Preface ix
Introduction and Motivations xi
Chapter 1. Metaheuristics for Controller Optimization 1
1.1. Introduction 1
1.2. Evolutionary approaches using differential evolution 2
1.2.1. Standard version 2
1.2.2. Perturbed version 7
1.3. Swarm approaches 8
1.3.1. Particle swarm optimization algorithm 8
1.3.2. Quantum particle swarm algorithm 14
1.3.3. Artificial bee colony optimization algorithm 20
1.3.4. Cuckoo search algorithm 25
1.3.5. Firefly algorithm 31
1.4. Summary 33
Chapter 2. Reformulation of Robust Control Problems for Stochastic Optimization 35
2.1. Introduction 35
2.2. H infinity synthesis 35
2.2.1. Full H infinity synthesis 35
2.2.2. Fixed-structure H infinity synthesis 45
2.2.3. Formulating H infinity synthesis for stochastic optimization 67
2.2.4. Conclusion 105
2.3. mu-Synthesis 105
2.3.1. The problem of performance robustness 105
2.3.2. mu-Synthesis 110
2.4. LPV/LFT synthesis 140
2.4.1. Introduction 140
2.4.2. The LPV/LFT controller synthesis problem 141
2.4.3. Reformulation for stochastic optimization 147
Chapter 3. Optimal Tuning of Structured and Robust H infinity Controllers Against High-level Requirements 171
3.1. Introduction and motivations 171
3.2. Loop-shaping H infinity synthesis 180
3.2.1. Approach principle 180
3.2.2. Generalized gain and phase margins 184
3.2.3. Four-block interpretation of the method 185
3.2.4. Practical implementation 186
3.2.5. Implementation of controllers 190
3.3. A generic method for the declination of requirements 194
3.3.1. General principles 194
3.3.2. Special cases 196
3.3.3. Management of requirement priority level 197
3.4. Optimal tuning of weighting functions 198
3.4.1. Optimization on nominal plant 198
3.4.2. Multiple plant optimization 202
3.4.3. Applicative example - inertial stabilization of line of sight 207
3.5. Optimal tuning of the fixed-structure and fixed-order final controller 238
3.5.1. Introduction 238
3.5.2. Toward eliminating weighting functions 240
3.5.3. Extensions to the approach 259
3.5.4. Link with standard control problems 277
Chapter 4. HinfStoch: A Toolbox for Structured and Robust Controller Computation Based on Stochastic Optimization 279
4.1. Introduction 279
4.2. Structured multiple plant H infinity synthesis 280
4.2.1. Principle 280
4.2.2. Formalism 280
4.3. Structured mu-synthesis 284
4.3.1. Principle 284
4.3.2. Formalism 285
4.4. Structured LPV/LFT synthesis 288
4.4.1. Principle 288
4.4.2. Formalism 289
4.5. Structured and robust synthesis against high-level requirements with HinfStoch_ControllerTuning 292
4.5.1. Principle 292
4.5.2. Formalism 293
4.5.3. Examples 311
Appendices 351
Appendix A. Notions of Coprime Factorizations 353
Appendix B. Examples of LFT Form Used for Uncertain Systems 359
Appendix C. LFT Form Use of an Electromechanical System with Uncertain Flexible Modes 365
Appendix D. FTM (1D) Computation from a Time Signal 383
Appendix E. Choice of Iteration Number for CompLeib Tests 385
Appendix F. PDE versus DE 393
Bibliography 399
Index 407
Introduction and Motivations
I.1. Developing control engineering in an industrial framework
The problem of inertial stabilization involves creating an image in which orientation and quality do not depend on its carrier. To do this, optronic sensors are carried on a mechatronic servo device inertially stabilized with gyrometers or gyroscopes, which gives the viewfinder particular features for observation, detection and identification. As an example, we list viewfinders for tanks, helicopters, periscopes, missile guidance, etc.
Owing to the fundamental principle of the dynamic for a solid in rotation, the absolute rate Oa of the line of sight with inertia J is governed by:
[I.1] Therefore, inertial stabilization is a disturbance rejection problem: with a servo-loop, a useful torque Cmot is produced (generally provided by an electric motor) to compensate external disturbances Cext being applied to the load at each moment. Several architectural solutions can be found in [MAS 08] and [HIL 08].
To be able to use its features, the viewfinder should have an adequate range. This range is directly linked to the viewfinder's stabilization performance, especially in mechanical environments and relatively harsh conditions of use and should be compatible with the optronic sensors' definition; the latter are characterized by an integration time (Ti) (the time during which the image is acquired and electronically generated) and by the size of their pixel detectors (IFOV)1.
Hence, to have an adequate range, the absolute angular performance of the line of sight2 has to be compatible with the IFOV and the integration time Ti, in response to all disturbances.
The problem of inertial stabilization consists of rejecting mainly two types of external disturbance:
- - The first is the friction torque Gf created during relative motion between the viewfinder carrier and the line of sight. This disturbance appears when the viewfinder carriers' orientation changes at low frequencies. In order to avoid image shaking during observation (and therefore provide visual comfort), we should make the line of sight angular dynamic movement in response to Gf compatible with the IFOV. Similarly, the steady-state error caused should be null. More particularly, the dynamic of the friction rejection should be compatible with the dynamic of the host vehicle's oscillation regime, which is only a few hertz, ten at the very most. The error in response to the friction should therefore be null (or at least very weak) after a time Tf. To validate the performances against Gf, an oscillating table simulating the host vehicle's angular motion at low frequencies is generally used.
