Modified integralbased identification for the long arm rover without delay
Consider the normalised second order differential equation:
$$\begin{aligned} \theta '' + c \theta ' + k \theta = b u(t), \end{aligned}$$
(1)
where:

\(\theta \equiv \theta (t)\) represents angular movement,

c is the damping constant,

k is the spring constant,

u is the input.
To designate some degree of flexibility for the model of damping as well as stiffness, these variables can be made timevarying. The simplest such timevarying function is the piecewise constant function defined as follows:
$$\begin{aligned} c(t) &= c_{1},\qquad T_{0}< t< T_{1} \\ &\vdots \\ &= c_{n},\qquad T_{n1} < t < T_{\mathrm{end}} \end{aligned}$$
(2)
and
$$\begin{aligned} k(t) &= k_{1},\qquad T_{0}< t< T_{1} \\ &\vdots \\ &= k_{n},\qquad T_{n1} < t < T_{\mathrm{end}}. \end{aligned}$$
(3)
The system of Equation (1) can now be rewritten as follows:
$$\begin{aligned} \theta '' + c(t) \theta ' + k(t) \theta = b u(t), \end{aligned}$$
(4)
where the functions \(c(t)\) and \(k(t)\) are defined in Equations (2) and (3). Furthermore, the following quantities are also defined:
$$\begin{aligned} &\Delta t= T_{i1}  T_{i},\qquad T_{0}=0, \end{aligned}$$
(5)
$$\begin{aligned} &\Delta t= \text{User defined time interval}. \end{aligned}$$
(6)
For future references, let the measurement times \(t_{i}^{(j)}\) be defined as follows:
$$\begin{aligned} t_{i}^{(j)} \equiv \text{ Measurement instant $t_{i}$ of section $j$, $i=1,\ldots,N$ and $j=1,\ldots,n$}. \end{aligned}$$
(7)
Defining also the operator:
$$\begin{aligned} I_{T_{i},t}^{(k)} F = \underbrace{ \int _{T_{i}}^{t}\cdots \int _{T_{i}}^{t}}_{ \text{k times}} F \,d t\cdots d t. \end{aligned}$$
(8)
Applying the operator of Equation (8) onto Equation (4) yields
$$\begin{aligned} \theta (t)  \theta _{i1}  \alpha _{i1} (t  T_{i1}) +c_{i} I_{T_{i1},t}^{(1)} \theta (t) +k_{i} I_{T_{i1},t}^{(2)} \theta (t) = b I_{T_{i1},t}^{(2)} u(t). \end{aligned}$$
(9)
Here, the initial conditions are defined as follows:
$$\begin{aligned} \theta _{i1} = \theta (T_{i1}),\qquad d\theta _{i1} = \theta '(T_{i1}). \end{aligned}$$
(10)
The integral reconstruction model for the vibration system can now be written as follows:
$$\begin{aligned} \theta _{\mathrm{model},i}(t) = \theta _{i1} + \alpha _{i} ( t  T_{i})  c_{i} I_{T_{i1},t}^{(1)} \theta (t)  k_{i} I_{T_{i1},t}^{(2)} \theta (t) + b I_{T_{i1},t}^{(2)} u(t), \end{aligned}$$
(11)
where the parameters \(\alpha _{i}\) are
$$\begin{aligned} \alpha _{i1} = d \theta _{i1}  c_{i} \theta _{i1}. \end{aligned}$$
(12)
Figure 1 shows a possible scenario at the joins in the neighbourhood of \(t=T_{i}, i=1,\ldots,n\). In practice, the angle measurements \(\theta _{\mathrm{meas}}\) will be obtained from an encoder, whose additive quantisation noise implies that the value of \(\theta _{\mathrm{meas}}\) may not equal the \(\theta _{\mathrm{model}}\) at the joins. This phenomenon introduces possible discontinuities at the joins. A simple method of resolving this discontinuity is to proceed to identify the sections by piece and resolve the discontinuity at the end of every section. The integral reconstructor for the first section is written as follows:
$$\begin{aligned} &\theta _{\mathrm{model},0}(t)= \theta _{0} + \alpha _{0} (tT_{0})  c_{1} I_{T_{0},t}^{(1)} \theta k_{1} I_{T_{0},t}^{(2)} \theta + b I_{T_{0},t}^{(2)} u, \end{aligned}$$
(13)
$$\begin{aligned} &\alpha _{0}= \text{Equation (13) with $i=0$}. \end{aligned}$$
(14)
Substituting \(\theta _{\mathrm{meas}}=\theta (t)\) in Equation (13) for \(t \in \{t_{1}^{(1)},\ldots,t_{N}^{(1)} \}\) yields a system of equations which can be summarised into a matrix equation
$$\begin{aligned} \mathbf{A} \mathbf{p}_{0} = \mathbf{b}, \end{aligned}$$
(15)
where
$$\begin{aligned} &\mathbf{A}= \begin{bmatrix} 1 & t_{1}^{(1)}  T_{0} & I_{T_{0},t_{1}^{(1)}}^{(1)} \theta _{\mathrm{meas}}(t) &I_{T_{0},t_{1}^{(1)}}^{(2)} \theta _{\mathrm{meas}}(t) &I_{T_{0},t_{1}^{(1)}}^{(2)} u \\ 1 & t_{2}^{(1)}  T_{0} & I_{T_{0},t_{2}^{(1)}}^{(1)} \theta _{\mathrm{meas}}(t) &I_{T_{0},t_{2}^{(1)}}^{(2)} \theta _{\mathrm{meas}}(t)&I_{T_{0},t_{2}^{(1)}}^{(2)} u \\ \vdots &\vdots &\vdots &\vdots \\ 1 & t_{N}^{(1)}  T_{0} & I_{T_{0},t_{N}^{(1)}}^{(1)} \theta _{\mathrm{meas}}(t) &I_{T_{0},t_{2}^{(1)}}^{(2)} \theta _{\mathrm{meas}}(t)&I_{T_{0},t_{N}^{(1)}}^{(2)} u \end{bmatrix}, \end{aligned}$$
(16)
$$\begin{aligned} &\mathbf{p}_{0}= \begin{bmatrix} \theta _{0} \\ \alpha _{0} \\ c_{1} \\ k_{1} \\ b \end{bmatrix},\qquad \mathbf{b}= \begin{bmatrix} \theta _{\mathrm{meas}}(t_{1}^{(1)}) \\ \theta _{\mathrm{meas}}(t_{2}^{(1)}) \\ \vdots \\ \theta _{\mathrm{meas}}(t_{N}^{(1)}) \end{bmatrix}. \end{aligned}$$
(17)
Equations (15)–(17) are solved by linear least squares subjected to the constraints
$$\begin{aligned} c_{1} >0,\qquad k_{1} >0. \end{aligned}$$
(18)
The result yields the unknown parameters that are the elements of vector \(\mathbf{p}_{0}\). Substituting the elements of \(\mathbf{p}_{0}\) into Equation (13) yields an integral reconstructor model for the angle \(\theta (t)\) of the first section from the data segmentation.
