8 Chapter 2 2.1 Model Selection Overview A Bayesian probabilistic approach is presented for selecting the most plausible class of models for a structure within some specified set of mode[r]

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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 **dynamical** **systems**. However, combined **model** **selection** 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.

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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

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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).

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The results given here may be used in **model**-based advanced **control** of complex **systems**, such as adaptive **control**, **robust** **control**, 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 **robust** **control**, stability, problems of **control** synthesis for dynamic **systems** of various types including problems of forecasting financial results in economic planning and other fields.

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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 **dynamical** **systems** 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.

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In this paper a modified **model** reference adaptive **control** (MRAC) technique is presented which can be used to **control** **systems** 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 **robust** **control**, which has significantly improved performance over linear fixed gain **control**.

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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 **robust** **control**, expressive task specification languages that are also computationally efficient, and algorithms that scale to **dynamical** **systems** with more than ten continuous states.

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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. [9]). 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.

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We express the quotient space in terms of a non-transitive subshift of finite type, give a necessary and sufficient condition for the existence of a local product structure and evaluate [r]

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The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item.. Where records identify the[r]

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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** ) ,

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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.

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Both recursive and direct algorithms are developed using either an unbiased or minimum mean square error criterion to obtain estimates of the Fourier transform and the power spectrum den[r]

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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.

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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.

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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 [43] 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 [9]. 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 [16] to tune the prediction step size.

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