- - The second disturbance Gv comes from the mechanical distortions of the viewfinder's structure which, through its flexibility, transmits to the line of sight part of the mechanical environment to which the viewfinder is subject and indeed amplifies it. In order to enable the viewfinder to operate its observation features, we should limit the blurring caused by this high-frequency disturbance3 to a value compatible with the IFOV during the integration time Ti. To validate performances against Gv, a shaker simulating the host vehicle's high-frequency vibrations is generally used.
In addition to these performance requirements, other requirements should be taken into consideration when designing the servo-loop:
- - because the viewfinder is an embedded device, and/or to preserve the integrity of the motorization, a requirement on the stabilization stage's energy consumption is necessary (on the maximum instantaneous power, on the maximum current or tension, etc.);
- - because the viewfinder is a mass system used in variable environments, the design for the control law should be made with robustness constraints (stability robustness, performance robustness) considering uncertainties.
Thus, the automation engineer should design a single control law that they will validate on a single prototype, with a degree of robustness sufficient to satisfy a complex specification on a large number of systems. In reality, this is the objective sought by any automation engineer working in an industrial framework, wishing to develop the most effective and robust control law possible in the shortest possible length of time.
The general methodology for developing a servo controller on a prototype viewfinder can be summarized in Figure I.1. In reality, it is a good example of the methodology used for solving automation problems in an industrial context. In fact, it has the four usual phases of servo-loop development:
- - declining high-level complex specifications using simplifications, linearization, etc. in order to build some linear frequency shapes;
- - modeling the system and its uncertainties;
- - synthesizing control laws;
- - experiments.
However, this methodology is sub-optimal and time-consuming, and therefore expensive, as:
- - The final control law, which is obtained at stage 7 through inevitable simplifications (linearization, etc.) and a repetitive (stages 5 and 6) and expensive experimental process, should satisfy an often complex high-level specification (stage 1). These simplifications are needed to use control techniques (stage 4), which in fact require the engineer to have a degree of expertise.
- - Based on the standard form for control, modern control techniques such as H8 synthesis used at stage 4 enable a (potentially multi-variable) controller to be determined, which is optimal (in the sense of the H8 norm) to make linear closed-loop or open-loop shapes being satisfied (stage 3). These shapes are materialized via setting the frequency weights on the signals to be monitored and on the exogenous inputs (the standard approach), or directly on the system's inputs and outputs (the loop-shaping approach). Thus, the most complex task of stage 4 is not computing the controller itself, but declining the simplified specifications into a judicious choice of weights; this adjustment is in itself a repetitive process requiring the engineer to have a high level of expertise, especially as they should ask a number of questions, for example:
- - For a SISO4 problem, how should we choose the structure of the weights to be set, the number of poles/zeros and the order?
- - In the MIMO5 case where we generally add as many weights as available measures and commands, the number of transfers to be shaped becomes very high so that we generally proceed to a sequential and decoupled tuning of the weights of the different channels by assuming that the weights are diagonal structured: what about the optimality of the tuning under these simplifying structural hypotheses? Which is the most adapted structure, diagonal or full?
- - What is the best trade-off attainable between performance and robustness?
- - We note that some very powerful robust synthesis techniques such as µ-synthesis become sub-optimal through their resolution method, which can naturally restrict their use.
- - Finally, modern synthesis methods generate an often high-order controller; in general, we proceed to a post-reduction of the latter before its implementation. This post-reduction can degrade performance and robustness so much that it is necessary to take account of the order constraint beforehand. Moreover, there is also the question of finding the best controller structure for the problem posed: is it a cascaded multi-loop structure, or a multi-variable feedback correction, or a decentralized correction?
Here, we have listed some examples of questions that the engineer is led to ask. This list is not exhaustive. In reality, the right answer to these questions depends on the engineer's level of expertise and on his or her ability, and most of the time, we have to proceed to simplifications in order to move forward in the process of synthesis, and simplifications making the final solution costly and sub-optimal for the primary objective, which is to satisfy the high-level system specification of stage 1.
Thus, the aim of this work is to make servo controller synthesis in the industrial framework better adapted through being more direct and therefore less time-consuming to develop, by computing a final (structured) controller by directly tackling the high-level specification system.
Figure I.1. Method for developing controllers in viewfinders
I.2. The place of optimization
Optimization always plays a major role in solving automation problems, which most of the time means calculating a stabilizing controller that meets specifications. The problem to be solved can often be expressed in the general form of an optimization problem in which an objective or cost function (or indeed several) f(x) that needs to be minimized in relation to the parameters involved x is defined. The definition of such a problem is often completed by constraints on x. The decision variables (the parameters to be determined) are therefore the variables that define the searched controller (for example, the matrices of its state-space representation or the coefficients of its transfer function).
[SCH 97] shows that multiple technical...
