Paper: The internal model principle of control theory
TL;DR: The Internal Model Principle is extended to weakly nonlinear systems subjected to step disturbances and reference signals and is shown that, in the frequency domain, the purpose of the internal model is to supply closed loop transmission zeros which cancel the unstable poles of the disturbance andreference signals.
About: This article is published in Automatica.The article was published on 1976-09-01. It has received 2613 citations till now. The article focuses on the topics: Repetitive control & Internal model.
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01 Jul 1992
TL;DR: A theoretical framework for designing interfaces for complex human-machine systems, based on the skills, rules, and knowledge taxonomy of cognitive control, is proposed, and three prescriptive design principles are suggested to achieve this objective.
Abstract: A theoretical framework for designing interfaces for complex human-machine systems is proposed. The framework, called ecological interface design (EID), is based on the skills, rules, and knowledge taxonomy of cognitive control. The basic goals of EID are not to force processing to a higher level than the demands of the task require, and to support each of the three levels of cognitive control. Thus, an EID interface should not contribute to the difficulty of the task, and at the same time, it should support the entire range of activities that operators will be faced with. Three prescriptive design principles are suggested to achieve this objective, each directed at supporting a particular level of cognitive control. Particular attention is paid to presenting a coherent deductive argument justifying the principles of EID. Support for the EID framework is discussed. Some issues for future research are outlined. >
1,072 citations
TL;DR: Using techniques from control and dynamical systems theory, it is demonstrated that integral control is structurally inherent in the Barkai-Leibler model and identified and characterize the key assumptions of the model.
Abstract: ‡Integral feedback control is a basic engineering strategy for ensuring that the output of a system robustly tracks its desired value independent of noise or variations in system parameters. In biological systems, it is common for the response to an extracellular stimulus to return to its prestimulus value even in the continued presence of the signal—a process termed adaptation or desensitization. Barkai, Alon, Surette, and Leibler have provided both theoretical and experimental evidence that the precision of adaptation in bacterial chemotaxis is robust to dramatic changes in the levels and kinetic rate constants of the constituent proteins in this signaling network [Alon, U., Surette, M. G., Barkai, N. & Leibler, S. (1998) Nature (London) 397, 168 ‐171]. Here we propose that the robustness of perfect adaptation is the result of this system possessing the property of integral feedback control. Using techniques from control and dynamical systems theory, we demonstrate that integral control is structurally inherent in the Barkai‐ Leibler model and identify and characterize the key assumptions of the model. Most importantly, we argue that integral control in some form is necessary for a robust implementation of perfect adaptation. More generally, integral control may underlie the robustness of many homeostatic mechanisms.
1,058 citations
TL;DR: It is shown that the individuals (autonomous agents or biological creatures) will form a cohesive swarm in a finite time and an explicit bound on the swarm size is obtained, which depends only on the parameters of the swarm model.
Abstract: In this note, we specify an "individual-based" continuous-time model for swarm aggregation in n-dimensional space and study its stability properties. We show that the individuals (autonomous agents or biological creatures) will form a cohesive swarm in a finite time. Moreover, we obtain an explicit bound on the swarm size, which depends only on the parameters of the swarm model.
929 citations
TL;DR: An internal model requirement is necessary and sufficient for synchronizability of the network to polynomially bounded trajectories and the resulting dynamic feedback couplings can be interpreted as a generalization of existing methods for identical linear systems.
886 citations
TL;DR: An inversion procedure is introduced for nonlinear systems which constructs a bounded input trajectory in the preimage of a desired output trajectory which leads to a simple geometric connection between the unstable manifold of the system zero dynamics and noncausality in the nonminimum phase case.
Abstract: An inversion procedure is introduced for nonlinear systems which constructs a bounded input trajectory in the preimage of a desired output trajectory. In the case of minimum phase systems, the trajectory produced agrees with that generated by Hirschorn's inverse dynamic system; however, the preimage trajectory is noncausal (rather than unstable) in the nonminimum phase case. In addition, the analysis leads to a simple geometric connection between the unstable manifold of the system zero dynamics and noncausality in the nonminimum phase case. With the addition of stabilizing feedback to the preimage trajectory, asymptotically exact output tracking is achieved. Tracking is demonstrated with a numerical example and compared to the well-known Byrnes-Isidori regulator. Rather than solving a partial differential equation to construct a regulator, the inverse is calculated using a Picard-like interaction. When preactuation is not possible, noncausal inverse trajectories can be truncated resulting in the tracking-error transients found in other control schemes.