To ensure that the \(C_{0}\) continuity is ensured at join \(k=i1\), the initial condition \(\theta _{i}\) can be computed thus:
$$\begin{aligned} \theta _{i1} \equiv \text{Equation (13) with $t=T_{i}$ for $i=2,\ldots,n$}. \end{aligned}$$
(19)
The value of b, which was identified for the first section, is assumed to be constant for all sections. This assumption is based on the fact that the arm is balanced and thus no sudden change of inertia is possible across the different time sections. The knowledge of the initial conditions for the ith section now leaves only three unknowns to be identified in the model of Equation (11). In this respect, setting \(\theta (t) \equiv \theta _{\mathrm{model},i}(t)\) and input \(u(t) \equiv u_{\mathrm{data},i}(t)\) for the time instants \(t \in \{t_{1}^{(i)},\ldots,t_{N}^{(i)}\}\) yields a system of N equations in three unknowns:
$$\begin{aligned} &\mathbf{A}_{i} \mathbf{p} = \mathbf{b}_{i}, \end{aligned}$$
(20)
$$\begin{aligned} &\mathbf{A}_{i} = \begin{bmatrix} t_{1}^{(i)}T_{i1} & I_{T_{i1},t_{1}^{(i)}}^{(1)} \theta _{\mathrm{meas}}(t) & I_{T_{i1},t_{1}^{(1)}}^{(2)} \theta _{\mathrm{meas}}(t) \\ t_{2}^{(i)}T_{i1} & I_{T_{i1},t_{2}^{(i)}}^{(1)} \theta _{\mathrm{meas}}(t) & I_{T_{i1},t_{2}^{(i)}}^{(2)} \theta _{\mathrm{meas}}(t) \\ \vdots & \vdots &\vdots \\ t_{N}^{(i)}T_{i1} & I_{T_{i1},t_{N}^{(i)}}^{(1)} \theta _{meas}(t) & I_{T_{i1},t_{N}^{(i)}}^{(2)} \theta _{meas}(t) \end{bmatrix}, \end{aligned}$$
(21)
$$\begin{aligned} &\mathbf{b}_{i}= \begin{bmatrix} \theta _{i}(t_{1})  b I_{T_{i1},t_{1}^{(i)}}^{(2)} u(t) \\ \theta _{i}(t_{2})  b I_{T_{i1},t_{2}^{(i)}}^{(2)} u(t) \\ \vdots \\ \theta _{i}(t_{N})  b I_{T_{i1},t_{2}^{(i)}}^{(2)} u(t) \end{bmatrix}. \end{aligned}$$
(22)
Equations (20)–(22) can now be solved by linear least squares subjected to the constraints
$$\begin{aligned} c_{i} >0, \qquad k_{i} >0. \end{aligned}$$
(23)
The result of such an operation will now provide the values for the damping \(c_{i}\) and stiffness \(k_{i}\) along with the integral reconstructor model for the angular movement \(\theta _{\mathrm{model}}(t)\). Figure 2 shows the algorithm for identifying the timevarying nondelay model.
