**3.5 Access to justice for all and build effective, accountable and inclusive institutions at all levels**

This fragment of substainable development goal 16, is become a serious point of concern due to the affaire of the day-care surtax. A damning judgment by a parliamentary investigation commission has led to the resignation of the Dutch government at January 15th of 2021. The report was titled 'Unprecedented Injustice' [69]. Here, we only discuss facts and judgments in relation to the sustainable development goal. It is also an example of how a resilience promoting policy had led to the inflicting of disadvantages on citizens and the creation of institutional distrust.

#### *An Explorative Perspective on the Resilience in Neighborhoods in the Netherlands DOI: http://dx.doi.org/10.5772/intechopen.98818*

The day-care surtax originates from 2005. It is introduced to promote participation of parents, particularly of the mothers in the labor market. The surtax depends on the proportion between income and costs of purchased day-care. In that way, the surtax was aimed to supports houselholds which have insufficient income for use of day-care. From the beginning, legality checks on the intake of applications were absent or, later on, provisional. The Dutch tax authorities relied on the article in the day-care surtax act, which allowed this state agency to reclaim advances for a period of five years. Arbitrarily, the tax authorities have figured out an 'everything-ornothing' principle, meaning the response of an complete reclaim on any error in the application forms. The princinple excluded the option of a restatement of the level of surtax and the leniency of redemption arrangements. The principle is judged by the commission as being disproportional [69]. The effectuation of the reclaims has caused a cascade of household debts [70].

The title of the report is a reference to the fact that up to and including the highest national court the practice rules of the tax authorities have been confirmed. Therefore, the maltreated citizens have been excluded from access to justice as well [69]. Moreover, this crisis in the Dutch institutions is not yet revealed in full extent. It is reported and confirmed that clients with a double passport, being an indicator of a non-western migrant ancestry, is used to red flag surtax receiver on a personal attribute [71]. A following disclosure is the fact that tax authorities kept up and shared across three social domains a blacklist of 240,000 households or people of which attributes would predict fraud risks [72]. As the surtaxes serve to support the provision of basic needs of the household of which the incomes do not suffice, these people have been deliberately and multiply disadvantaged in their daily life. This state practice has, in spite of its inclusive goals, contributed to more inequality in the Dutch society.


**Abstract**

In control theory, the inverted pendulum is a class of dynamic systems widely used as a benchmarking for evaluating several control strategies. Such a system is characterized by an underactuated behavior. It is also nonlinear and presents openloop unstable and integrating modes. These dynamic features make the control more difficult, mainly when the controller synthesis seeks to include constraints and the guarantee of stability of the closed-loop system. This chapter presents a stabilizing model predictive control (MPC) strategy for inverted pendulum-like behaved systems. It has an offset-free control law based on an only optimization problem (one-layer control formulation), and the Lyapunov stability of the closedloop system is achieved by adopting an infinite prediction horizon. The controller feasibility is also assured by imposing a suitable set of slacked terminal constraints associated with the unstable and integrating states of the system. The effectiveness of the implementable and stabilizing MPC controller is experimentally demonstrated in a commercial-didactic rotary inverted pendulum prototype, considering both cases of stabilization of the pendulum in the upright position and the output tracking of the rotary arm angle.

**Keywords:** rotary inverted pendulum, model predictive control, nonlinear system, Lyapunov stability, feasible-optimization problem

## **1. Motivation**

Dynamic inverted pendulum-featured apparatuses are widespread in systems and control theory. These represent a class of nonlinear and underactuated electromechanical systems, which, in turn, are composed of open-loop unstable and integrating modes. The scale-up of inverted pendulum-based conceptual sketches in practical mechanisms and real applications has been in progress, of which one can cite stabilization of rocket launch, robot balance, and segway-like means of transportation, among others [1, 2].

In control theory, inverted pendulum-type systems have enabled extensive studies concerning controller architectures ranging from proportional-integrativederivative (PID)-like classical strategies to more advanced technique ones, such as optimal and adaptive strategies. In the middle of the advanced control strategies, the so-called class of model predictive control (MPC) strategies has been preferred by systematically handling system constraints. In fact, any MPC algorithm makes explicit use of a system model to predict its outputs and to obtain an optimal control law that minimizes the prediction error and control efforts [3–5]. Since it requires an online solution to its associated optimization problem, it first became popular in applications of slow dynamic systems, such as those in petroleum refineries and petrochemical industries [6]. However, with advances in hardware development and optimization techniques, MPC applications have been extended to fast dynamic systems, including inverted pendulum-like mechatronic systems [7–9].

On the other hand, when one seeks to control open-loop unstable systems such as inverted pendulum-like behaved ones, the guarantee of stability associated with control laws plays a crucial role concerning practical implementation purposes. In particular, the synthesis of stabilizing MPC laws deals essentially with terminal state constraints. One of the most heavily studied stability approach is one based on a dual-mode framework, which is composed of two distinct control modes: in the first mode, a conventional MPC law forces the system states to converge to a certain invariant set at the end of the finite prediction horizon, while in the second control mode, a local state-feedback controller takes over and drives the state to the desired operating point within this set [10]. This approach, however, requires the computation of the invariant set parameters, which is obtained from an offline numerical procedure. Although this set can be obtained offline with standard algorithms, undesired convergence and numerical issues may appear as the system dimension increases. Furthermore, the control horizon should be large enough such that the system states at the end of this horizon lie in the invariant set; otherwise the resulting optimization problem becomes infeasible, thus compromising both feasibility and control performance.

Another way to guarantee the closed-loop system stability of MPC controllers is to adopt an infinite prediction horizon, the so-called infinite-horizon model predictive control (IHMPC). Since the infinite-horizon problems cannot be directly handled by an optimization algorithm, the realization of IHMPC controllers is obtained from the combination between a terminal cost term and terminal equality constraints [11]. The terminal cost, associated with open-loop stable modes of the system, is calculated through the solution of the Lyapunov equation, whereas the terminal constraints are necessary to limit the objective function when the system is composed of integrating and unstable modes. However, stability proof is only achieved if the constrained optimization problem is feasible. The feasibility is also a critical issue of this approach, particularly because the domain of attraction of the controller becomes quite reduced by virtue of the associated hard constraints. Although there is already a rich theory in this field, the applications are heavily limited to theoretical works [12]. Among the methods developed to circumvent this issue so far, the approach based on slacked terminal constraints seems to be more adequate for practical implementation purposes, with recent applications reported in the literature, one implementation in a crude oil distillation [13] and the other in an inverted pendulum mechatronic-like fast dynamic system, namely, customized engine control unit [14].

This class of controllers formulates optimization problems that are always feasible through the suitable inclusion of slack variables in the control laws, without compromising their convergence and stabilizing properties. Also, these IHMPC controllers make use of the customized state-space models, obtained from an analytical expression of the step response of the system. Because of this, their formulations have been gradually developed over time. Odloak [15] focused on systems

#### *An Implementable and Stabilizing Model Predictive Control Strategy for Inverted… DOI: http://dx.doi.org/10.5772/intechopen.91629*

with open-loop stable poles, his work being extended to contemplate simple integrating poles as well in [6], commonly found in systems of the process industry. Then, Santoro and Odloak [16] encompassed time delays to the formulation proposed in [6]. For open-loop stable and unstable time-delay processes, Martins and Odloak [12] synthesized their IHMPC controller. More recently, the master's dissertation [17] included integrating poles in the last work formulation [12], such that its associated IHMPC controller can be directly applied to rotary inverted pendulum-behaved systems, the case understudy of this chapter. The implementation in a real system of the feasible-optimization problem-based stabilizing MPC controller proposed in [17] has not yet been documented in the literature. This gap will be filled in the present work.

