CONTROL
The position error with controller KDKW,3 decreased to the half or third of that with controller KDK. Closed-loop experiment data with controller KDKW,3 not invalidates the model, so the final controller can be accepted.
It can also be concluded that the strategy of braking only the front wheels is con-firmed: a satisfactory steering performance (yaw-rate of0.2rad/s) can be achieved with moderate braking effort (below5bar).
6.7. SUMMARY AND THESIS 3 on dry asphalt and in normal driving situations to steer a heavy vehicle and, for this reason, appropriate for use in an emergency lane-keeping system.
The contribution of this chapter can be summarized in the following thesis:
Thesis 3
A control strategy, including the process of modelling, identification and control design, is elaborated for steering a vehicle by using the electronic brake system. The design involves the following steps:
I. Velocity scheduled linear model construction
1. Physically parameterized continuous-time state-space models are derived de-scribing yaw dynamics, steering system and wheel dynamics
2. A structure estimation and model reduction method is developed which is based on physical considerations on model reduction, parameter identification and pole analysis of models with frozen velocity
II. Robust control design
1. By using linear matrix-inequalities, unfalsified uncertainty models are parame-terized in a numerically tractable simple structure involving multiplicative dy-namic perturbation and additive disturbance
2. Robust performance bound is minimized iteratively based on µ synthesis and shaping of uncertainty weighting functions
It is shown that 1.) the individual braking of the front wheels alone provide satisfactory steering performance in normal driving conditions with relatively low, applicable brak-ing effort and 2.) the elaborated control design method leads to improved robust control performance compared to pure µ synthesis.
Own publications related to the chapter and thesis: [Röd03, RB04b, RB04a, RB05b, RB05a, Röd07, RGSB08, Röd09, RBar, RGB10].
CONTROL
Chapter 7
Conclusions
The aim of this chapter is to sum up the results of the previous chapters. The central theme of the dissertation, which connects the three theses, is structured uncertainty modelling and robust control design. In the first thesis the basic concept is elaborated for LTI systems and an iterative algorithm is developed. The second thesis presents the algorithm for LPV systems. The concept is utilized in the third thesis where a concrete vehicle steering problem is elaborated. The three theses are summarized as follows.
1. An iterative algorithm has been elaborated for LTI models with the aim of shaping structured uncertainty models and designing robust controllers based on measure-ment data. The algorithm handles unstable experimeasure-ments ensuring safe improve-ment of guaranteed robust performance on the true, unknown system. For additive uncertainty structures, the algorithm is formulated as a series of LMI problems.
The steps of skewµsynthesis has been elaborated as part of the algorithm. It has been proved that performance degradation beyond the guaranteed level implies falsification of the actual model, which forces the continuation of the algorithm.
2. An iterative uncertainty modelling and robust control design algorithm is elabo-rated for LPV models with LFT dependence on scheduling parameters, structured dynamic uncertainty and disturbances. Based on time- and frequency-domain IQCs, guaranteed robust quadratic performance level is minimized by searching for an unfalsified uncertainty model and a robust LPV controller. For additive uncertainty structures, the algorithm is formulated as a series of LMI problems.
Due to the advantageous properties of the algorithm, unstable experiments are handled and improvement of guaranteed robust performance on the true, unknown system is ensured.
3. A control strategy, including the process of modelling, identification and control design, is elaborated for steering a vehicle by using the electronic brake system.
The design involves the following steps:
I. Velocity scheduled linear model construction
1. Physically parameterized continuous-time state-space models are derived describing yaw dynamics, steering system and wheel dynamics
2. A structure estimation and model reduction method is developed which is based on physical considerations on model reduction, parameter iden-tification and pole analysis of models with frozen velocity
II. Robust control design
1. By using linear matrix-inequalities, unfalsified uncertainty models are pa-rameterized in a numerically tractable simple structure involving multi-plicative dynamic perturbation and additive disturbance
2. Robust performance bound is minimized iteratively based on µsynthesis and shaping of uncertainty weighting functions
It is shown that 1.) the individual braking of the front wheels alone provide satisfactory steering performance in normal driving conditions with relatively low, applicable braking effort and 2.) the elaborated control design method leads to improved robust control performance compared to pureµ synthesis.
The work is motivated by complex dynamical systems in safety critical applications.
