The key question being investigated in this paper is that of how to accurately recover the correct equations of motion of a dynamical system together with the associated parameters. Individually, both of these tasks have received a significant amount of attention, both within the remit of structural dynamics as well as in the more general context of dynamicalsystems. However, combined modelselection and parameter estimation is a significantly more challenging task. Models of higher complexity tend to also be better predictors and successful model comparison requires one to take this into account and balance complexity against quality of fit. Bayesian inference has emerged as a powerful tool to address exactly this type of problem; it has been studied in the field of system identification owing to its ability to quantify uncertainty in parameter estimates [3, 4]. This uncertainty quantification leads directly to the idea of Bayesian model comparison [5, 6], where one seeks to compare the quality of fit of diﬀerent models according to posterior probability distributions (after observing evidence) over them.
Before analyzing the results, it should be pointed out that the poles/zeros with large stability margin are not shown in the range of the real axis (x-axis) since their variation are negligible. From Fig. 6, it can be seen that for high SNR, i.e., case 1, the uncertainties regions on each pole and zero can be distinguished. However, for low SNR (case 3), the regions show interference and an explicit variance estimation of the system poles/zeros are not possible. Unlike the SISO case (not reported here due to the lack of space), the variation of the identified model is not necessarily close to the nominal values (see the uncertainty regions associated with shaker zeros). This issue is closely related to Tustin transformation that is involved in the identification and indicates the sensitivity of the algorithm w.r.t. distortions of poles/zeros close to the imaginary axis in discrete frequency-domain. As reported in (Vuerinckx et al. 2001), the 95 % confidence interval, although may be used as an approximation, represents an inaccurate estimation of the uncertainty bounds. Having the variance of BLA in Fig. 2 in mind, it is naturally
Identification of nonlinear systems is a challenge since several different model structures as well as estimation approaches can be used. The choice of model structure plays an important role in identification of all kinds of models. Classical, simple first and second order linear models reveal directly information of the dynamical behavior to the user of the models, e.g. the size of gain and time constants. Also complex nonlinear system models can reveal useful information if the structure has been chosen with care. Since there are several possibilities to express complex nonlinear models it is important to choose a structure that is compatible with the dynamic behavior of the system and tells as much as possible about the dynamical behavior to the user of the model. It is easier to work with black box models that have a model structure that reveals certain information of system dynamics than to work with abstract black box models.
is satised (see Theorem 3.2). The control law (1.2) provides a sub-optimal control law for the uncertain plant (1.1) in the sense of nding an optimal control law for the reference model (1.3) and an asymptotic regulator via the adaptive feedback law (1.4). Thus, the problem of constructing a feedback law for the uncertain plant (1.1) can be decomposed into (i) the optimal control problem of r ( t ) for the reference model (1.3) and (ii) the asymptotic tracking problem in the sense of (1.6).
The results given here may be used in model-based advanced control of complex systems, such as adaptive control, robustcontrol, sliding-mode control, H-infinite control, etc. [1, 3–6, 23, 25, 30]. Methods and schemes proposed in the paper possess such features as reliability, sufficient simplicity of computational algorithms and relatively high speed of their processing, so these schemes allow using them in real time e.g. in problems of robustcontrol, stability, problems of control synthesis for dynamic systems of various types including problems of forecasting financial results in economic planning and other fields.
Abstract—this paper introduces a method to design a robust adaptive predictive control based on Fuzzy model. The plant to be used as predictive model is simulated by Takagi- Sugeno Fuzzy Model, and the optimization problem is solved by a Genetic Algorithms or Branch and Bound. The method to tune parameters of the model predictive controller based on Lyapunov stability theorem is presented in this paper to bring higher control performance and guaranty Global Asymptotical Stable (GAS) for the closed- loop system. This method is used for nonlinear systems with non-minimum phase (CSTR), uncertain dynamicalsystems and nonlinear DC motor. The simulation results for the Continuous Stirrer Tank Reactor (CSTR), nonlinear uncertain dynamical system and nolinear DC motor are used for verifying the proposal method.