#### **2. System description**

The objective of the MPC controller to be explored in this work is to stabilize the pendulum rod in the upright position while it leads the rotary arm angle to the desired positions. To this end, the rotary inverted pendulum used here will be a commercial-didactic prototype manufactured by Quanser. This prototype is installed in the Control Laboratory of the Center for Technological Training in Industrial Automation (CTAI) at the Federal University of Bahia (UFBA). **Figure 1** illustrates the features of such a system [18].

The rotary inverted pendulum prototype consists of a servomotor system, whose voltage *Vm* applied to it is responsible for generating torque in the rotary arm of angle (*θ*). The long pendulum rod is connected to the end of the rotary arm, and its angle, *α*, is zero when it is upright in the vertical position (cf. **Figure 1**).

The governing mathematical model of this system can be obtained by the Euler-Lagrange formalism, resulting in the following well-known equations [18]:

$$\begin{aligned} \ddot{\theta} &= \frac{-bc\sin(a)(\dot{a})^2 + bd\sin(a)\cos\left(a\right) - ce\left(\dot{\theta}\right) + cfV\_m}{ac - b^2\cos^2(a)},\\ \ddot{a} &= \frac{ad\sin(a) - b^2\sin\left(a\right)\cos\left(a\right)\left(\dot{a}^2\right) - bec\cos\left(a\right)\left(\dot{\theta}\right) + bf\cos\left(a\right)V\_m}{ac - b^2\cos^2(a)},\end{aligned} \tag{1}$$

**Figure 1.** *Rotary inverted pendulum prototype manufactured by Quanser (left) and the schematic diagram (right).*

#### *Automation and Control*

where €*θ*, \_ *θ*, *α*€, and *α*\_ represent angular accelerations and velocities associated with rotary arm angle and inverted pendulum angle, respectively. In addition, the parameters ð Þ *a*, *b*,*c*, *d*,*e*, *f* are constants related to the physical dimensions of the various components that make up the inverted pendulum prototype. Information about the modeling and physical parameters of the system can be referred to [18].

Since one of the control objectives aims to the stabilization of the pendulum in the upright position, it is quite adequate to assume that *α* will suffer small variations, which implies that sin ð Þ *α* ≈*α*, cosð Þ *α* ≈0, and *α*\_ 2 ≈0. Then, after some algebraic manipulations in Eq. (1), applying the Laplace transform as well, one turns out to be the following transfer function matrix (*G s*ð Þ):

$$\begin{bmatrix} \theta(\varsigma) \\ a(\varsigma) \end{bmatrix} = \underbrace{\begin{bmatrix} f\varsigma^2 - fd \\ s\left( (ac - b^2)\varsigma^3 + ec\varsigma^2 - ad\varsigma - ed\right) \\\\ \hline b\mathfrak{f}\mathfrak{s} \\ \hline \overline{(ac - b^2)\varsigma^3 + ec\varsigma^2 - ad\varsigma - ed}}^{G\text{-}(c)} \end{bmatrix}}\_{G(\varsigma)} V\_m(\varsigma). \tag{2}$$

This model representation of the system in terms of transfer functions is useful to obtain the state-space formulation to be used in the stabilizing MPC control law, as will be shown in the next section.

### **3. Stabilizing MPC formulation**

The stabilizing MPC control law used in this work seeks to solve an infinitehorizon optimization problem, such that its objective function is composed of the following terms:

$$\begin{split} J\_k &= \sum\_{j=1}^{\infty} \left\| \mathfrak{y}(k+j|k) - \mathfrak{y}\_{\text{sp}} - \mathfrak{d}\_{\text{y},k} - \Psi\_{\text{un}} \mathbf{F}\_{\text{un}}^{(\
j-m)} \delta\_{\text{un},k} - (j-m) \Delta t \delta\_{i,k} \right\|\_{Q\_{\text{p}}}^2 \\ &+ \sum\_{j=0}^{m-1} \left\| \left. \Delta \mathfrak{u}(k+j|k) \left\| \mathbf{I}\_{\text{R}}^2 + \left\| \left. \delta\_{j,k} \right\| \right\|\_{\mathbf{S}\_p}^2 + \left\| \left. \delta\_{\text{un},k} \right\| \right\|\_{\mathbf{S}\_{\text{uv}}}^2 + \left\| \left. \delta\_{i,k} \right\| \right\|\_{\mathbf{S}\_i}^2, \end{split} \tag{3}$$

where *m* is the control horizon, Δ*u*ð Þ *k* þ *j*j*k* ∈ *nu* is the vector of input moves at time step *k* þ *j*, **Q***y* ∈ *ny*�*ny* is a positive-definite weighting matrix of the controlled outputs, **R** ∈ *nu*�*nu* is a positive-definite weighting matrix of the input moves, *ysp* ∈ *ny* is the vector of references of the controlled variables, and *y*ð Þ *k* þ *j*j*k* ∈ *ny* is the vector of the predicted outputs at time step *k* þ *j* computed at time step *k*, considering a state-space model obtained from an analytical expression of the step response of the system described as in Eq. (2), namely:

$$
\begin{bmatrix} \mathbf{x}^{\varepsilon}(k+1) \\\\ \mathbf{x}^{\varepsilon}(k+1) \\\\ \mathbf{x}^{\mathrm{un}}(k+1) \\ \end{bmatrix} = \underbrace{\begin{bmatrix} \mathbf{I}\_{\mathrm{ny}} & \mathbf{O}\_{\mathrm{ny}\times\mathrm{nd}} & \mathbf{O}\_{\mathrm{ny}\times\mathrm{man}} & \Delta\mathbf{f}\_{\mathrm{ny}} \\ \mathbf{0}\_{\mathrm{nd}\times\mathrm{ny}} & \mathbf{F}\_{\mathrm{st}} & \mathbf{O}\_{\mathrm{nd}\times\mathrm{man}} & \mathbf{O}\_{\mathrm{nd}\times\mathrm{ny}} \\ \mathbf{0}\_{\mathrm{mu}\times\mathrm{ny}} & \mathbf{0}\_{\mathrm{mu}\times\mathrm{nd}} & \mathbf{F}\_{\mathrm{nn}} & \mathbf{O}\_{\mathrm{mu}\times\mathrm{ny}} \\ \mathbf{0}\_{\mathrm{ny}} & \mathbf{0}\_{\mathrm{ny}\times\mathrm{nd}} & \mathbf{0}\_{\mathrm{my}\times\mathrm{man}} & \mathbf{I}\_{\mathrm{ny}} \end{bmatrix} \begin{bmatrix} \mathbf{x}^{\prime}(k) \\\\ \mathbf{x}^{\mathrm{st}}(k) \\\\ \mathbf{x}^{\mathrm{un}}(k) \\\\ \mathbf{x}^{\prime}(k) \end{bmatrix} + \underbrace{\begin{bmatrix} \mathbf{B}\_{\mathrm{t}} \\\\ \mathbf{B}\_{\mathrm{tt}} \\\\ \mathbf{B}\_{\mathrm{uu}} \\\\ \mathbf{B}\_{\mathrm{t}} \end{bmatrix}}\_{\mathbf{B}} \Delta\mathbf{u}(k), \tag{4}
$$