The strategy is to design robust controllers based on low order nominal models equipped with carefully tuned sophisticated (structured) dynamic uncertainty models. The basic concept for the design of uncertainty models is that the uncertainty model is, first, augmented by the disturbance model and, then, all weighting functions (characterizing the size and shape of the uncertainty set) are designed along with a fixed structure.
The structure can be chosen based on physical modelling. The weight selection is based on measurement data of, mainly, closed-loop experiments, in order to capture the systems’s behaviour during the target operation. In order to decrease conservatism of the modelling, all unfalsified (consistent) models are considered and that one is chosen that minimizes the robust performance criterion of the control design. On the same time, robustness is to be guaranteed against the set of systems that is represented by thechosen unfalsified model. This concept implies a joint (iterative) design of controller and uncertainty model (D-K-W iteration). The convergence of the iteration cannot be guaranteed, however, examples show its practical usefulness.
The problem is formalized as a robust control problem for an unknown set of true systems. The performance is defined in terms ofL2-norm of given performance signals.
Important statements of the thesis (Lemma 4.1 and Theorem 4.2) establish connection between the consistency of the actual model and control performance in future exper-iments: if the model is consistent with any new data then performance is guaranteed for the true system, and if a measured performance is greater than the guaranteed skew µ performance, then the model is inconsistent. Assuming that performance degrada-tion or instability can be detected during the experiments (e.g. by monitoring levels of measured variables), a safe iterative algorithm involving closed-loop experiments and D-K-W iteration is elaborated where the iteration is supervised based on monitoring the consistency.
Naturally, the consistency of the model with the unknown set of (true) systems cannot be validated. This is the principal problem of all validation methods as well, in-cluding any model and controller validation methods. However, the introduced method provides a picture on what is the ideal shape of a structured uncertainty model in a given control problem.
In the first two theses the method is elaborated respectively for LTI and LPV sys-tems. LPV systems are widely applied in safety critical control problems for modelling a class of nonlinear systems, because of the advantage that robustness guarantees can be provided based on LMIs. This motivated Thesis 2.
In the third thesis an emergency steer-by-brake problem is elaborated for heavy vehicles. It is assumed that a high level algorithm exists that, based on visual informa-tion, generates yaw-rate reference signal that the vehicle should track. The objective of the thesis is to provide a control law that guarantees robust tracking performance by manipulating the electronic brake system. The following approach is elaborated which is the main contribution of the thesis. Using only the front wheel brakes the steering system is turned aside and so the vehicle is governed. The efficiency of this approach is proved by measurements on a real heavy vehicle [RB05a]. A nominal model is de-veloped describing the system between brake cylinder pressure and yaw-rate. Then, the robust control problem is composed by defining the augmented closed-loop system.
Uncertainty modelling and control design is performed by using the iterative algorithm of Thesis 1. The efficiency of the algorithm is demonstrated on a high-fidelity vehicle simulator.
Appendix A
Simple numerical example
The following simple example is aimed at demonstrating the D-K-W inner loop of Algorithm 1 through both analytical and numerical calculations. An output sensitivity problem is to solve with the help of standard µ synthesis ([137, 13]) where nominal model is subject to additive dynamic perturbation and additive disturbance. In this two scalar block µ problem, standard µ criterion can be calculated analytically, so numerical calculations can be validated.
A.1 Problem formulation
Figure A.1 shows a simplified output sensitivity problem. Suppose that input-output measurements (u, y) ∈ E (or, equivalently, (u, e) ∈ E with e := y−Gnu, where Gn
denotes the nominal model) are given in the frequency-domain. The goal is to design a controllerK that minimizes the outputzp of the closed-loop system subject to all possi-ble additive uncertainty. The uncertainty consists of additive perturbation and additive disturbance and must be modelled based on data. In the output sensitivity problem, measured output y is penalized by performance weighting function Wz, which defines performance output zp = Wzy. The performance input is the additive disturbance wp =d.
Robust performance is equivalent with condition γ := kTzdk∞ < 1, whenever k∆k∞≤1, where the transfer function of the closed-loop system is defined by
Tzd=Wz(1−(Gn+W∆∆)K)−1Wdd
This shows that γ can be made arbitrary small by decreasing the gain of Wd to the detriment ofW∆ (uncertainty modelling problem) while simultaneously decreasing the controller gain as well.