# 金融代写|金融工程作业代写Financial Engineering代考|IE 2097

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## 金融代写|金融工程作业代写Financial Engineering代考|Control Based on Lyapunov Stability Analysis

As far as methods of nonlinear control of the Lyapunov type are concerned, one comes against problems of minimization of Lyapunov functions so as to assure the asymptotic stability of the control loop. For the development of Lyapunov type controllers one can either exploit a model about the system’s dynamics or can proceed in a model-free manner, as in the case of indirect adaptive control. In the latter case, the system’s dynamics is taken to be completely unknown and can be approximated by adaptive algorithms which are suitably designed so as to assure the stabilization and robustness of the control loop.
A single-input differentially flat dynamical system is considered next:
$$\dot{x}=f_{s}(x, t)+g_{s}(x, t)(u+\tilde{d}), x \in R^{n}, u \in R, \tilde{d} \in R$$
where $f_{s}(x, t), g_{s}(x, t)$ are nonlinear vector fields defining the system’s dynamics, $u$ denotes the control input and $\tilde{d}$ denotes additive input disturbances. Knowing that the system of Eq. (2.63) is differentially flat, the next step is to try to write it into a Brunovsky form. It has been shown that, in general, transformation into the Brunovsky (canonical) form can be succeeded for systems that admit static feedback linearization [158]. Single input differentially flat systems, admit static feedback linearization, and can be transformed into the Brunovksy form.

## 金融代写|金融工程作业代写Financial Engineering代考|Adaptive Control Law for Nonlinear Systems

For the measurable state vector $x$ of the system of Eqs. (2.66) and (2.67), and for uncertain functions $f(x, t)$ and $g(x, t)$ an appropriate control law is
$$u=\frac{1}{\hat{g}(x, t)}\left[y_{d}^{(n)}-\hat{f}(x, t)-K^{T} e+u_{c}\right]$$
with $e=\left[e, \dot{e}, \ddot{e}, \ldots, e^{(n-1)}\right]^{T}$ and $e=y-y_{d}, K^{T}=\left[k_{n}, k_{n-1}, \ldots, k_{1}\right]$, such that the polynomial $e^{(n)}+k_{1} e^{(n-1)}+k_{2} e^{(n-2)}+\cdots+k_{n} e$ is Hurwitz. The term $u_{c}$ denotes the supervisory (supplementary) control input that is used for unmodeled dynamics and external perturbations. The control law of Eq. (2.69) results into $$e^{(n)}=-K^{T} e+u_{c}+[f(x, t)-\hat{f}(x, t)]+[g(x, t)-\hat{g}(x, t)] u+g(x, t) \tilde{d}$$
where the supervisory control term $u_{c}$ aims at the compensation of the approximation error
$$w=[f(x, t)-\hat{f}(x, t)]+[g(x, t)-\hat{g}(x, t)] u$$
as well as of the additive disturbance term $d_{1}=g(x, t) \tilde{d}$. The above relation can be written in a state-equation form. The state vector is taken to be $e^{T}=\left[e, \dot{e}, \ldots, e^{(n-1)}\right]$, which after some operations yields
$$\dot{e}=\left(A-B K^{T}\right) e+B u_{c}+B\left{[f(x, t)-\hat{f}(x, t)]+[g(x, t)-\hat{g}(x, t)] u+d_{1}\right}$$

# 金融工程代写

## 金融代写|金融工程作业代写Financial Engineering代考|Control Based on Lyapunov Stability Analysis

$$\dot{x}=f_{s}(x, t)+g_{s}(x, t)(u+\tilde{d}), x \in R^{n}, u \in R, \tilde{d} \in R$$

Brunovsky (规范) 形式[158]。单输入差分平坦系统，允许静态反绩线性化，并且可以转 换为 Brunovksy 形式。

## 金融代写|金融工程作业代写Financial Engineering代考|Adaptive Control Law for Nonlinear Systems

$$u=\frac{1}{\hat{g}(x, t)}\left[y_{d}^{(n)}-\hat{f}(x, t)-K^{T} e+u_{c}\right]$$

$$e^{(n)}=-K^{T} e+u_{c}+[f(x, t)-\hat{f}(x, t)]+[g(x, t)-\hat{g}(x, t)] u+g(x, t) \tilde{d}$$

$$w=[f(x, t)-\hat{f}(x, t)]+[g(x, t)-\hat{g}(x, t)] u$$

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