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State of the art
Control engineering is considered as a mature technology in many different ways, being able of dealing with almost any kind of application in the industrial context. The literature is rich in algorithms to design control systems, even highly complex control problems. But, in practice, several problems appear for its implementation, especially when the system is exposed to dynamics, instrumental or environmental changes. In short, a lot of tools exists to design feedback controllers for a system with a known structure, but they are not providing proper responses when the structure of the system to be controlled changes over time. The problem of designing a controller able to deal with these changes is not new: Fault Tolerant Control (FTC) specializes in the case of components that fail. The work in this area, however, is usually limited to a prespecified amount of faults, and the problem of handling new components is not addressed. Another field that considers changing systems, it is the adaptive control area, which allows tracking changes that can be defined as parameters in the controlled system. Changes for controller reconfiguration need to be identified somehow. Since those changes are already set as predefined parameters, structural changes or new dynamics introduction are not considered either. On the other hand, robust control considers a system that changes their characteristics over time through uncertainties. These changes are somewhat bounded, so a fixed controller can be designed, guaranteeing an acceptable behavior. However, robust control design is not possible in scenarios where changes in the system are large. Hierarchical structures to deal with running structural changes have been also widely studied in the areas of decentralized, distributed, hierarchical or networked controls. Handling these structural changes also involves dealing with the transients when changing from one system to the other. Considerations about transient behavior when doing controller reconfiguration can be found on the bumpless transfer control area. In short, there are many different solutions depending on the nature of the problem, and a control/supervision structure would be necessary to deal with all the types of changes that may come.
Youla-Jabr-Bongiorno-Kucera (YK) parameterization is a control framework that appeared simultaneously in [Kučera, 1975, Youla et al., 1976a, Youla et al., 1976b]. YK parameterization provides all stabilizing controllers for a given system. All stablizing controllers are parameterized based on the transfer function called YK parameter Q, so K(Q). It can be used to perform stable controller reconfiguration when some change occurs. The type of controller could be any– classical, adaptive, optimal or robust control. Mixing different types of controller is allowed in this controller reconfiguration. The dual theory, dual YK parameterization, provides all the plants stabilized for a given controller. The class of all the plant stabilized by a controller depends on the transfer function called dual YK parameter S, so G(S). This parameter could represent any plant variations, uncertainties, parameter variations, change of operation point, etc. This is employed for dynamics identification and/or identification of new sensors/actuators connected to a system. Finally, both can be used together, so a control structure that changes based on identified dynamics is obtained; as sensors/actuators are identified, hierarchical and fault tolerant control structures are also supported by YK. Controller is changed depending on new dynamics with some performance/stability criteria. With these premises, YK is able to encompass all the solutions proposed at the beginning of this chapter within the same theoretical framework and with stability guarantees. Thus, it could serve as a general control framework to deal with systems exposed to dynamics, instrumental or environmental changes.
The present chapter gives an overview of the YK parameterization research field. The origins of this technique are explained. Important groups worldwide are reviewed, focusing on the different type of control applications by using this mathematical framework. Applications are divided depending on whether Q or S are used, or both.
This idea of reparameterizing a set plant-controller in order to obtain linearity reappeared in [Zames, 1981]; and is well known as internal model control in chemical control process [Morari and Zafiriou, 1989]. But, it was not able to be applied for Multi-Input-Multi-Output (MIMO) systems. [Kučera, 1975] and [Youla et al., 1976a,Youla et al., 1976b] proposed simultaneusly discrete and continous solutions to deal with MIMO unstable plants– so-called Youla-Kucera parameterization. There are two key points in the solutions: First, an initial stabilizing controller is considered; and second, plants are described using stable polynomial fractional transformations. Its use permited to see the plant as the combination of two stable transfer functions– e.g. an unstable plant G(s) = 1/(s − 5) is represented by X (s)Y (s)−1 with X (s) = 1/(s + 1) and Y (s) = (s − 5)/(s + 1). These factors were employed in order to obtain an equivalent to Q(s) in Eq. 2.1. This new Q(s), called YK parameter, characterizes the class of all stabilizing controllers depending on stable polynomial fractional factors for G(s) and an initial K(s). Linearity was preserved even if stable polynomial fractional factors were used.
This approach was updated with coprime factors in order to avoid algebraic difficulties as noticed by [Desoer et al., 1980] and [Vidyasagar, 1985] for SISO and MIMO systems. An efficent method for obtaining these factors is based on a state-space representation [Nett et al., 1984]. As those coprime factors are the basis for obtaining the class of all stabilizing controllers, this state-space representation is preserved in almost every future application.
The linearity of Q within the Closed-Loop (CL) function facilitates optimization over the class of all stabilizing controllers. Every single controller could be augmented with Q. This Q is seen as a stable filter that can be optimize offline or online in order to improve system’s performace. An adaptive Q technique could be no longer useful when systems variations or uncertainites are large. A controller solution for such situations is provided by the dual YK parameterization.