*An Implementable and Stabilizing Model Predictive Control Strategy for Inverted… DOI: http://dx.doi.org/10.5772/intechopen.91629*

$$\mathbf{y}(k) = \underbrace{\begin{bmatrix} \mathbf{I}\_{\text{wy}} & \Psi\_{\text{st}} & \Psi\_{\text{un}} & \mathbf{0}\_{\text{wy}\times\text{nu}} \end{bmatrix}}\_{\mathbf{C}} \begin{bmatrix} \mathbf{x}^{\text{s}}(k) \\\\ \mathbf{x}^{\text{st}}(k) \\\\ \mathbf{x}^{\text{nn}}(k) \\\\ \mathbf{x}^{\text{i}}(k) \end{bmatrix}. \tag{5}$$

In the state-space model defined in the pair of Eqs. (4) and (5), *xs* ð Þ*k* ∈ *ny* are the artificial integrating states introduced by the incremental form of inputs, *xst*ð Þ*k* ∈ *nst* are the stable states of the system, *xun*ð Þ*k* ∈ *nun* are the unstable states of the system, and *xi* ð Þ*k* ∈ *ny* are the true integrating states of the system. **I***n* and **0***n* are identity and null matrices of *n* � *n* dimension, respectively. The remaining matrices (**F***st*, **F***un*,**B***s*, **B***st*, **B***un*, **B***i*, Ψ**st e** Ψ**un**) are obtained from step-response coefficients of the transfer function matrix of the system, and the details can be referred to [19, 20].

In the objective function, there are also *δy*,*k*, *δun*,*k*, and *δi*,*k* that are slack variables introduced into the control law so as to provide additional degrees of freedom to the resulting optimization problem, thus assuring the feasibility of the controller. These slack variables are weighted by positive defined matrices **Sy** ∈ *ny*�*ny*, **S***un* ∈ *nun*�*nun*, and **S***i* ∈ *ni*�*ni*, respectively. In fact, the set of slack variables adopted in the problem formulation is responsible for softening, when necessary, terminal constraints that are imposed to limit the infinite-horizon objective function, owing to the existence of open-loop unstable and integrating modes.

It should be kept in mind that the objective function defined in Eq. (3) can be rewritten as follows:

$$J\_k = \sum\_{j=1}^{m} \left\| \left\| \mathbf{y}(k+j|k) - \mathbf{y}\_{sp} - \boldsymbol{\mathfrak{G}}\_{\mathcal{I},k} - \boldsymbol{\mathfrak{V}}\_{\mathcal{U}} \mathbf{F}\_{\mathcal{U}}^{-(j-m)} \boldsymbol{\mathfrak{G}}\_{\mathcal{U},k} - (j-m) \boldsymbol{\Delta t} \boldsymbol{\mathfrak{G}}\_{i,k} \right\|\_{\mathbf{Q}\_{\mathcal{I}}}^2$$

$$+ \sum\_{j=1}^{\infty} \left\| \left\| \mathbf{y}(k+m+j|k) - \mathbf{y}\_{sp} - \boldsymbol{\mathfrak{G}}\_{\mathcal{I},k} - \boldsymbol{\mathfrak{V}}\_{\mathcal{U}} (\mathbf{F}\_{\mathcal{U}})^{\dagger} \boldsymbol{\mathfrak{G}}\_{\mathcal{U},k} - j \boldsymbol{\Delta t} \boldsymbol{\mathfrak{G}}\_{i,k} \right\|\_{\mathbf{Q}\_{\mathcal{I}}}^2 \right\|\_{\mathbf{Q}\_{\mathcal{I}}}^2$$

$$+ \sum\_{j=0}^{m-1} \left\| |\Delta \boldsymbol{\mathfrak{u}}(k+j|k)| \right\|\_{\mathbf{R}}^2 + \left\| |\boldsymbol{\mathfrak{G}}\_{\mathcal{I},k}| \right\|\_{\mathbf{S}\_{\mathcal{I}}}^2 + \left\| |\boldsymbol{\mathfrak{G}}\_{\mathcal{U},k}| \right\|\_{\mathbf{S}\_{\mathcal{U}}}^2 + \left\| |\boldsymbol{\mathfrak{G}}\_{i,k}| \right\|\_{\mathbf{S}\_{\mathcal{I}}}^2.$$

Then, with the aid of the state-space model used to carry out the prediction of the system, it is possible to demonstrate that the objective function becomes:

$$\begin{split} J\_{k} &= \sum\_{j=1}^{m} \left\| \left. \mathbf{y}(k+j|k) - \mathbf{y}\_{cp} - \boldsymbol{\delta}\_{\mathbf{y},k} - \boldsymbol{\Psi}\_{\mathbf{un}} \mathbf{F}\_{\mathbf{un}} \right| \boldsymbol{\delta}\_{\mathbf{un},k} - (j-m)\Delta t \boldsymbol{\delta}\_{i,k} \right\|\_{\mathbf{Q}\_{\mathbf{y}}}^{2} \\ &+ \sum\_{j=1}^{\infty} \left\| \left. \begin{array}{l} \mathbf{x}^{\boldsymbol{\prime}}(k+m|k) + j\Delta t \mathbf{x}^{\boldsymbol{\prime}}(k+m|k) + \boldsymbol{\Psi}\_{\boldsymbol{x}t} (\mathbf{F}\_{\mathbf{u}})^{\boldsymbol{\prime}} \mathbf{x}^{\boldsymbol{\prime}}(k+m|k) + \\\\ \boldsymbol{\Psi}\_{\mathbf{un}} (\mathbf{F}\_{\mathbf{un}})^{j} \mathbf{x}^{\boldsymbol{m}}(k+m|k) - \mathbf{y}\_{cp} - \boldsymbol{\delta}\_{\mathbf{y},k} - \boldsymbol{\Psi}\_{\mathbf{un}} (\mathbf{F}\_{\mathbf{un}})^{j} \boldsymbol{\delta}\_{\mathbf{un},k} - j\Delta t \boldsymbol{\delta}\_{i,k} \end{array} \right\|\_{\mathbf{Q}\_{\mathbf{y}}} \\ &+ \sum\_{j=0}^{m-1} \left\| \left. \Delta \mathbf{u}(k+j|k) \right\|\_{\mathbf{R}}^{2} + \left\| \left. \boldsymbol{\delta}\_{\mathbf{y},k} \right\|\_{\mathbf{S}\_{y}}^{2} + \left\| \left. \boldsymbol{\delta}\_{\mathbf{un},k} \right\|\_{\mathbf{S}\_{w}}^{2} + \left\| \left. \boldsymbol{\delta}\_{i,k} \right\|\_{\mathbf{S}\_{z}}^{2} \right. \right. \end{split}$$