Integralbased identification for the long arm rover with delay
As a comparison, consider now the normalised second order delay differential equation given by
$$\begin{aligned} \theta ''(t\tau ) + c \theta '(t\tau ) + k \theta (t\tau ) = u(t), \end{aligned}$$
(24)
where τ represents the delay, and the variable θ and the associated parameters c and k retain their meaning from the nondelay case in Sect. 2.1. Inserting also the timevarying models for the damping and stiffness as provided by Equations (2) and (3) gives
$$\begin{aligned} \theta ''(t\tau ) + c(t) \theta '(t\tau ) + k(t) \theta (t\tau ) = u(t). \end{aligned}$$
(25)
Note that the time segmentation intervals for Equation (25) are again given by Equations (5) and (6). The time delay functions \(\theta (t\tau )\) and their derivatives are usually difficult to model. However, under the assumption that the time delay τ is small, it is possible to approximate \(\theta (t\tau )\) and their derivatives by Taylor approximations:
$$\begin{aligned} &\theta (t\tau )= \theta (t)  \tau \theta '(t), \end{aligned}$$
(26a)
$$\begin{aligned} &\theta '(t\tau )= \theta '(t)  \tau \theta ''(t), \end{aligned}$$
(26b)
$$\begin{aligned} &\theta ''(t\tau )= \theta ''(t)  \tau \theta '''(t). \end{aligned}$$
(26c)
Substituting Equation (26a)–(26c) into Equation (25), we obtain:
$$\begin{aligned} \theta ''' + a_{1,i} \theta '' + a_{2,i} \theta ' + a_{3,i} \theta = b u(t), \end{aligned}$$
(27)
where
$$\begin{aligned} &a_{1,i}= c_{i}  \frac{1}{\tau },\qquad a_{2,i} = k_{i}  \frac{c_{i}}{\tau }, \end{aligned}$$
(28)
$$\begin{aligned} &a_{3,i}=\frac{k_{i}}{\tau },\qquad b = \frac{1}{\tau }. \end{aligned}$$
(29)
Applying the operator of Equation (8) to Equation (27) gives
$$\begin{aligned} &\theta (t)= \theta _{i1} + \beta _{1,i} ( t T_{i}) + \beta _{2,i} (tT_{i})^{2}  a_{1,i} I_{i1,t}^{(1)} \theta \\ &\phantom{\theta (t)=}{}a_{2,i} I_{i1,t}^{(2)} \theta a_{3,i} I_{i1,t}^{(3)} \theta + b I_{i1,t}^{(3)} u(t), \end{aligned}$$
(30)
$$\begin{aligned} &\beta _{1,i}= d \theta _{i1}+a_{1,i} \theta _{i1}, \end{aligned}$$
(31)
$$\begin{aligned} &\beta _{2,i}= \frac{dd\theta _{i1}}{2} + a_{1,i} \frac{d\theta _{i1}}{2} + a_{2,i} \theta _{i}, \end{aligned}$$
(32)
where the initial conditions are
$$\begin{aligned} dd\theta _{i1} = \theta ''(T_{i1}),\qquad d\theta _{i1} = \theta '(T_{i1}). \end{aligned}$$
(33)
The discontinuities at every section joins are resolved using the method presented in Sect. 2.1. In this light the integral reconstructor model for the first section is written as follows:
$$\begin{aligned} \theta _{\mathrm{model},0}(t)= {}&\theta _{0} + \beta _{1,1} (tT_{0})  \beta _{1,2} (tT_{0})^{2} a_{1,1} I_{T_{0},t}^{(1)} \theta \\ &{} a_{2,1} I_{T_{0},t}^{(2)} \theta a_{3,1} I_{T_{0},t}^{(3)} \theta +b I_{T_{0},t}^{(3)} u(t). \end{aligned}$$
(34)
Substituting the values of \(\theta \equiv \theta _{\mathrm{data}}(t)\) and \(u(t) \equiv u_{\mathrm{applied}}(t)\) for the times of \(t \in \{ t_{0}^{(1)},\ldots,t_{N}^{(1)} \}\) will give a matrix equation
$$\begin{aligned} &\mathbf{M} \mathbf{p}_{0}= \mathbf{b}_{0}, \end{aligned}$$
(35)
$$\begin{aligned} &\mathbf{M}= \begin{bmatrix} 1&(t_{1}^{(1)}T_{0})&(t_{1}^{(1)}T_{0})^{2}&I_{T_{0},t_{1}^{(1)}}^{(1)} \theta & I_{T_{0},t_{1}^{(1)}}^{(2)} \theta & I_{T_{0},t_{1}^{(1)}}^{(3)} \theta & I_{T_{0},t_{1}^{(1)}}^{(3)} u \\ 1&(t_{2}^{(1)}T_{0})&(t_{2}^{(1)}T_{0})^{2}&I_{T_{0},t_{2}^{(1)}}^{(1)} \theta & I_{T_{0},t_{2}^{(1)}}^{(2)} \theta & I_{T_{0},t_{2}^{(1)}}^{(3)} \theta & I_{T_{0},t_{2}^{(1)}}^{(3)} u \\ \vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\ 1&(t_{N}^{(1)}T_{0})&(t_{N}^{(1)}T_{0})^{2}&I_{T_{0},t_{N}^{(1)}}^{(1)} \theta & I_{T_{0},t_{N}^{(1)}}^{(2)} \theta & I_{T_{0},t_{N}^{(1)}}^{(3)} \theta & I_{T_{0},t_{N}^{(1)}}^{(3)} u \end{bmatrix}, \end{aligned}$$
(36)
$$\begin{aligned} &\mathbf{b}_{0}= \begin{bmatrix} \theta _{\mathrm{data}}({t_{1}^{(1)}}) \\ \theta _{\mathrm{data}}({t_{2}^{(1)}}) \\ \vdots \\ \theta _{\mathrm{data}}({t_{N}^{(1)}}) \end{bmatrix}, \end{aligned}$$
(37)
where \(\theta _{\mathrm{data}}({t_{j}^{(1)}})\) denotes the angular data at \(t=t_{j}\) of the first section. The system identification algorithm begins with the solving of Equations (35)–(37) by linear least squares subjected to the conditions
$$\begin{aligned} a_{1,1} >0,\qquad a_{2,1} >0,\qquad a_{3,1} >0,\qquad b >0. \end{aligned}$$
(38)
The result of this process yields the unknown parameters that belong to the elements of \(\mathbf{p}_{0}\), whose vector is then substituted into Equation (34) to obtain the integral reconstructor model for the angle \(\theta (t)\) for the first section of the segmentation.