Coprime factors of an inital plant connected to a stabilizing controllers are used in order to obtain the class of all plants stabilized by a controller. The connection between the dual YK and YK parameterizations was first developed by [Tay et al., 1989a], giving robust stability results. This dual YK parameter was used to suppress CL identification difficulties in [Hansen et al., 1989, Schrama, 1991]. The identification of a plant in the presence of a feeedback loop could be complex due to the noise. Given an initial model and controller, by identifying the dual YK S instead of G(s), the CL problem is transformed into an Open-Loop (OL) like problem. This is called in the literature Hansen scheme. The resulting S is used to carefully redesign the filter Q such that a better performance is achieved without loosing the stability of the system.
Performance enhancement techniques working with an adaptive Q are seen particularly in the work of the Australian National University. It is also there that the dual YK parameterization and robust stability results were first developed. Later, the Technical University of Denmark analysed Q as a fixed filter, focusing in stable controller reconfiguration properties. Stable controller recon-figuration through Q is combined with a fault-detection system through the dual YK parameter S, obtaining a Fault-Tolerant-Control (FTC) solution. Finally, some results are in proceeding with structural changes through S at the University of Aalborg. A more detailed review of the YK de-sign methodologies at the Australian National University, in the Technical University of Denmark and at the Aalborg University is in the following three sections.
Australian National University
The department of System Engineering, Research School of Information Sciences and Engineering at the Australian National University was the pioneer in using YK parameterization, to get what they called high performance control. The concept of high performance control is to use the tools of classical, optimal, robust and adaptive control in order to deal with complexity, uncertainty and variability of the real world. They aimed to find a mathematical framework able to join performance and robustness.
First steps in this direction were made by John Moore at 1970s, on a high order NASA flexible wing aircraft model with flutter mode uncertaintiess. Least square identification was used in order to have an adaptive loop based on linear quadratic optimal control able to achieve robustness to these uncertainties. However, the blending betweeen adaptive and robust control lacked a mathematical framework. A collaboration with Keith Glover at Cambridge University allowed them to discover the interpretation of the YK parameterization as a general solution to optimal control problem provided by [Doyle, 1983]. Doyle characterized the class of all stabilizing controller as an initial Linear-Quadratic-Gaussian (LQG) controller with a stable filter Q, what fits the adaptive filter that they put in their solution for the aircraft model. A graduate student, Teng Tiow Tay started working on how to use that theory, obtaining really encouraging results. Initial point of his thesis with Moore as supervisor [Tay et al., 1989b]. Different applications related to YK formed a new field of research. We refer to book [Tay et al., 1997] and article [Anderson, 1998] as principal referees for undestanding how to use YK parameterization towards high performance control. Different techniques and applications related to that are detailed below:
Q offline control design
An offline optimization of Q was carried out in order to achieve various performance objectives. The idea is to design a controller in the class of all stabilizing controllers instead of over the class of all possible controllers (which includes destabilizing controllers). Different control performance objectives can be set in order to optimize the YK filter Q. Performance requirements can be described in time or frequency domain. System norms in the frequency domain is directly related to optimal control. H∞ is concerned primarily with the peaks in the frequency response, while H2 is related to the overall response of the system. The idea is simple, once a transfer function between different signal of interest is determined, H∞ control designs a stabilizing controller that ensures that the peaks in the transfer function are knocked down; on the contrary, H2 or LQG control designs a stabilizing controller that reduces the H2 of the transfer function as much as possible. Penalization of the energy of the tracking error and control energy are examples of LQG control; while penalizing the maximum tracking error subject to control limits is an example of H∞ control.
The design of an LQG controller with loop transfer recovery was analysed. This LQG controller uses a state estimator with the aim of estimating the non-accesible states of the plant. Its aim is to minimize error tracking and control effort. The controller will be optimal if a good model of the plant has been considered, otherwise the performance could be poor. Loop transfer recovery refers to the idea of reconfiguring the initial LQG controller to achieve full or partial loop transfer recovery of the original feedback loop. This is usually done through a scalar parameter as a trade-off between performance and robustness. In [Moore and Tay, 1989], loop recovery was achieved by augmenting the original LQG controller with the additional YK filter Q. They showed how full or partial loop recovery may be obtained depending if minimum or non-minimum phase plants are considered. The technique was ilustrated for the case of minimum and non-minimum phase plants through simulation. Improvements over standard loop recovery techniques were obtained.
The CL transfer function including Q from a disturbance input to a tracking error with a H∞ norm is in chapter 4 of [Tay et al., 1997]. This equation is useful to keep the tracking error within a given tolerance. However, minimizing the tracking error could lead sometimes to large control efforts, what would be unnaceptable. As weighting factors between tracking error and control effort in a H∞ setup is not allowed [Dahleh and Pearson, 1986], a l1 equivalent was proposed in [Teo and Tay, 1995]. This algorithm allows to choose the correct weighting factors in a l1 manner. A curve with all the possible solutions is generated for a simulation example, analysing the limitations and choosing the best weighting factors. This strategy is used in a hard disk servo system to minimize the maximum position error signal [Teo and Tay, 1996], which is the deviation of the read/write head from the center of the track.