It is worth emphasizing that if constraints are not imposed at the end of the control horizon, the objective function value will increase unbounded. To this end, the following terminal constraints are imposed on the optimization control problem:

$$
\mathfrak{x}^s(k+m|k) - \mathfrak{y}\_{sp} - \mathfrak{d}\_{\mathfrak{y},k} = \mathbf{0},\tag{8}
$$

$$\mathbf{x}^{un}(k+m|k) - \delta\_{un,k} = \mathbf{0},\tag{9}$$

$$\mathbf{x}^i(k+m|k) - \delta\_{i,k} = \mathbf{0}.\tag{10}$$

Furthermore, the term associated with stable modes of the system comprises a convergent series, giving rise to the so-called terminal cost, namely:

$$\sum\_{j=1}^{\infty} \left\| \left| \Psi\_{\mathfrak{sl}} (\mathbf{F}\_{\mathfrak{sl}})^{\dot{j}} \mathfrak{z}^{\mathfrak{sl}} (k+m|k) \right| \right\|\_{\mathbf{Q}\_{\mathfrak{J}}}^2 = \left\| |\mathfrak{z}^{\mathfrak{sl}} (k+m|k) \right\|\_{\mathbf{Q}}^2,\tag{11}$$

where **Q** is the terminal weighting matrix obtained from the solution to the Lyapunov equation of the system. In symbols:

$$\overline{\mathbf{Q}} = \left(\mathbf{F}\_{\mathfrak{sl}}\right)^{\mathsf{T}} \left(\boldsymbol{\Psi}\_{\mathfrak{sl}}\right)^{\mathsf{T}} \mathbf{Q}\_{\mathfrak{g}} \boldsymbol{\Psi}\_{\mathfrak{sl}} \mathbf{F}\_{\mathfrak{sl}} + \left(\mathbf{F}^{\mathfrak{sl}}\right)^{\mathsf{T}} \overline{\mathbf{Q}} \left(\mathbf{F}\_{\mathfrak{sl}}\right). \tag{12}$$

Therefore, the feasible-optimization problem-based stabilizing MPC control law is summarized as follows:

**Problem 1.**

$$\begin{split} \min\_{\boldsymbol{\Delta}\mathbf{u}\_{k},\boldsymbol{\delta}\_{j,k},\boldsymbol{\delta}\_{m,k},\boldsymbol{\delta}\_{i,k}} & J\_{k} = \sum\_{j=1}^{m} \left\| \left\| \boldsymbol{\mathcal{y}}(k+j|k) - \boldsymbol{\mathcal{y}}\_{\mathbf{y}} - \boldsymbol{\delta}\_{\mathbf{y},k} - \boldsymbol{\Psi}\_{\mathbf{un}} \mathbf{F}\_{\mathbf{un}} \boldsymbol{\delta}^{\ (j-m)} \boldsymbol{\delta}\_{\mathbf{un},k} - (j-m) \boldsymbol{\Delta} \boldsymbol{\delta}\_{i,k} \right\|\_{\mathbf{Q}\_{\mathbf{y}}}^{2} \right\|\_{\mathbf{Q}\_{\mathbf{y}}} \\ &+ \left\| \left\| \mathbf{x}^{\mathbf{t}}(k+m|k) \right\|\_{\mathbf{Q}}^{2} + \sum\_{j=0}^{m-1} \left\| \left\| \boldsymbol{\Delta} \mathbf{u}(k+j|k) \right\|\_{\mathbf{R}}^{2} + \left\| \left\| \boldsymbol{\delta}\_{j,k} \right\|\_{\mathbf{S}\_{\mathbf{y}}}^{2} + \left\| \left\| \boldsymbol{\delta}\_{m,k} \right\|\_{\mathbf{S}\_{\mathbf{u}}}^{2} + \left\| \left\| \boldsymbol{\delta}\_{i,k} \right\|\_{\mathbf{S}\_{\mathbf{y}}}^{2} \right\|\_{\mathbf{S}\_{\mathbf{u}}}^{2} \right\|\_{\mathbf{S}\_{\mathbf{u}}}^{2} \right\|\_{\mathbf{S}\_{\mathbf{u}}}^{2} . \end{split}$$

subject to Eqs. (8), (9), and (10), and

$$\Delta\mathfrak{u}(k+j|k)\in\mathbb{U},\quad j=0,\ldots,m-1,\tag{13}$$

$$\mathbf{U}=\left\{\begin{array}{c}-\Delta\mathfrak{u}\_{\max}\leq\Delta\mathfrak{u}(k+j|k)\leq\Delta\mathfrak{u}\_{\max}\\\\ \Delta\mathfrak{u}(k+j|k)=\mathbf{0},j\geq m\\\\ \mathfrak{u}\_{\min}\leq\mathfrak{u}(k-\mathbf{1})+\sum\_{i=0}^{j}\Delta\mathfrak{u}(k+i|k)\leq\mathfrak{u}\_{\max}\end{array}\right\},\tag{14}$$

where Δ*uk* ¼ Δ*u*ð Þ *k*j*k* ⊤ ⋯ Δ*u*ð Þ *k* þ *m* � 1j*k* T � �⊤ is the sequence of control moves along the control horizon.

**Remark 1.** The slack variables play a remarkable role with respect to the feasibility of the control formulation, i.e., the control law of **Problem 1** will always provide a feasible solution, either the nominal case (linear model) or plant-model mismatch, an object under study of this work.

**Remark 2.** The weighting matrices **S***y*, **S***un*, and **S***i* (additional tuning parameters when compared to conventional MPC strategies) should be carefully selected. For instance, the values of **S***y* should be chosen sufficiently large, e.g., orders of magnitude larger than **Q***y* (≥ 103 **Q***y*), to guarantee that the solution of the slacked optimization problem will only use the slack vector when the terminal constraints need to be softened. While *δun*,*k* and *δi*,*k* do not need to be minimized a priori, by issues of achieving the closed-loop stability as fast as possible, one seeks their minimization weighted by large enough values of positive-definite **S***un* and **S***in* (≥102 **Q***y*) in order to enforce them to zero in a finite number of steps.

*An Implementable and Stabilizing Model Predictive Control Strategy for Inverted… DOI: http://dx.doi.org/10.5772/intechopen.91629*

**Remark 3.** From the stability point of view, the master's dissertation [17] demonstrates the conditions necessary to prove that the objective function behaves as a Lyapunov function, thus assuring that the control actions obtained from the solution of **Problem 1** drive the system asymptotically to the reference value (desired steady state), if it is reachable; otherwise, the system will converge to an equilibrium point (reachable steady state) lying at a minimum distance from the desired steady state.

#### **4. Results and discussion**

This section is devoted to present the implementation results of the feasibleoptimization problem-based stabilizing MPC controller (**Problem 1**) in the rotary inverted pendulum prototype described in Section 2. The ultimate goal of the controller is to maintain the pendulum rod in the upright position after it has been swung up to this position by the energy-based swing-up control scheme embedded in the system. In addition, the IHMPC controller is simultaneously designed to track the desired positions to be configured for rotary arm angles. In Quanser apparatus, an unconstrained linear-quadratic regulator (LQR) controller makes up the control system, besides the swing-up control strategy. The existing LQR strategy will be replaced by the IHMPC controller, and this scheme is depicted in **Figure 2**.

The architecture used for this real-time implementation of the IHMPC controller is summarized in **Figure 3**. From this figure, it is possible to note the information exchange among software-hardware-equipment mechanisms of the prototype. The control law is solved at each sampling time on the computer i7-8550H with 1.80GHz processor and 16GB of RAM, using Matlab script and *Quarc real-time control* toolbox. The software-hardware interface is done via USB communication through the Q8-USB acquisition board. This acquisition system acts, in turn, as an interface between the digital part of the system (controller) and the analogic one that is composed of the amplifier (VoltPAQ-X1).

For the experimental results presented as follows, we consider a scenario of square wave-type tracking on the rotary arm, while the controller must maintain the pendulum rod around the upright vertical position, even in the existing unmeasured disturbance scenarios. The constraints associated with the control signal and control actions (decision variables) are those established in **Table 1**.