825 citations
Cites background from "Paper: The internal model principle..."
...Also, for linear systems, the asymptotic regulation and tracking of signals generated by finite-dimensional linear systems has been studied in a general framework by Francis and Wonham [ 2 ]....
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References
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Book•
01 Jan 1955
TL;DR: The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable as discussed by the authors, which is a useful text in the application of differential equations as well as for the pure mathematician.
Abstract: The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable. It has been developed from courses given by the authors and probably contains more material than will ordinarily be covered in a one-year course. It is hoped that the book will be a useful text in the application of differential equations as well as for the pure mathematician.
7,071 citations
Book•
03 Dec 1974
TL;DR: In this article, the authors present an approach to controlability, feedback assignment, and pole shifting in a single linear functional model, where the observer is assumed to be a dynamic observer.
Abstract: 0 Mathematical Preliminaries.- 0.1 Notation.- 0.2 Linear Spaces.- 0.3 Subspaces.- 0.4 Maps and Matrices.- 0.5 Factor Spaces.- 0.6 Commutative Diagrams.- 0.7 Invariant Subspaces. Induced Maps.- 0.8 Characteristic Polynomial. Spectrum.- 0.9 Polynomial Rings.- 0.10 Rational Canonical Structure.- 0.11 Jordan Decomposition.- 0.12 Dual Spaces.- 0.13 Tensor Product. The Sylvester Map.- 0.14 Inner Product Spaces.- 0.15 Hermitian and Symmetric Maps.- 0.16 Well-Posedness and Genericity.- 0.17 Linear Systems.- 0.18 Transfer Matrices. Signal Flow Graphs.- 0.19 Rouche's Theorem.- 0.20 Exercises.- 0.21 Notes and References.- 1 Introduction to Controllability.- 1.1 Reachability.- 1.2 Controllability.- 1.3 Single-Input Systems.- 1.4 Multi-Input Systems.- 1.5 Controllability is Generic.- 1.6 Exercises.- 1.7 Notes and References.- 2 Controllability, Feedback and Pole Assignment.- 2.1 Controllability and Feedback.- 2.2 Pole Assignment.- 2.3 Incomplete Controllability and Pole Shifting.- 2.4 Stabilizability.- 2.5 Exercises.- 2.6 Notes and References.- 3 Observability and Dynamic Observers.- 3.1 Observability.- 3.2 Unobservable Subspace.- 3.3 Full Order Dynamic Observer.- 3.4 Minimal Order Dynamic Observer.- 3.5 Observers and Pole Shifting.- 3.6 Detectability.- 3.7 Detectors and Pole Shifting.- 3.8 Pole Shifting by Dynamic Compensation.- 3.9 Observer for a Single Linear Functional.- 3.10 Preservation of Observability and Detectability.- 3.11 Exercises.- 3.12 Notes and References.- 4 Disturbance Decoupling and Output Stabilization.- 4.1 Disturbance Decoupling Problem (DDP).- 4.2 (A, B)-Invariant Subspaces.- 4.3 Solution of DDP.- 4.4 Output Stabilization Problem (OSP).- 4.5 Exercises.- 4.6 Notes and References.- 5 Controllability Subspaces.- 5.1 Controllability Subspaces.- 5.2 Spectral Assignability.- 5.3 Controllability Subspace Algorithm.- 5.4 Supremal Controllability Subspace.- 5.5 Transmission Zeros.- 5.6 Disturbance Decoupling with Stability.- 5.7 Controllability Indices.- 5.8 Exercises.- 5.9 Notes and References.- 6 Tracking and Regulation I: Output Regulation.- 6.1 Restricted Regulator Problem (RRP).- 6.2 Solvability of RRP.- 6.3 Example 1 : Solution of RRP.- 6.4 Extended Regulator Problem (ERP).- 6.5 Example 2: Solution of ERP.- 6.6 Concluding Remarks.- 6.7 Exercises.- 6.8 Notes and References.- 7 Tracking and Regulation II: Output Regulation with Internal Stability.- 7.1 Solvability of RPIS: General Considerations.- 7.2 Constructive Solution of RPIS: N= 0.- 7.3 Constructive Solution of RPIS: N Arbitrary.- 7.4 Application: Regulation Against Step Disturbances.- 7.5 Application: Static Decoupling.- 7.6 Example 1 : RPIS Unsolvable.- 7.7 Example 2: Servo-Regulator.- 7.8 Exercises.- 7.9 Notes and References.- 8 Tracking and Regulation III: Structurally Stable Synthesis.- 8.1 Preliminaries.- 8.2 Example 1: Structural Stability.- 8.3 Well-Posedness and Genericity.- 8.4 Well-Posedness and Transmission Zeros.- 8.5 Example 2: RPIS Solvable but Ill-Posed.