The initial conditions for the beginning of the ith segmentation are evaluated thus:
$$\begin{aligned} \theta _{i1}&= \text{ Equation (30) for $t=T_{i}$ with $i=2,\ldots,n$. } \end{aligned}$$
(39)
Again note that the variable b is assumed to be constant throughout all sections. The knowledge of the initial conditions implies that only six parameters are now required to be identified in the model of Equation (30). Setting \(\theta (t) \equiv \theta _{\mathrm{model},i}(t)\) and input \(u(t) \equiv u_{\mathrm{data},i}(t)\) for the time instants \(t \in \{t_{1}^{(i)},\ldots,t_{N}^{(i)}\}\) yields the matrix equation
$$\begin{aligned} &\mathbf{M}_{i} \mathbf{p}_{i}= \mathbf{b}_{i}, \end{aligned}$$
(40)
$$\begin{aligned} &\mathbf{M}_{i}= \begin{bmatrix} (t_{1}^{(i)}T_{i1})&(t_{1}^{(i)}T_{i1})^{2}&I_{T_{i1},t_{1}^{(i)}}^{(1)} \theta & I_{T_{i1},t_{1}^{(i)}}^{(2)} \theta & I_{T_{i1},t_{1}^{(i)}}^{(3)} \theta \\ (t_{2}^{(i)}T_{i1})&(t_{2}^{(i)}T_{i1})^{2}&I_{T_{i1},t_{2}^{(i)}}^{(1)} \theta & I_{T_{i1},t_{2}^{(i)}}^{(2)} \theta & I_{T_{i1},t_{2}^{(i)}}^{(3)} \theta \\ \vdots &\vdots &\vdots &\vdots &\vdots \\ (t_{N}^{(i)}T_{i1})&(t_{2}^{(i)}T_{i1})^{2}&I_{T_{i1},t_{N}^{(i)}}^{(1)} \theta & I_{T_{i1},t_{N}^{(i)}}^{(2)} \theta & I_{T_{i1},t_{N}^{(i)}}^{(3)} \theta \end{bmatrix}, \end{aligned}$$
(41)
$$\begin{aligned} &\mathbf{b}_{i}= \begin{bmatrix} \theta _{\mathrm{data}}({(t_{1}^{(i)}})  b I_{T_{i1},t_{1}^{(i)}}^{(3)} u(t) \\ \theta _{\mathrm{data}}({(t_{2}^{(i)}})  b I_{T_{i1},t_{2}^{(i)}}^{(3)} u(t) \\ \vdots \\ \theta _{\mathrm{data}}({(t_{N}^{(i)}}) b I_{T_{i1},t_{N}^{(i)}}^{(3)} u(t) \end{bmatrix}. \end{aligned}$$
(42)
Equations (40)–(42) can now be solved by linear least squares subjected to the constraints
$$\begin{aligned} &a_{i,1} >0,\qquad a_{i,2} >0, \qquad a_{i,3} >0, \end{aligned}$$
(43a)
$$\begin{aligned} &\vert a_{i,1}  a_{i1,1} \vert < \gamma _{1} \vert t_{i}  t_{i1} \vert , \end{aligned}$$
(43b)
$$\begin{aligned} &\vert a_{i,2}  a_{i1,2} \vert < \gamma _{2} \vert t_{i}  t_{i1} \vert , \end{aligned}$$
(43c)
$$\begin{aligned} &\vert a_{i,3}  a_{i1,3} \vert < \gamma _{3} \vert t_{i}  t_{i1} \vert . \end{aligned}$$
(43d)
The result of such an operation will now provide the values for the parameters \(a_{i,1}\), \(a_{i,2}\) and \(a_{i,3}\) along with the integral reconstructor model for the angular movement \(\theta _{\mathrm{model}}(t)\). Figure 3 shows the algorithm for identifying the time delay model. Note that Equations (43b)–(43d) place the Lipschitz constraints on the derivatives of \(a_{i,1}\), \(a_{i,2}\) and \(a_{i,3}\) to make sure that they are bounded. Were these constraints not placed on the derivatives, it would be possible to choose very small ϵ such that the modelled response \(y_{\mathrm{model}}(t)\) gets very close to \(y_{\mathrm{true}}(t)\), yet the values of \(a_{i,1}\), \(a_{i,2}\) and \(a_{i,3}\) do not resemble the true function. In fact, previous work by Wongvanich et al. [35] has shown that the identified parameters significantly oscillate without bound about the true values, yet the modelled response matches very well with the true data.
Analyses
This section gives the theoretical analyses for the vibration model, both without the delay and with the delay.