In this paper a modified model reference adaptive control (MRAC) technique is presented which can be used to controlsystems with nonsmooth characteristics. Using unmodified MRAC on (noisy) nonsmooth systems leads to destabilization of the controller. A localized analysis is presented which shows that the mechanism behind this behavior is the presence of a time invariant zero eigenvalue in the system. The modified algorithm is designed to eliminate this zero eigenvalue, making all the system eigenvalues stable. Both the modified and unmodified strategies are applied to an experimental system with a nonsmooth deadzone characteristic. As expected the unmodified algorithm cannot control the system, whereas the modified algorithm gives stable robustcontrol, which has significantly improved performance over linear fixed gain control.
Modelselection is a challenging problem that is of importance in many branches of the sciences and engineering, particularly in structural dynamics. By definition, it is intended to select the most likely model among a set of competing models that best matches the dynamic behaviour of a real structure and better predicts the measured data. The Bayesian approach which is based essentially on the evaluation of a likelihood function is one of the most popular approach to deal with modelselection and parameter estimation issues. However, in some circumstances, the likelihood function is either intractable or not available even in a closed form. To overcome this issue, the likelihood-free or approximate Bayesian computation (ABC) algorithm has been introduced in the literature, which relaxes the need for an explicit likelihood function to measure the level of agreement between model predictions and measurements. However, ABC algorithms suffer from a low accep- tance rate of samples which is actually a common problem with the traditional Bayesian methods. To overcome this shortcoming and alleviate the computational burden, a new variant of the ABC algorithm based on an ellipsoidal Nested Sampling (NS) technique is introduced in this paper; it has been called ABC-NS. Through this paper, it will be shown how the new algorithm is a promising alternative to deal with parameter estimation and modelselection issues. It promises drastic speedups and provides a good approximation of the posterior distributions. To demonstrate its robust computational efficiency, four illustrative examples are given. Firstly, the efficiency of the algorithm is demonstrated to deal with parameter estimation. Secondly, two examples based on simulated and real data are given to demonstrate the efficiency of the algorithm to deal with modelselection in structural dynamics.
Formal methods for the specification and verification of software have enjoyed enormous success in both academia and industry. However, formal synthesis of high- performance controllers for hybrid systems carrying out complex tasks in uncertain and adversarial environments remains an open problem. This lack of progress for hybrid systems compared to discrete systems (e.g., software) is largely due to the interaction of the non-convex state constraints arising from the specifications with the continuous dynamics of the system. Uncertainties in the system model and the desire for optimal solutions further complicate the issue. Major problems include the computation of robust controllers, the scalable and optimal synthesis of discrete supervisory controllers, and the computation of controllers (both feasible and optimal) for high-dimensional, nonlinear systems. We have developed new techniques that help overcome these problems. The contributions of this thesis include techniques for optimal and robustcontrol, expressive task specification languages that are also computationally efficient, and algorithms that scale to dynamicalsystems with more than ten continuous states.
Over the last few decades model predictive control (MPC) has been applied successfully in many applications. MPC consists of a step-by-step optimization technique: at each sample a new value of the control signal is calculated on the basis of the current measurement and the prediction of the future states and outputs (see e.g. ). The predictive control technique is very popular since it is possible to handle constraints on the input and output signals. The design of the control law is usually based on two assumptions: a) there is no uncertainty and b) the disturbance, which includes the effect of uncertainties, has a well defined behavior (the most common assumption is that the disturbance remains constant over the prediction horizon). The resulting control law will therefore be optimal for the nominal plant model and the disturbance model assumed during the design. Thus, the closed-loop performance may be rather poor and constraints may be violated, when there are uncertainties and / or disturbances in the system.