**Figure 2.** *Schematic representation of the application of the IHMPC controller in the rotary inverted pendulum prototype.*

#### **Figure 3.**

*The architecture used in communication among software-hardware-equipment mechanisms.*

#### **Table 1.**

*Constraints on system inputs.*

Note that there is a strict condition of �1 V on the control actions. Also, the IHMPC tuning parameters considered were sampling period Δ*t* = 2 milliseconds, **Q***y* ¼ *diag* 1, 6 � 102 , **R** ¼ 9*:*8 � 10�2 , **S***y* ¼ *diag* 105 , 104 , **S***un* ¼ *diag* 102 , 6 � 104 , and **S***i* ¼ 102 . The state estimator used here was the Kalman filter, whose tuning parameters associated with process noise and measurement noise were the following covariance matrices **Q***Kalman* ¼ **I**9�9 and **R***Kalman* ¼ 2*:*4 � 10�6**I**2�2, respectively. Finally, a control horizon of *m* ¼ 9 has been adopted as an appropriate value to attain the desired control performance, which was chosen from a sensitivity analysis, as will also be shown here.

The closed-loop system results are depicted in **Figures 4** and **5**. From **Figure 4**, one can see that after about 5.5 seconds, the time necessary that the swing-up control acted to lead the pendulum rod to its upright position, the IHMPC controller takes over and performs quite well both tasks associated with the rotary arm angle tracking and the stabilization of the pendulum rod within an acceptable range lying at about �2∘ . It is also noteworthy that after the execution of the square-wave trajectory on the rotary arm angle, the controller had a great performance concerning impulse-like external disturbances inserted in the pendulum rod since the controlled variables are momentarily moved away from their set points, but soon they are brought back to their original positions.

Even though the stability of the IHMPC controller is only related to the nominal case (linear model), it proved to be very sufficient in a realistic plant-model mismatch scenario, including nonlinearities existing in the rotary inverted pendulum apparatus, such as dead zone, friction, backlash, hysteresis, and so on. This model uncertainty scenario was responsible for non-prohibitive oscillations, within a practical implementation purpose, on the constrained control signal (cf. **Figure 5**), which were reflected in the controlled outputs.

*An Implementable and Stabilizing Model Predictive Control Strategy for Inverted… DOI: http://dx.doi.org/10.5772/intechopen.91629*

**Figure 4.** *Controlled variables: rotary arm angle (θ) and pendulum rod angle (α).*

**Figure 5.** *Behavior of the control signal (tension applied to the servomotor).*

Furthermore, it is worth mentioning that fulfilling a tighter constraint (1V) by a conventional stabilizing MPC controller, e.g., [11, 21], could result in an unfeasibility scenario; however, since IHMPC controller used here is based on a feasible-optimization formulation, its control law always will provide a feasible solution while the system is controllable, thus becoming it implementable in practice. The IHMPC controller uses its additional degrees of freedom (slack variables), when necessary, in order to comply with the terminal constraints. **Figure 6** illustrates the use of the slack variables in the control problem. It is observed that the controller makes use of these variables immediately after a perturbation in the system occurred namely, set-point changes and unmeasured disturbance entrance, situations in which it can be hard to comply with non-slacked terminal constraints.

**Figure 6.** *Behavior of the slack variables associated with the feasible-optimization formulation of the IHMPC controller.*

However, as the system goes to an acceptable cyclic steady state around its set point, due to the noise degree intrinsic to the system, the slack variables converge to the origin very fast and systematically.

On the other hand, in order to obtain a satisfactory performance as in the results presented earlier, an effort with respect to the controller tuning was necessary. This tuning task is easier in the stabilizing MPC controllers than the conventional finite-horizon MPC ones [13], as demonstrated in what follows. In this case, it was sufficient to handle only the control horizon. In inverted pendulum-like fast dynamic systems, when one applies a more aggressive control policy, i.e., small control horizon, it can cause undesired overshoots, while adopting large control horizons, it cannot have time sufficient to act with control action properly, thus bringing unnecessary oscillations or even causing the instability of the closed-loop system.

To work around this trade-off, we proceeded with sensitivity analysis on the control horizon, keeping the same remaining tuning parameters shown in the

*An Implementable and Stabilizing Model Predictive Control Strategy for Inverted… DOI: http://dx.doi.org/10.5772/intechopen.91629*

**Figure 7.** *Sensitivity analysis on the control horizon of the closed-loop system.*

preceding experimental results. **Figure 7** summarizes the aforementioned analysis. Note that as the control horizon increases, the oscillations decrease until the control horizon *m* ¼ 9. However, a value greater than *m* ¼ 9 makes the closed-loop system go back to having undesired and larger oscillations, thus jeopardizing the use of energy associated with the control signal. Therefore, the use of the IHMPC controller enabled a simple analysis concerning only one tuning parameter, which yielded an appropriate value to meet the desired control performance in the real case.

### **5. Conclusions**

In this chapter, we have investigated the application of an implementable and stabilizing model predictive control model strategy in a commercial-didactic rotary inverted pendulum apparatus, hitherto unexplored in the literature. Although the

guarantee of stability of the controller is devoted to the nominal case (linear model), its formulation based on a feasible-optimization problem allows it to be used in any plant-model mismatch scenario in practice, such as one series of nonlinearities existing in the real system, namely, dead zone, friction, and backlash, among other unmodelled dynamics. The experimental results showed the effectiveness and robustness of the controller in the aforementioned plant-model mismatch setting by performing quite well its task in the rotary arm angle tracking and stabilization of the pendulum rod around the upright position as well as in the optimum use of energy associated with control efforts.

A simple tuning procedure was adopted by virtue of using a stabilizing MPC controller, which allowed us to handle only one tuning parameter through sensitivity analysis on the control horizon. The value found was quite adequate to attain the control objectives in terms of the trade-off existing between the performance on the controlled variables and the use of energy related to the control signal.

The future direction for this research is to further extend this controller to guarantee the stability of the nonlinear case, including the energy-based swing-up control schemes.


**2. Surgical techniques for primary flexor tendon repair**

### **2.1 General considerations**

Usually flexor tendon repair is performed in an emergency setup. In cases such as dirty trauma or crush injuries, debridement should be done to convert contaminated wound into a cleaner wound. All the injuries (fracture, skin loss, neurovascular bundles) are repaired simultaneously along with flexor tendon repair. However, if the surgeon does not possess enough expertise to treat such lesions, it's advisable to delay the repair till next appropriate time [12]. All injured flexor tendons should be repaired using proper instruments and under magnification in an operating room thus allowing atraumatic repair of such tendons. Cleaning of wound before tenorrhaphy and in certain circumstances administration of intravenous antibiotics just before, during and 6 hours after surgery is indicated. The tendon repair

is performed under axillary block anesthesia along with a pneumatic tourniquet applied at the level of the arm. The tourniquet is released Just before the wound closure in order to perform hemostasis and hence preventing formation of hematoma, infection, potential adhesion and fibrosis.

### **2.2 Incisions for exposure**

In order to provide better visibility and to allow atraumatic repair of injured tendons, the skin wound should be debrided and enlarged. The position of the finger and the shape of the initial lesion govern the method of extension of wound for exposure; a palmar zigzag approach (Bruner's incision) can be used to extend an oblique skin laceration; a midlateral approach can be used to extend transverse skin laceration. A palmar zigzag incision provides an excellent exposure but at the same time can lead to adhesions as well as scar tissue formation over the repaired tendon. The midlateral approach allows a direct repair of the injured flexor tendon and also preserves vascular transverse branches of neurovascular bundle. Midpalmar incisions and straight incisions that cross flexor creases should be avoided, as sharp angle in the raised skin flaps can result in tip necrosis. The wound needs to be extended in distal direction if tendon injury has occurred while fingers where in flexion and similarly wound needs to be extended proximally if at time of injury fingers where in extension.