Model without delay
Lemma 1
Consider the homogeneous second order differential equation
$$\begin{aligned} \theta ''(t) + c(t) \theta '(t) + k(t) \theta (t) =0. \end{aligned}$$
(44)
The required Lyapunov function can be written as follows:
$$\begin{aligned} V &= \frac{A}{2} z_{1}^{2} + \frac{B}{2} z_{2}^{2} + G z_{1} z_{2}, \end{aligned}$$
(45)
where
$$\begin{aligned} A = \frac{\alpha _{2} k(t) + \alpha _{1}}{c(t)} + \frac{\alpha _{1} c(t)}{k(t)},\qquad B = \frac{\alpha _{2} k(t) + \alpha _{1}}{c(t)}, \qquad G = \frac{\alpha _{1}}{k(t)} \end{aligned}$$
(46)
and
$$\begin{aligned} z_{1} \equiv z_{1}(t) = \theta,\qquad z_{2} \equiv z_{2}(t) = \theta '. \end{aligned}$$
(47)
Proof
The proof of this lemma is similar to the one given in [36]. □
Theorem 2
Consider the homogeneous second order differential equation defined in Equation (44). The system will have global asymptotic stability if there exists a number Q such that
$$\begin{aligned} \max \Bigl[\sup_{t\in [t,T_{\mathrm{end}} ]} c(t), \sup_{t\in [t,T_{\mathrm{end}} ]} k(t) \Bigr] < Q. \end{aligned}$$
(48)
Proof
Consider the Lyapunov candidate function of Equation (45) with parameters A, B and G as defined in Equation (46). To ensure that A, B and G are finite, select \(Q_{1}\) and \(Q_{2}\) such that
$$\begin{aligned} \sup_{t\in [t,T_{\mathrm{end}} ]} c(t) < Q_{1}, \qquad\sup_{t \in [t,T_{\mathrm{end}} ]} k(t) < Q_{2}. \end{aligned}$$
Hence the resulting upper bound Q is
$$\begin{aligned} \max \Bigl[\sup_{t\in [t,T_{\mathrm{end}} ]} c(t), \sup_{t\in [t,T_{\mathrm{end}} ]} k(t) \Bigr] < Q. \end{aligned}$$
□
Model with delay
Lemma 3
Consider the third order homogeneous system
$$\begin{aligned} \theta '''(t) + a(t) \theta ''(t) + b(t) \theta '(t) + r(t) \theta (t) =0, \end{aligned}$$
(49)
where
$$\begin{aligned} &a \equiv a(t) \quad\textit{and}\quad 0< a(t)< a_{m}, \\ &b \equiv b(t) \quad\textit{and}\quad 0< b(t)< b_{m}, \\ &r \equiv r(t) \quad\textit{and}\quad 0< r(t)< r_{m}. \end{aligned}$$
The required Lyapunov function is written as follows:
$$\begin{aligned} V &= \frac{1}{2} a r \biggl(z_{1} + \frac{z_{2}}{a} \biggr)^{2} + \frac{1}{2} \biggl(b\frac{r}{a} \biggr) z_{2}^{2} + (z_{3} + a z_{2})^{2}, \end{aligned}$$
(50)
where the states are chosen as follows:
$$\begin{aligned} z_{1} \equiv z_{1}(t) = \theta, \qquad z_{2} \equiv z_{2}(t) = \theta ',\qquad z_{3} \equiv z_{3} = \theta ''. \end{aligned}$$
(51)
Proof
The proof of this lemma is similar to the one given in [37]. □
Theorem 4
Consider the third order differential equation
$$\begin{aligned} \theta ''' + F_{1,\mathrm{true}} \theta '' + F_{2,\mathrm{true}} \theta ' + \frac{K_{\mathrm{true}}}{D} \theta = 0, \end{aligned}$$
(52)
where \(F_{1,\mathrm{true}} \equiv F_{1,\mathrm{true}}(t)\), \(F_{2,\mathrm{true}} \equiv F_{2,\mathrm{true}}(t)\) and \(K_{\mathrm{true}} \equiv K_{\mathrm{true}}(t)\). The system of Equation (52) will have global asymptotic stability if there exists a number M such that
$$\begin{aligned} \max \biggl[\sup_{t\in [t,T_{\mathrm{end}} ]} \biggl( \frac{F_{1,\mathrm{true}} K_{\mathrm{true}}}{D} \biggr), \sup_{t\in [t,T_{\mathrm{end}} ]} \biggl( \frac{F_{1,\mathrm{true}} K_{\mathrm{true}} D  K_{\mathrm{true}}  F_{2,\mathrm{true}}^{2}}{D} \biggr) \biggr] < M. \end{aligned}$$
(53)
Proof
The following Lyapunov function is written by applying Lemma 3:
$$\begin{aligned} V={}& \frac{1}{2} \biggl[ \frac{F_{1,\mathrm{true}} K_{\mathrm{true}}}{D} \biggl(z_{1}  \frac{z_{2}^{2}}{F_{1,\mathrm{true}}} \biggr)^{2} + \biggl(K_{\mathrm{true}} \biggl(1 \frac{1}{D F_{2,\mathrm{true}}} \biggr)\frac{F_{2,\mathrm{true}}}{D} \biggr) z_{2}^{2} \biggr] \end{aligned}$$
(54)
$$\begin{aligned} &{}+\frac{1}{2} (z_{3} + F_{1,\mathrm{true}} z_{2})^{2}. \end{aligned}$$
(55)
To keep the Lyapunov function of Equation (55) finite, first choose \(M_{1}\) and \(M_{2}\) so that the coefficients of \((z_{1}  \frac{z_{2}^{2}}{F_{1,\mathrm{true}}} )^{2}\) and \(z_{2}^{2}\) are finite:
$$\begin{aligned} &\sup_{t\in [t,T_{\mathrm{end}} ]} \frac{F_{1,\mathrm{true}} K_{\mathrm{true}}}{D} < M_{1}, \\ &\sup_{t\in [t,T_{\mathrm{end}} ]} \biggl(K_{\mathrm{true}} \biggl(1 \frac{1}{D F_{2,\mathrm{true}}} \biggr)\frac{F_{2,\mathrm{true}}}{D} \biggr) < M_{2}. \end{aligned}$$
The resulting upper bound M is thus:
$$\begin{aligned} \max \biggl[\sup_{t\in [t,T_{\mathrm{end}} ]} \biggl( \frac{F_{1,\mathrm{true}} K_{\mathrm{true}}}{D} \biggr), \sup_{t\in [0,T_{\mathrm{end}} ]} \biggl( \frac{F_{1,\mathrm{true}} K_{\mathrm{true}} D  K_{\mathrm{true}}  F_{2,\mathrm{true}}^{2}}{D} \biggr) \biggr] < M. \end{aligned}$$
□
Having established the global asymptotic stability for the system of Equation (27), it is now possible to establish the convergence of our integral reconstructor model. In this respect, we propose the following theorem.