- 8.6 Structurally Stable Synthesis.- 8.7 Example 3: Well-Posed RPIS: Strong Synthesis.- 8.8 The Internal Model Principle.- 8.9 Exercises.- 8.10 Notes and References.- 9 Noninteraeting Control I: Basic Principles.- 9.1 Decoupling: Systems Formulation.- 9.2 Restricted Decoupling Problem (RDP).- 9.3 Solution of RDP: Outputs Complete.- 9.4 Extended Decoupling Problem (EDP).- 9.5 Solution of EDP.- 9.6 Naive Extension.- 9.7 Example.- 9.8 Partial Decoupling.- 9.9 Exercises.- 9.10 Notes and References.- 10 Noninteraeting Control II: Efficient Compensation.- 10.1 The Radical.- 10.2 Efficient Extension.- 10.3 Efficient Decoupling.- 10.4 Minimal Order Compensation: d(?) = 2.- 10.5 Minimal Order Compensation: d(?) = k.- 10.6 Exercises.- 10.7 Notes and References.- 11 Noninteraeting Control III: Generic Solvability.- 11.1 Generic Solvability of EDP.- 11.2 State Space Extension Bounds.- 11.3 Significance of Generic Solvability.- 11.4 Exercises.- 11.5 Notes and References.- 12 Quadratic Optimization I: Existence and Uniqueness.- 12.1 Quadratic Optimization.- 12.2 Dynamic Programming: Heuristics.- 12.3 Dynamic Programming: Formal Treatment.- 12.4 Matrix Quadratic Equation.- 12.5 Exercises.- 12.6 Notes and References.- 13 Quadratic Optimization II: Dynamic Response.- 13.1 Dynamic Response: Generalities.- 13.2 Example 1 : First-Order System.- 13.3 Example 2: Second-Order System.- 13.4 Hamiltoman Matrix.- 13.5 Asymptotic Root Locus: Single Input System.- 13.6 Asymptotic Root Locus: Multivariable System.- 13.7 Upper and Lower Bounds on P0.- 13.8 Stability Margin. Gain Margin.- 13.9 Return Difference Relations.- 13.10 Applicability of Quadratic Optimization.- 13.11 Exercises.- 13.12 Notes and References.- References.- Relational and Operational Symbols.- Letter Symbols.- Synthesis Problems.
2,571 citations
"Paper: The internal model principle..." refers background in this paper
...Later, Otto Smith [4] incorporated an internal model in his scheme of predictive control; while from the 1970s the study of parameter-insensitive perfect asymptotic tracking led to the recognition of both error feedback and the internal model as necessary and sufficient structural features of “robust” linear multivariable systems; for references see Wonham [5]....
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...THE INTERNAL MODEL PRINCIPLE OF CONTROL THEORY W. M. Wonham, 2018.06.17 I. INTRODUCTION The Internal Model Principle (IMP) of control theory states (informally) that “a good controller incorporates a model of the dynamics that generate the signals which the control system is intended to track.”...
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TL;DR: In this paper, structural stability of linear multivariable regulators is defined and necessary and sufficient structural criteria are obtained for linear multi-variable regulators which retain loop stability and output regulation in the presence of small perturbations, of specified types, in system parameters.
Abstract: Necessary structural criteria are obtained for linear multivariable regulators which retain loop stability and output regulation in the presence of small perturbations, of specified types, in system parameters. It is shown that structural stability thus defined requires feedback of the regulated variable, together with a suitably reduplicated model, internal to the feedback loop, of the dynamic structure of the exogenous reference and disturbance signals which the regulator is required to process. Necessity of these structural features constitutes the ‘internal model principle’.
1,090 citations
TL;DR: Today, as a step towards the control of complex dynamic systems, models are being used ubiquitously, being modelled, for instance, are the air traffic flows around New York, the endocrine balances of the pregnant sheep, and the flows of money among the banking centres.
Abstract: Today, as a step towards the control of complex dynamic systems, models are being used ubiquitously. Being modelled, for instance, are the air traffic flows around New York, the endocrine balances of the pregnant sheep, and the flows of money among the banking centres.
1,017 citations