Model predictive control (MPC) [1, 2] is an important method to handle control problems with systems having input, state, and output constraints [3–7]. Its current con- trol action is obtained by solving an online, at each time instant, open-loop constrained infinite horizon optimization problem. The current state of the system is treated as the initial state of optimal control problems , and only the first optimal control sequence is implemented . In this line of the research, a model is used to predict the future behavior of the real system and obtain an optimal control sequence which satisfies the input, state, and output constraints. So the model quality is very vital in the robust MPC.
5X5 measurement data. Although this measurement data storage requirement is very modest, there is a need for a very large amount o f space for matrix manipulation as i n dicated by F i g . 2 — 3 . For the example considered here, a total of 21 Kbytes of memory ( I BM — 4 331 , 3 2 —bit word length data ) was required for data storage and numerical computation. The total CPU tim e needed to compute the identification for. this example was around 1.4 secon d s. For small computers e.g. I B M —PC ( I B M — 5550 model ) ,
The problem of adaptive robust state observers has been considered for a class of uncertain systems with time- varying delays. A new method has been presented whereby a class of continuous memoryless adaptive robust state observers with a rather simpler structure is constructed. Since our adaptive state observers do not involve the upper bound of uncertainties, such upper bound is not required to be known for the system designer. It has been also shown that the proposed adaptive robust state observers can guarantee that the observation error between the observer state estimate and the true state converges uniformly exponentially towards a ball which can be as small as desired.
The saturation applied to the F-8 aircraft is the limitations on the elevator deflection angle, which has restricted motion of ±30 degrees. The linear system model gives the high speed aircraft more maneuverability and a higher tolerance (extended controllable region) for flights that are entering into stall. Since stall is a function of the angle of attack, it represents the output performance. The high-order approximation of nonlinear phenomena produces an accurate model for the aircraft around the trim conditions even as its flight enters stall. After the flight enters stall, the highly unstable aircraft dynamics becomes very difficult to model for large angles of attack. Therefore, the angle of attack will be the sole output performance for the system, which will allow the controller to stabilize the high speed aircraft with additional outside disturbances. The disturbance model is represented by a gust of wind hitting the aircraft. The gust of wind is modeled as a step response that lasts for two seconds and causes an increase of 5 degrees of pitch.
In general, the discrete epidemic models obtained by Mickens-type discretization have the same features as the original continuous-time model [, , ]. For the Rössler system , the diﬀerence equations obtained by the non-standard or Mickens-type method also show that the solutions to the discrete models are topologically equivalent to the solutions of the continuous-time system as long as the time step is less than a threshold value. For the discrete population models [–] approached by the forward Euler scheme, there existed a ﬂip bifurcation, a Hopf bifurcation and chaos dynamical behaviors which are diﬀerent from the dynamical behaviors in the corresponding continuous-time models. In  the authors used the forward Euler scheme to obtain a class of discrete SIRS epidemic models. They claimed that when the time step h is small (h < h ∗ ) the dynamical behaviors are similar with the continuous-time model, and when the time step h is increasing (h > h ∗ ) in the discrete epidemic model appears a ﬂip bifurcation, a Hopf bifurcation, chaos, and more complex dynamical behaviors by the numerical simulations.
Next, the designs and setting of the four comparative controllers were considered. With the PID controller, the PID gains were derived through a two-step procedure in Matlab/Simulink: first, the EMLS model developed in  was employed to represent the real system and their model parameters were optimized using the parameter estimation toolbox, and second, a closed-loop control simulation with the optimized model and the PID controller was performed to optimize the PID gains using the PID tuning toolbox. The last FPID, and OTGFPID1 and OTGFPID2 controllers were constructed with the same fuzzy PID design as that of the RPTC except the use of the robust learning mechanism (Section 3.2). In addition, the fuzzy PID parameters of the OTGFPID2 was online tuned by the delta rule-based learning mechanism in . For the prediction functions, the typical grey model, GM(1,1), of OTGFPID1 and the SAUIGM model of OTGFPID2 used the same method proposed in  to tune the prediction step size.