# Bounded-Error Parameter Estimation Using Integro-Differential Equations for Hindmarsh-Rose Model
## 2 Parameter Estimation Method
### Estimation Procedure  
A numerical method, deduced from (2), was proposed to estimate the unknown constant parameters in [17] and is first recalled.  
In the numerical applications, the measurement \(y\) is supposed to be described by \(y(t)=y(t,\theta^{*})+\eta(t)\), where the measurement noise \(\eta(t)\) is supposed to belong to \([\eta(t)]\), and \(\theta^{*}\) represents the \"true\" parameter vector value. Denote \(\{y_{k}=y(t_{k}),k=1,\ldots,N\}\), the set of measurements at \((t_{k})_{1\leq k\leq N}\) and \(u_{k}=u(t_{k})\) the associated inputs.  
The parameter vector \(\theta\) belongs to \(\Theta\), where \(\Theta\) is an interval vector. Consider \(\Gamma_{k}(\Theta)\), the associated expression of \(\gamma_{k}(\theta)\) defined in the polynomial (2), where \(\theta\) is substituted by \(\Theta\). Then, the following system, whose interval vector \(\Theta\) is unknown, can be deduced:  
\[\forall j=1,\ldots,N,\,0\in m_{0}(y_{j},u_{j})+\sum_{l=1}^{q}\Gamma_{l}( \Theta)m_{l}(y_{j},u_{j}). \tag{3}\]  
Notice that (3) is linear, with respect to \(\Gamma_{1}(\Theta),\ldots,\Gamma_{q}(\Theta)\). Solving the previous system comes back to solving \(0\in[A][x]-[b]\) or \([A][x]=[b]\), where \([A]_{j}=([m_{l}(y_{j},u_{j})])_{l=1,\ldots,q}\) is the \(i^{th}\) line of the interval matrix \([A]\), and \([b]_{j}=[-m_{0}(y_{j},u_{j})]\) is the \(j^{th}\) line of the interval vector \([b]\).  
However, some derivatives of high order can be involved in the linear system in using elimination algorithms, since the IO polynomials are deduced from the model equations in using addition and multiplication by any polynomials in \(x\), \(u\), \(y\) and \(\theta\) as well as differentiation in time. Integrating these relations will permit not only to decrease the derivative order but also to attenuate the structured noise, whose amplitude is unknown (see [18], for more details).  
Afterwards, we present an improvement of the method proposed in [17], which is based on the integrated relations obtained from (2).  
Let \(f\), a real-valued function, and \(I_{\
u}(f)\), \(\
u\geq 0\), the integrated function, such that  
\[I_{\
u}(f)=\int_{l-\tau}^{t}\int_{\eta_{1}-\tau}^{\tau_{1}}\cdots\int_{\tau_{ \tau_{\tau}-1}-\tau}^{\tau_{v}}f(\tau_{v})d\tau_{v}\ldots d\tau_{1}.\]  
Using the linearity of the integral, a new relation is obtained from \(P\) and can be rewritten:  
\[I_{\
u}(P)=I_{\
u}(\hat{m}_{0}(y,u))+\sum_{l=1}^{q}\hat{\tau}_{l}(\theta)I_{ \
u}(\hat{m}_{l}(y,u)).\]  
\(I_{\
u}(P)\) is called the _ID-IO polynomial_. The approximated value of \(I_{\
u}(P)\), by a numerical procedure at the measurement points \(t_{k}\) (\(k=1,\ldots,N\)), will be denoted \(I_{\
u}^{k}(P)\).  
In the same way, as previously, evaluating the expression \(I_{\
u}(P)\) at each \(t_{k}\) leads to the linear systems.  
\[0\in[\tilde{A}][\xi]-[\tilde{b}]\;\;\text{and}\;\;[\tilde{A}][\xi]=[\tilde{b}] \tag{4}\]  
where the jth line of \([\tilde{A}]\) and \([\tilde{b}]\) are, respectively, given by \([\tilde{A}]_{j}=[I_{\
u}^{j}(\tilde{m}_{l}(y,u))]_{l=1,\ldots,q}\) and \([\tilde{b}_{j}]=[-I_{\
u}^{j}(\tilde{m}_{0}(y,u))]\).  
In the numerical applications, System (4) will be solved using the algorithm SIVIA, as presented in the following section.