Theorem 5
Consider the following third order nonlinear differential equation:
$$\begin{aligned} \theta '''+ F_{1,\mathrm{true}} \theta '' + F_{2,\mathrm{true}} \theta ' + \frac{K_{1,\mathrm{true}}}{D} \theta = u(t), \quad t \in [T_{0},T_{\mathrm{end}}], \end{aligned}$$
(56)
where
$$\begin{aligned} &\theta _{\mathrm{true}}(0)= \theta _{0},\qquad \theta '_{\mathrm{true}}(0) = d\theta _{0}, \\ &F_{i,\mathrm{true}} \equiv F_{i,\mathrm{true}}(t)>0,\qquad K_{\mathrm{true}} \equiv K_{\mathrm{true}}(t) >0, \qquad F_{i,\mathrm{true}} \in C^{0}, K_{\mathrm{true}} \in C^{0}, \\ &\sup_{t\in [0,T_{\mathrm{end}} ]} \vert F_{i,\mathrm{true}} \vert \quad \textit{is finite} \quad\textit{and}\quad \sup_{t\in [0,T_{\mathrm{end}} ]} \vert K_{i,\mathrm{true}} \vert \quad\textit{is finite}. \end{aligned}$$
(57)
Define also the following functions:
$$\begin{aligned} &F_{i,\mathrm{model},k}(t)= \sum_{j=1}^{k} a_{i,j,\mathrm{model}}^{(k)} \bigl[ u\bigl(t \Delta t (j1)\bigr)u(tj\Delta t) \bigr],\quad i=1,2, \\ &K_{\mathrm{model},k}(t)= \sum_{j=1}^{k} a_{3,j,\mathrm{model}}^{(k)} \bigl[ u\bigl(t \Delta t (j1)\bigr)u(tj\Delta t) \bigr], \\ &u(tt_{k})= \textit{ unit step function at }t=t_{k}. \end{aligned}$$
(58)
If the parameter \(a_{i,j,\mathrm{model}}^{k}, i=1,2,3\), are functions satisfying the Lipschitz condition, that is,
$$\begin{aligned} &\bigl\vert a_{1,\mathrm{model}}^{(j+1)}a_{1,\mathrm{model}}^{(j)} \bigr\vert < \gamma _{1} \vert t_{j}t_{j1} \vert ,\qquad \gamma _{1} = \max_{t\in [t,T_{\mathrm{end}} ]} \bigl\vert \dot{F}_{1,\mathrm{true}}(t) \bigr\vert , \end{aligned}$$
(59a)
$$\begin{aligned} &\bigl\vert a_{2,\mathrm{model}}^{(j+1)}a_{2,\mathrm{model}}^{(j)} \bigr\vert < \gamma _{2} \vert t_{j}t_{j1} \vert ,\qquad \gamma _{1} = \max_{t\in [t,T_{\mathrm{end}} ]} \bigl\vert \dot{F}_{2,\mathrm{true}}(t) \bigr\vert , \end{aligned}$$
(59b)
$$\begin{aligned} &\bigl\vert a_{3,\mathrm{model}}^{(j+1)}a_{3,\mathrm{model}}^{(j)} \bigr\vert < \gamma _{3} \vert t_{j}t_{j1} \vert ,\qquad \gamma _{1} = \max_{t\in [t,T_{\mathrm{end}} ]} \bigl\vert \dot{F}_{3,\mathrm{true}}(t) \bigr\vert . \end{aligned}$$
(59c)
The limit of \(\theta _{\mathrm{model},n}\) will approach the true angular function \(\theta _{\mathrm{true}}\). In addition,
$$\begin{aligned} &\lim_{k\to \infty } F_{i,\mathrm{model},k}(t) = F_{i,\mathrm{true}}(t),\quad i=1,2, \end{aligned}$$
(60a)
$$\begin{aligned} &\lim_{k\to \infty } K_{\mathrm{model},k}(t) = K_{\mathrm{true}}(t). \end{aligned}$$
(60b)
Proof
The integral reconstructor model for the system of Equation (56) is as follows:
$$\begin{aligned} \theta _{\mathrm{model}}(t)= {}&\theta _{i} + \beta _{i,1} (tT_{i1}) +\beta _{i,2} (tT_{i1})^{2}  I_{T{i1},t}^{(1)} F_{1,\mathrm{true}} \theta \\ &{}  I_{T{i1},t}^{(2)} F_{2,\mathrm{true}} \theta  I_{T{i1},t}^{(3)} K_{\mathrm{true}} \theta + I_{T{i1},t}^{(3)} u(t). \end{aligned}$$
We will firstly consider the case where \(t=0\). In this case, there exist N̄ and \(\delta >0\) such that, for \(\bar{N}>k\), \(F_{1,\mathrm{model},k}(0)F_{1,\mathrm{true}}(0) > \delta _{1}\), \(F_{2,\mathrm{model},k}(0)F_{2,\mathrm{true}}(0) > \delta _{2}\) and \(F_{3,\mathrm{model},k}(0)F_{3,\mathrm{true}}(0) > \delta _{3}\).