Mid palmar transverse incision become sometimes necessary when proximal end of tendon has retracted to palmar level and if flexor tendon massage or flexion of wrist fails to deliver retracted proximal tendon end (**Figure 1**). The silicone tube is passed in retrograde direction into the palm, from superficialis chiasma up to proximal stump. The proximal end of injured tendon is attached to the silicone tube and pulled distally, delivering it back into wound. This procedure helps in avoiding traumatic injury to digital sheath and hence preventing adhesion formation (**Figure 2**) [13].

With an L-shaped incision (Lister's technique) the digital flexor tendon sheath is opened in between the annular pulleys [9]. to prevent bowstringing Annular pulleys (especially A2 and A4) should be spared and repaired if they are traumatized. The sheath should be closed using a fine suture material after completion of repair of severed flexor tendon. In case of severe damage to the sheath, it may be necessary to excise the portion of the sheath over the repaired tendon site to prevent trigger finger or an impingement. In cases where tendon injury is not a result of sharp cut, the tendon ends needs to be refreshed using a sharp blade, this debridement should be minimal to avoid any tension over the tenorrhaphy site. A needle can be used to fix proximal end in place while performing tenorrhaphy, this allows tension free approximation of two injured tendon ends (**Figure 3**).

**127**

**2.3 Suturing technique**

**Figure 2.**

*adhesions.*

**Figure 3.**

Various suturing techniques have been defined. Among these the modified Kessler suture using two sutures [14] and a 'grasping' suture [15, 16] having knots inside the cross-section have been widely accepted. A running fine epitendinous sutures increase the tensile strength and also allows smooth gliding of the repaired tendon within the digital sheath. However, immediate active rehabilitation is not possible after using these suturing techniques. Therefore, number of studies has been carried out to improve the suturing technique as well as the suturing material. The 'ideal' suturing material should be strong, pliable but non-reactive, and of small caliber. It is advisable to use Nylon 3/0 for the central suture and for the epitendinous running suturing its recommended to use nylon 5/0 or 6/0 [17]. The 'locking' or a 'grasping' suture [18, 19] with four or six strand sutures [20, 21] is considered to be ideal for central core suturing along with running epitendinous locking sutures [22]. This allows an immediate active rehabilitation programme as this suturing technique provides double strength than usual traditional suturing methods. Tsuge's suture [23] is easy method of performing 'locking' sutures, but it leaves a knot outside over the tendon repair site, which in turn can affect smooth gliding of flexor tendons within the synovial sheath or the pulley system. This new suturing method by virtue of strong repair, allows early active motion with minimal risk of gap formation or early tendon rupture. A new material has been reported which is characterized by its high traction resistance: it consists of two intratendinous, stainless steel anchors that are joined by a multifilament stainless steel suture.

*Temporary fixation of the proximal tendon stump with a needle to facilitate repair.*

*A silicone tube to bring the proximal tendon stump to the desired level to avoid potential formation of*

*Management of Flexor Tendon Injuries in Hand DOI: http://dx.doi.org/10.5772/intechopen.83483*

**Figure 1.** *Bruner's incision with palmar incision to expose the tendon stumps.*

*Management of Flexor Tendon Injuries in Hand DOI: http://dx.doi.org/10.5772/intechopen.83483*

### **Figure 2.**

*Tendons*

infection, potential adhesion and fibrosis.

**2.2 Incisions for exposure**

fingers where in extension.

is performed under axillary block anesthesia along with a pneumatic tourniquet applied at the level of the arm. The tourniquet is released Just before the wound closure in order to perform hemostasis and hence preventing formation of hematoma,

In order to provide better visibility and to allow atraumatic repair of injured tendons, the skin wound should be debrided and enlarged. The position of the finger and the shape of the initial lesion govern the method of extension of wound for exposure; a palmar zigzag approach (Bruner's incision) can be used to extend an oblique skin laceration; a midlateral approach can be used to extend transverse skin laceration. A palmar zigzag incision provides an excellent exposure but at the same time can lead to adhesions as well as scar tissue formation over the repaired tendon. The midlateral approach allows a direct repair of the injured flexor tendon and also preserves vascular transverse branches of neurovascular bundle. Midpalmar incisions and straight incisions that cross flexor creases should be avoided, as sharp angle in the raised skin flaps can result in tip necrosis. The wound needs to be extended in distal direction if tendon injury has occurred while fingers where in flexion and similarly wound needs to be extended proximally if at time of injury

Mid palmar transverse incision become sometimes necessary when proximal end of tendon has retracted to palmar level and if flexor tendon massage or flexion of wrist fails to deliver retracted proximal tendon end (**Figure 1**). The silicone tube is passed in retrograde direction into the palm, from superficialis chiasma up to proximal stump. The proximal end of injured tendon is attached to the silicone tube and pulled distally, delivering it back into wound. This procedure helps in avoiding traumatic injury to

With an L-shaped incision (Lister's technique) the digital flexor tendon sheath is opened in between the annular pulleys [9]. to prevent bowstringing Annular pulleys (especially A2 and A4) should be spared and repaired if they are traumatized. The sheath should be closed using a fine suture material after completion of repair of severed flexor tendon. In case of severe damage to the sheath, it may be necessary to excise the portion of the sheath over the repaired tendon site to prevent trigger finger or an impingement. In cases where tendon injury is not a result of sharp cut, the tendon ends needs to be refreshed using a sharp blade, this debridement should be minimal to avoid any tension over the tenorrhaphy site. A needle can be used to fix proximal end in place while performing tenorrhaphy, this allows tension free

digital sheath and hence preventing adhesion formation (**Figure 2**) [13].

approximation of two injured tendon ends (**Figure 3**).

*Bruner's incision with palmar incision to expose the tendon stumps.*

**126**

**Figure 1.**

*A silicone tube to bring the proximal tendon stump to the desired level to avoid potential formation of adhesions.*

**Figure 3.** *Temporary fixation of the proximal tendon stump with a needle to facilitate repair.*

### **2.3 Suturing technique**

Various suturing techniques have been defined. Among these the modified Kessler suture using two sutures [14] and a 'grasping' suture [15, 16] having knots inside the cross-section have been widely accepted. A running fine epitendinous sutures increase the tensile strength and also allows smooth gliding of the repaired tendon within the digital sheath. However, immediate active rehabilitation is not possible after using these suturing techniques. Therefore, number of studies has been carried out to improve the suturing technique as well as the suturing material.

The 'ideal' suturing material should be strong, pliable but non-reactive, and of small caliber. It is advisable to use Nylon 3/0 for the central suture and for the epitendinous running suturing its recommended to use nylon 5/0 or 6/0 [17]. The 'locking' or a 'grasping' suture [18, 19] with four or six strand sutures [20, 21] is considered to be ideal for central core suturing along with running epitendinous locking sutures [22]. This allows an immediate active rehabilitation programme as this suturing technique provides double strength than usual traditional suturing methods. Tsuge's suture [23] is easy method of performing 'locking' sutures, but it leaves a knot outside over the tendon repair site, which in turn can affect smooth gliding of flexor tendons within the synovial sheath or the pulley system. This new suturing method by virtue of strong repair, allows early active motion with minimal risk of gap formation or early tendon rupture. A new material has been reported which is characterized by its high traction resistance: it consists of two intratendinous, stainless steel anchors that are joined by a multifilament stainless steel suture.

This permanent implant is intended to hold the repaired ends of tendon in close approximation until healing is completed. Protected passive tendon mobilization exercises are carried out after completion of tendon repair.