Direct adaptive Q-control
In the previous section, an offline optimization of the YK parameter Q has been explained. [Wang et al., 1991] presented the first results in online optimization of Q without an identification process. The method is valid when the uncertainty is limited but unknown, and the plant-model mistmatch is not important. The optimization process is based on root-mean-square signals measures. A state-space relationship between a nominal plant with disturbances and a observer-based feedback controller K(Q) is obtained. The order of Q should be fixed depending on the application. A steepest descent algorithm is used to obtain the parameter values of the predefined YK parameter Q, so the error is minimized from the disturbances on the system. Simulation results of the direct adaptive-Q controller were presented in [Tay and Moore, 1991] to ilustrate their performance enhancement capabilities when disturbances appear on the system. Part of these results were previouysly validated in a 55th order aircraft model with a controller design via LQG with Q augmentations for achieving resonance suppression in [Moore et al., 1989].
In chapter 6 of [Tay et al., 1997], this method was analyzed to discover its limitations. First, a perfect plant model with disturbances was considered, achieving again without problems an optimal control. Then, the model-plant mistmach case was analysed, seeing how the adaptive mechanism breaks down under severe model-plant mismatch. An identification algorithm would be needed when a large model-plant mistmach is present. Section below presents an extension of the first YK-based CL identification algorithm– Hansen scheme.
CL identification
CL identification provided by [Hansen et al., 1989] is extended when connected to a controller with the YK filter Q in chapter 5 of [Tay et al., 1997]. Robust stabilization results in [Tay et al., 1989a] connecting K(Q) and G(S) are used to obtained an unbiased identification of S when a YK parameter Q is applied. A time-invariance property of Q is considered in the result. The unbiased
CL identification of is done through the identification of ˆ = ( )−1 , which includes .
S S SI−QS Q In order to obtain the real value of S, Q needs to be known. This CL identification method is the basis of the iterated (Q, S) control design shown below.
On the other hand, the original Hansen scheme is also extended with a non-linear initial model G(s) connected to a stabilizing controller K(s) in [Linard and Anderson, 1996, Linard and Anderson, 1997]; and [De Bruyne et al., 1998] presented a modification able to tune the order of the resulting model given by the Hansen scheme.
Table of contents :
Acknowledgments
1 Introduction
1.1 Motivation
1.2 Objectives
1.3 Manuscript organization
1.4 Contributions
1.5 Publications
1.5.1 Journal articles
1.5.2 Conference papers
2 State of the art
2.1 Origins
2.2 Australian National University
2.2.1 Q offline control design
2.2.2 Direct adaptive Q-control
2.2.3 CL identification
2.2.4 Iterated/Nested (Q, S) control design
2.2.5 Indirect adaptive (Q, S)-control
2.3 Technical University of Denmark
2.4 Aalborg University
2.5 Discussion
3 Youla-Jabr-Bongiorno-Kucera parameterization
3.1 System description
3.1.1 The nominal plant model
3.1.2 The stabilizing controller
3.2 Doubly coprime factorization
3.3 All stabilizing controllers/Controller reconfiguration
3.3.1 From a initial stabilizing controller to a final stabilizing controller
3.3.2 From a initial stabilizing controller to several stabilizing controllers
3.3.3 Controller structures
3.4 Numerical examples
3.4.1 Stable transition
3.4.2 Root locus evaluation
3.4.3 Transient behavior
3.5 Conclusions
4 Dual Youla-Jabr-Bongiorno-Kucera parameterization
4.1 System variations
4.2 Doubly coprime factorization
4.3 All systems stabilized by a controller
4.3.1 From a nominal model to a real model
4.4 Adaptive control design
4.4.1 Multi model adaptive control
4.5 Dynamics identification
4.5.1 Open-loop identification
4.5.2 Closed-loop identification
4.6 Conclusions
5 Applications
5.1 Experimental platform and simulation models
5.1.1 Cycab
5.1.2 Nissan Infinity M56
5.2 YK controller reconfiguration
5.2.1 Cooperative adaptive cruise control
5.2.2 Advanced cooperative adaptive cruise control
5.2.3 Cut-in/cut-out transitions in CACC systems
5.3 Closed-loop identification
5.3.1 Online closed-loop identification for longitudinal vehicle dynamics
5.4 Adaptive control design
5.4.1 Multi model adaptive control for cooperative adaptive cruise control applications
5.5 Conclusions
6 Conclusions
6.1 Contributions to the state of the art
6.1.1 Youla-Kucera parameterization
6.1.2 CACC
6.1.3 Autonomous driving
6.2 Future research directions
Bibliography