Since the constituent functions of \(F_{i,\mathrm{model},k}, i=1,2\), and \(K_{i,\mathrm{model},k}\) are \(a_{i,\mathrm{model},k},i=1,2,3\), which are Lipschitz functions, there will exist a time \(t \in [0,dt^{*}]\) regardless of k such that \(F_{i,\mathrm{model},k}\) and \(K_{i,\mathrm{model},k}\) will intersect with the true functions \(F_{i,\mathrm{true}}\) and \(K_{\mathrm{true}}\) respectively. Therefore,
$$\begin{aligned} &F_{1,\mathrm{model},k}(t)  F_{1,\mathrm{true}}(t)> \delta _{1}^{*}\quad \text{for all } t \in \bigl[0,dt^{*}\bigr], \\ &F_{2,\mathrm{model},k}(t)  F_{2,\mathrm{true}}(t)> \delta _{1}^{*}\quad \text{for all } t \in \bigl[0,dt^{*}\bigr], \\ &K_{\mathrm{model},k}(t)  K_{\mathrm{true}}(t)> \delta _{1}^{*}\quad \text{for all } t \in \bigl[0,dt^{*}\bigr]. \end{aligned}$$
And if \(\theta '(0) \leq 0\), then it is possible to choose \(\tilde{dt}^{*} < dt^{*}\) such that the value of \(\theta _{\mathrm{true}}(t) \leq 0, t \in [0,dt^{*}]\). Hence the value of \(F_{i,\mathrm{model},k}  F_{i,\mathrm{true}}(t)\) and \(K_{\mathrm{model},k}K_{\mathrm{true}}\) or \(F_{\mathrm{true}}F_{i,\mathrm{model},k}\) and \(K_{\mathrm{true}}K_{i,\mathrm{model},k}\) cannot change sign in that time. Thus, the error between the integral reconstruction function and the true value is as follows:
$$\begin{aligned} \epsilon \bigl(dt^{*}\bigr) &= \bigl\vert \theta _{\mathrm{true}} \bigl(dt^{*}\bigr)  \theta _{\mathrm{model}}\bigl(dt^{*} \bigr) \bigr\vert \\ &= I_{0,t}^{(3)} \vert F_{1,\mathrm{true}}a_{i,1} \vert + I_{0,t}^{(3)} \vert F_{2,\mathrm{true}}a_{i,2} \vert + I_{0,t}^{(3)} \vert K_{\mathrm{true}}a_{i,3} \vert \\ & > \delta ^{**} >0. \end{aligned}$$
(61)
Equation (61) contradicts the assumption that the limit of \(\theta _{\mathrm{model},k}\) will approach \(\theta _{\mathrm{true}}\). Hence it follows that \(F_{1,\mathrm{model},k} \to F_{1,\mathrm{true}}(0)\), \(F_{2,\mathrm{model},k} \to F_{2,\mathrm{true}}(0)\), \(K_{\mathrm{model},k} \to K_{\mathrm{true}}(0)\).
For the case of \(t>0\), we also prove by contradiction. Suppose now that there exists a time \(t_{0}>0\), which is the smallest time such that \(F_{1,\mathrm{model},k}\) does not approach \(F_{1,\mathrm{true}}(0)\), \(F_{2,model,k}\) does not approach \(F_{2,true}(0)\) and \(K_{model,k}\) does not approach \(K_{true}(0)\). Hence,
$$\begin{aligned} &\vert F_{\mathrm{model},i}  F_{\mathrm{true},i} \vert > \delta _{F,i},\quad \text{for $k>N$}, i=1,2, \end{aligned}$$
(62)
$$\begin{aligned} &\vert K_{\mathrm{model}}  K_{\mathrm{true}} \vert > \delta _{K} \quad\text{for $k>N$}. \end{aligned}$$
(63)
Using the same concept as for the case of \(t=0\), since the functions \(a_{i,j,\mathrm{model}}, i=1,2,3\) are Lipschitz function, there exists \(dt^{*}>0\) such that
$$\begin{aligned} &F_{1,\mathrm{model},k}  F_{1,\mathrm{true}} < \delta _{1}^{*}, \\ &F_{2,\mathrm{model},k}  F_{2,\mathrm{true}} < \delta _{2}^{*}, \\ &K_{\mathrm{model},k}  K_{\mathrm{true}} < \delta _{3}^{*}. \end{aligned}$$
The above statement implies that it is possible to find the smallest \(t< t_{0}\) such that the limit of \(F_{1,\mathrm{model},k}\) will not approach \(F_{1,\mathrm{true}}(t)\), the limit of \(F_{2,\mathrm{model},k}\) will not approach \(F_{2,\mathrm{true}}(t)\) and \(K_{\mathrm{model},k}\) will not approach \(K_{2,\mathrm{true}}(t)\). This contradicts the assumption that \(t_{0}\) is the smallest such value. Statement is thus proved for the case of \(t>0\). □
Identification with the pure linear model
The analyses given in the previous section means that the use of constraints on the identified parameters are possible. This conjecture is due to the fact that the nonlinear least squares problem has duly been converted to a corresponding linear least squares problem by integral reconstruction, that is guaranteed to yield a unique solution. Two possible constraints exist for the models considered in this work, one for the nondelay model and the other for the delay model.
