymptotic optimality for a direct adaptive controller is established based on multistep quadratic cost minimization with a receding horizon philosophy. The only substantial assumptions are that the controller designed when the parameters are known, stabilizes the plant and the exact knowledge of the first N y (N y -optimization horizon) impulse response coefficients of the plant. The technical hurdles for the development of a comprehensive convergence theory of this type of controller are also highlighted.