### **2.4 Postoperative management**

postoperatively, A dorsal plaster splint is applied, from the proximal forearm to the fingertips in 'intrinsic position': the wrist is kept with 20° of palmar flexion, the MP joints in 60° flexion while as PIP and DIP joints are placed in full extension in order to avoid development of any flexion contracture. Two possible options as post-operative protocol are: to immobilize operated hand for 4 weeks or to start early mobilization according to specific exercise protocol. Immobilization is better option in non-cooperative patients and in case of children, in these patients mobilization is started in fifth week, with combination of both active as well as passive motions with dorsal blocking splint in place; seventh week onwards mobilization against resistance is initiated. The complications like tendon rupture or gaping of repair are very low with this protocol, but there are increased chances of adhesion formation, which usually requires tenolysis. Strickland and Glogovac [24] and Lister et al. [25] studied the benefits of early mobilization for tendon healing with better final end results especially in zone II flexor tendon repairs. In these studies using controlled passive mobilization post-operatively excellent results were obtained in 36% patients, 24% patients had poor results and only 4% had tenorrhaphy ruptures; while as no excellent results were obtained in cases of immobilization protocol, poor results were observed in 44% patients and 16% had tenorrhaphy site ruptures. Gelberman et al. [26, 27] also obtained superior results with early mobilization protocol and additional advantages like: improved tendon gliding (due to low rate of soft tissue adherence), enhancement of intrinsic healing mechanism, with enhanced tensile strength thereby decreasing risk of gap formation. The highest risk of tenorrhaphy rupture is between 5th and 10th postoperative day, during this period the hand therapist should be very cautious while doing during active motion exercises.

Kleinert et al. [28] proposed active and passive mobilization with dorsal blocking plaster splint keeping wrist in flexion of 20°, MP joint in flexion of 70° and allowing complete extension of fingers. An elastic traction band is attached to a loop, which is fixed to nail, keeping fingers in flexion but at same time allowing active extension within the range of dorsal blocking splint (**Figure 4**). D first 4 weeks, the patients is asked to perform active extension of the fingers many times for half an hour periods every day at different intervals. For the rest of the day and during the night the rubber band traction is detached in order to prevent development of flexion contracture in interphalangeal joint. At the beginning, the exercises should be guided by the hand therapist keeping patients elbow flexed and pronated in order to relax the flexor muscles. Between the fifth and sixth post-operative weeks, active flexion is begun with dorsal blocking splint in place. This technique is excellent but highly demanding for the therapist, surgeon as well as patient, and a control at every step is necessary to prevent a rupture or a gap at the tendon repair site. A palmar pulley situated at the level of the distal palmar crease significantly improves the range of flexion of the fingers and hence better results have been reported. After repairing the flexor pollicis longus (FPL) in the thumb MP and IP joints are kept in 20° of flexion.

Duran and Hauser [29] proposed controlled passive motion for the post-operative flexor tendon repaired lesions in zone II. The wrist is kept in 20–30° of flexion, the MP joint in 60° of flexion while as PIP and DIP joints are placed in extension.

**129**

hand surgeon is very important.

in close collaboration with a hand surgeon.

*Management of Flexor Tendon Injuries in Hand DOI: http://dx.doi.org/10.5772/intechopen.83483*

*Kleinert's technique for passive flexion and active extension.*

**Figure 4.**

For first 4 weeks controlled passive motion is used, by the hand therapist, twice a day with each session of six to eight motions for each tendon. This method uses 3–5 minute exercise movements at the repair site for preventing any firm adhesion formation. For a week, a rubber band traction is attached to the wrist and active

Duran's technique was modified by Strickland [15]. He increased the duration and frequency of the passive daily exercises. The PIP and DIP joints are separately mobilized with repeated motions of full passive extension and flexion. An occupational therapist works closely with the operating hand surgeon and guides the controlled passive motion protocol during first 5 weeks. For starting the active flexion exercises after 5th week, the block technique advised by Bunnel [30] can be utilized: the PIP joint is actively flexed while the MP joint is blocked in extension; similarly while the DIP joint is actively flexed, the PIP joint is blocked in extension. Beyond 6 weeks, If the extension of the finger is limited, dynamic splinting may be necessary. Six months is the minimal period before considering any tenolysis and this is the time period that is required to obtain complete motion (especially in children). Excellent results were reported by Chow et al. [31] in a multicenter study carried

out for zone II flexor tendon injuries. They utilized rubber band traction with a palmar pulley at distal palmar crease level thus increasing passive flexion at the MP and PIP joints. This modification increases the differential gliding between superficialis and profundus tendon and in addition increases tendon excursion in the sheath as well. Full passive extension and flexion were performed for the first 4 weeks under the supervision of a hand therapist, in addition to the active extension exercise programme against the rubber band traction. The rubber band traction is removed for the fifth and sixth week and active and passive full flexion and extension exercises are performed. To prevent development of any contracture at the level of interphalangeal joints the supervision of both hand therapist and a

Many authors have reported their results using early active flexion exercises after performing flexor tendon repair in zone II [32–34]. to perform this rehabilitation programme it's important to Improve the quality as well as resistance of the suture, in order to prevent rupture or a gap of the tendon at the repair site. Indications for using early motion protocol is limited to motivated and intelligent patients having clean cut tendon injury, with a specialized hand therapist working

Magnetic resonance imaging (MRI) is very useful for diagnosing many postoperative complications especially in differentiating gap from adhesion formation,

especially in zone II after tenorrhaphy of FDS and FDP tendon injuries.

exercises are done for 2 weeks with dorsal blocking splint in place.

*Management of Flexor Tendon Injuries in Hand DOI: http://dx.doi.org/10.5772/intechopen.83483*

*Tendons*

This permanent implant is intended to hold the repaired ends of tendon in close approximation until healing is completed. Protected passive tendon mobilization

postoperatively, A dorsal plaster splint is applied, from the proximal forearm to the fingertips in 'intrinsic position': the wrist is kept with 20° of palmar flexion, the MP joints in 60° flexion while as PIP and DIP joints are placed in full extension in order to avoid development of any flexion contracture. Two possible options as post-operative protocol are: to immobilize operated hand for 4 weeks or to start early mobilization according to specific exercise protocol. Immobilization is better option in non-cooperative patients and in case of children, in these patients mobilization is started in fifth week, with combination of both active as well as passive motions with dorsal blocking splint in place; seventh week onwards mobilization against resistance is initiated. The complications like tendon rupture or gaping of repair are very low with this protocol, but there are increased chances of adhesion formation, which usually requires tenolysis. Strickland and Glogovac [24] and Lister et al. [25] studied the benefits of early mobilization for tendon healing with better final end results especially in zone II flexor tendon repairs. In these studies using controlled passive mobilization post-operatively excellent results were obtained in 36% patients, 24% patients had poor results and only 4% had tenorrhaphy ruptures; while as no excellent results were obtained in cases of immobilization protocol, poor results were observed in 44% patients and 16% had tenorrhaphy site ruptures. Gelberman et al. [26, 27] also obtained superior results with early mobilization protocol and additional advantages like: improved tendon gliding (due to low rate of soft tissue adherence), enhancement of intrinsic healing mechanism, with enhanced tensile strength thereby decreasing risk of gap formation. The highest risk of tenorrhaphy rupture is between 5th and 10th postoperative day, during this period the hand therapist should be very cautious while doing during active motion