Nondelay model
For the nondelay model, the simplest such constraint is simply the equality constraints \(c_{1}=c_{2}=\cdots=c_{n}\) and \(k_{1}=k_{2}=\cdots=k_{n}\). In other words, the damping and stiffness parameters are assumed constant across the entire data set. This constraint also represents a full linear model without delay. Setting \(\theta (t) = \theta _{\mathrm{model}}(t)\) and \(u(t) = u_{\mathrm{applied}}(t)\) for the times of \(t \in \{ t_{0},\ldots,t_{N} \}\) gives a matrix equation which is written as follows:
$$\begin{aligned} &\mathbf{M}_{\mathrm{lin}} \mathbf{p}_{\mathrm{lin}}= \mathbf{b}_{\mathrm{lin}}, \end{aligned}$$
(64)
$$\begin{aligned} &\mathbf{M}_{\mathrm{lin}}= \begin{bmatrix} 1 & t_{1}  t_{0} & I_{t_{0},t_{1}}^{(1)} \theta _{\mathrm{meas}}(t) & I_{t_{0},t_{1}}^{(2)} \theta _{\mathrm{meas}}(t) &I_{t_{0},t_{1}}^{(2)} u_{\mathrm{appl}}(t_{1}) \\ 1 & t_{2}  t_{0} & I_{t_{0},t_{2}}^{(1)} \theta _{\mathrm{meas}}(t) & I_{t_{0},t_{2}}^{(2)} \theta _{\mathrm{meas}}(t) &I_{t_{0},t_{2}}^{(2)} u_{\mathrm{appl}}(t_{2}) \\ \vdots & \vdots &\vdots & \vdots & \vdots \\ 1 & t_{N}  t_{0} & I_{t_{0},t_{N}}^{(1)} \theta _{\mathrm{meas}}(t) & I_{t_{0},t_{N}}^{(2)} \theta _{\mathrm{meas}}(t) &I_{t_{0},t_{N}}^{(2)} u_{\mathrm{appl}}(t_{N}) \end{bmatrix}, \end{aligned}$$
(65)
$$\begin{aligned} &\mathbf{b}_{\mathrm{lin}}= \begin{bmatrix} \theta _{\mathrm{meas}}(t_{1}) \\ \theta _{\mathrm{meas}}(t_{2}) \\ \vdots \\ \theta _{\mathrm{meas}}(t_{N}) \end{bmatrix}. \end{aligned}$$
(66)
Equations (64)–(66) can duly be solved by linear least squares to yield the linear model without delay.
Delay model
Applying the equality constraints to the delay case means that \(a_{1,1}=a_{1,2}=\cdots=a_{1,n}\), \(a_{2,1}=a_{2,2}=\cdots=a_{2,n}\) and \(a_{3,1}=a_{3,2}=\cdots=a_{3,n}\). Setting \(\theta (t) = \theta _{\mathrm{model}}(t)\) and \(u(t) = u_{\mathrm{applied}}(t)\) for the times of \(t \in \{ t_{0},\ldots,t_{N} \}\), as was done for the nondelay case, now gives a matrix equation which is written as follows:
$$\begin{aligned} \mathbf{M}_{\mathrm{lin},d} \mathbf{p}_{\mathrm{lin},d} = \mathbf{b}_{\mathrm{lin},d}, \end{aligned}$$
(67)
where
$$\begin{aligned} &\mathbf{M}_{\mathrm{lin},d} = \begin{bmatrix} \mathbf{1} & (\mathbf{t}t_{0}) &(\mathbf{t}t_{0})^{2} &  \mathbf{I}_{1,2,3} & \mathbf{u}_{\mathrm{appl}} \end{bmatrix}, \end{aligned}$$
(68)
$$\begin{aligned} &\mathbf{I}_{1,2,3} = \begin{bmatrix} I_{t_{0},t_{1}}^{(1)} \theta _{\mathrm{meas}}(t) & I_{t_{0},t_{1}}^{(2)} \theta _{\mathrm{meas}}(t)&I_{t_{0},t_{1}}^{(3)} \theta _{\mathrm{meas}}(t) \\ I_{t_{0},t_{2}}^{(1)} \theta _{\mathrm{meas}}(t) & I_{t_{0},t_{2}}^{(2)} \theta _{\mathrm{meas}}(t)&I_{t_{0},t_{2}}^{(3)} \theta _{\mathrm{meas}}(t) \\ \vdots &\vdots &\vdots \\ I_{t_{0},t_{N}}^{(1)} \theta _{\mathrm{meas}}(t) & I_{t_{0},t_{N}}^{(2)} \theta _{\mathrm{meas}}(t)&I_{t_{0},t_{N}}^{(3)} \theta _{\mathrm{meas}}(t) \end{bmatrix}, \end{aligned}$$
(69)
$$\begin{aligned} &\mathbf{t} = \begin{bmatrix} t_{1} \\ t_{2} \\ \vdots \\ t_{N} \end{bmatrix}, \qquad\mathbf{1} :=\text{ vector of ones}, \qquad\mathbf{u}_{\mathrm{appl}} = \begin{bmatrix} u_{\mathrm{appl}}(t_{1}) \\ u_{\mathrm{appl}}(t_{2}) \\ \vdots \\ u_{\mathrm{appl}}(t_{N}) \end{bmatrix}. \end{aligned}$$
(70)
In a similar fashion to the nondelay model, Equations (67)–(70) are again solved by linear least squares. The result from this solving process yields the unknown parameters as well as the model for the delay case. Figure 4 depicts the algorithm for identifying the linear model, both for the nondelay and the delay cases.