# Control of DC Motors to Guide Unmanned Underwater Vehicles
## 3 Results
### Persistent Excitation  
In order to contemplate whether relative performances were attributable to algorithm topology or the nature of the method to take advantage of parameter convergence, persistent excitation is briefly discussed in two theorems. These are compiled here to aid readers by assembling the disparate parts of the theorems into a few lines, as opposed to leaving them spread across many pages (particularly with respect to the proofs).  
**Theorem 1**.: _Taken from [8] (p. 64)--Persistent excitation: A signal \(u\) is called persistently exciting of order \(n\) if the limits (Equation (13)) and if the matrix \(C_{n}\) given by Equation (12) is positive definite, where no mean value is included in the definition of empirical covariance c(k). Estimation is consistent and variance of the estimates goes to zero as 1/t if the input signal is persistently exciting of order \(n\)._  
\[C_{n}=\lim_{t\to\infty}\frac{1}{t}\Phi^{T}\Phi=\left[\begin{array}{cccc}c(0) &c(1)&\cdots&c(n-1)\\ c(1)&c(0)&\cdots&c(n-2)\\ \vdots&\vdots&\cdots&\vdots\\ c(n-1)&c(n-2)&\cdots&c(0)\end{array}\right] \tag{12}\]  
\[c(k)=\lim_{t\to\infty}\frac{1}{t}\sum_{i=1}^{t}u(i)u(i-k) \tag{13}\]  
An alternative description of persistent excitation follows. A signal \(u\) is said to be persistently exciting of order \(n\) if for every \(t\) there exists an integer \(m\) such that Equation (14) holds.  
\[\rho_{i}I>\sum_{k=t}^{t+m}\varphi(k)\varphi^{T}(k)>\rho_{i}I \tag{14}\]  
where \(\rho_{1},\rho_{2}>0\) and the vector \(\rho(t)\) is given by Equation (15) permitting Equation (12) to be expressed as Equation (16).  
\[\varphi(t)\equiv\left[\begin{array}{cccc}u(t-1)&u(t-2)&\cdots&u(t-n)\end{array}\right] \tag{15}\]  
\[C_{n}=\lim_{t\to\infty}\frac{1}{t}\sum_{k=1}^{t}\varphi(k)\varphi^{T}(k) \tag{16}\]  
**Proof of Theorem 1** ([8]).: Assembled from pages 42, 48, and 65 if the parameter \(\theta\) is chosen to minimize the least squares loss function \(\frac{1}{2}\sum\limits_{i=1}^{t}\left(y(i)-\varphi^{T}(i)\theta\right)^{2}\), parameter estimates \(\hat{\theta}\) are such that Equation (17) holds. \(\square\)  
\[\Phi^{T}\Phi\hat{\theta}=\Phi^{T}\gamma \tag{17}\]  
If matrix \(\Phi^{T}\Phi\) is nonsingular, the minimum is unique and given by \(\hat{\theta}^{*}=\left(\Phi^{T}\Phi\right)^{-1}\Phi^{T}\gamma\) (also known as the pseudo-inverse solution). Assuming data is generated by an approximated system, as in equation (18), plus independent, random noise or errors \(e(i)\) with zero mean and variance \(\sigma^{2}\), the estimates are unbiased and converge to the true parameter value as the number of observations increases infinity, per Theorem 2.2 in [8] (p. 48).  
\[y(i)=\varphi^{T}(i)\theta^{0}+e(i) \tag{18}\]  
**Theorem 2** ([8]).: _Applied to this research theorem and proof as taken from p. 65: The signal \(u\) in Equation (13) is persistently exciting of order \(n\) if and only if Equation (19) holds for all nonzero polynomials \(A\) of degree \(n\)-1 or less._  
**Proof of Theorem 2.** Where \(C_{n}\) is defined by Equation (12) and polynomial \(A(q)\) defined by Equation (19), an uncomplicated calculation shows Equation (20) is true; if \(C_{n}\) is positive definite, the right-hand side and left-hand side of equation (12) are positive for all \(a\). \(\square\)  
\[A(q)=a_{0}q^{n-1}+a_{1}q^{n-2}+\cdots+a_{n-1} \tag{19}\]  
\[U=\lim_{t\to\infty}\frac{1}{t}\sum_{k=1}^{t}(A(q)u(k))^{2}>0 \tag{20}\]  
For periodic signals \(u(t)\) of period \(n\), it follows that \((q^{n}-1)u(t)=0\), thus the signal can at most be persistently exciting of order \(n\), the period.