Kleinert et al. [28] proposed active and passive mobilization with dorsal blocking plaster splint keeping wrist in flexion of 20°, MP joint in flexion of 70° and allowing complete extension of fingers. An elastic traction band is attached to a loop, which is fixed to nail, keeping fingers in flexion but at same time allowing active extension within the range of dorsal blocking splint (**Figure 4**). D first 4 weeks, the patients is asked to perform active extension of the fingers many times for half an hour periods every day at different intervals. For the rest of the day and during the night the rubber band traction is detached in order to prevent development of flexion contracture in interphalangeal joint. At the beginning, the exercises should be guided by the hand therapist keeping patients elbow flexed and pronated in order to relax the flexor muscles. Between the fifth and sixth post-operative weeks, active flexion is begun with dorsal blocking splint in place. This technique is excellent but highly demanding for the therapist, surgeon as well as patient, and a control at every step is necessary to prevent a rupture or a gap at the tendon repair site. A palmar pulley situated at the level of the distal palmar crease significantly improves the range of flexion of the fingers and hence better results have been reported. After repairing the flexor pollicis longus (FPL) in the thumb MP and IP

Duran and Hauser [29] proposed controlled passive motion for the post-operative

flexor tendon repaired lesions in zone II. The wrist is kept in 20–30° of flexion, the MP joint in 60° of flexion while as PIP and DIP joints are placed in extension.

exercises are carried out after completion of tendon repair.

**2.4 Postoperative management**

**128**

joints are kept in 20° of flexion.

exercises.

**Figure 4.** *Kleinert's technique for passive flexion and active extension.*

For first 4 weeks controlled passive motion is used, by the hand therapist, twice a day with each session of six to eight motions for each tendon. This method uses 3–5 minute exercise movements at the repair site for preventing any firm adhesion formation. For a week, a rubber band traction is attached to the wrist and active exercises are done for 2 weeks with dorsal blocking splint in place.

Duran's technique was modified by Strickland [15]. He increased the duration and frequency of the passive daily exercises. The PIP and DIP joints are separately mobilized with repeated motions of full passive extension and flexion. An occupational therapist works closely with the operating hand surgeon and guides the controlled passive motion protocol during first 5 weeks. For starting the active flexion exercises after 5th week, the block technique advised by Bunnel [30] can be utilized: the PIP joint is actively flexed while the MP joint is blocked in extension; similarly while the DIP joint is actively flexed, the PIP joint is blocked in extension. Beyond 6 weeks, If the extension of the finger is limited, dynamic splinting may be necessary. Six months is the minimal period before considering any tenolysis and this is the time period that is required to obtain complete motion (especially in children).

Excellent results were reported by Chow et al. [31] in a multicenter study carried out for zone II flexor tendon injuries. They utilized rubber band traction with a palmar pulley at distal palmar crease level thus increasing passive flexion at the MP and PIP joints. This modification increases the differential gliding between superficialis and profundus tendon and in addition increases tendon excursion in the sheath as well. Full passive extension and flexion were performed for the first 4 weeks under the supervision of a hand therapist, in addition to the active extension exercise programme against the rubber band traction. The rubber band traction is removed for the fifth and sixth week and active and passive full flexion and extension exercises are performed. To prevent development of any contracture at the level of interphalangeal joints the supervision of both hand therapist and a hand surgeon is very important.

Many authors have reported their results using early active flexion exercises after performing flexor tendon repair in zone II [32–34]. to perform this rehabilitation programme it's important to Improve the quality as well as resistance of the suture, in order to prevent rupture or a gap of the tendon at the repair site. Indications for using early motion protocol is limited to motivated and intelligent patients having clean cut tendon injury, with a specialized hand therapist working in close collaboration with a hand surgeon.

Magnetic resonance imaging (MRI) is very useful for diagnosing many postoperative complications especially in differentiating gap from adhesion formation, especially in zone II after tenorrhaphy of FDS and FDP tendon injuries.


*3.4. TiN*

The XRD patterns are shown in Figure 7 for unirradiated and irradiated TiN films on the SiO2 glass substrate. As already mentioned in Section 2, (111) and (200) diffraction peaks are observed and the XRD intensity decreases due to ion impact. Figure 8 shows XRD intensities normalized to those of unirradiated TiN films on SiO2 glass, C-Al2O3 and R-Al2O3 substrates as a function of the ion fluence. It is seen that the XRD intensity degradation is nearly the same for the diffraction planes of (111) and (200) on SiO2, and for (111) on C-Al2O3. The XRD intensity degradation is less sensitive to the ion impact for the diffraction plane (220) on the R-Al2O3 substrate (~30% smaller than that for (111) and (200) on SiO2, and for (111) on C-Al2O3). The XRD intensity degradation per unit fluence YXD for (111) and (200) diffractions is given in Table 5, together with sputtering yields and stopping powers (TRIM1997 and SRIM2013). No appreciable change in the lattice parameter is observed, as shown in Figure 7. Similarly to the SiO2, ZnO and Fe2O3 cases, the appropriate energy, E − Se`/2, ` = film thickness of ~170 nm is taken into account, and the energy is close to that for sputtering, in which the energy loss of the carbon foil of 100 nm is considered. The X-ray (Cu-kα) attenuation length LXA is obtained to be 11.8 µm [80], and the attenuation depth is 3.7, 4.3 and 6.0 µm for diffraction angles of 36.6◦ , ~43◦ and 61◦ , respectively; thus, the X-ray attenuation correction is insignificant.

**Figure 7.** XRD patterns of TiN film on SiO2 glass substrate: unirradiated (•) and irradiated by 100 MeV Xe at 0.72 × 1012 cm−2 (*+*).

**Figure 8.** XRD intensity normalized to unirradiated films of TiN as a function of ion fluence for 60 MeV Ar (, ♦), 90 MeV Ni (H, +, , T ), 100 MeV Xe (o, x, ) and 200 MeV Xe (N, •) ions. Diffraction plane (111) at diffraction angle of ~36.6◦ is indicated by , H, o and N for SiO2 substrate, (200) at ~43◦ by ♦, +, x and • for SiO2 substrate, (111) by for C-Al2O3 substrate and (220) at ~61◦ by T and for R-Al2O3 substrate. Linear fit is indicated by dotted lines. An estimated error of XRD intensity is 10%.

The characteristic length (LEQ) is estimated to be 4.5, 4.4, 4.2 and 4.0 nm for 60 MeV Ar+7, 90 MeV Ni+10, 100 MeV Xe+14 and 200 MeV Xe+14, respectively, from the empirical formula of the single-electron loss cross-section σ1L(10−16 cm2 ) of 0.43 (60 MeV Ar+7), 0.44 (90 MeV Ni+10), 0.46 (100 MeV Xe+14) and 0.48 (200 MeV Xe+14) [83,84]. Here, σ1L = σ1L(Ti) + σ1L(N), and the ionization potential IP and Neff are (IP = 143 eV and Neff = 1) for Ar+7 , with those described in Section 3.1 for Ni+10 and Xe+14. LEQ is much smaller than the film thickness, and hence the charge-state effect is insignificant.

It is found that sputtered Ti collected in the carbon foil is proportional to the ion fluence, as shown in Figure 9 for 60 MeV Ar, 90 MeV Ni, 100 MeV Xe and 200 MeV Xe ions. The sputtering yield of Ti is obtained using the collection efficiency of 0.34 in the carbon foil collector [47] and the results are given in Table 5. Sputtered N collected in the carbon foil is obtained to be 0.4 × 1014 and 0.44 × 1014 cm−2 with an estimated error of 20% for 200 MeV Xe at 0.22 × 1012 cm−2 and 60 MeV Ar at 2.8 × 1012 cm−2 , respectively, and this is comparable with the Ti areal density of 0.4 × 1014 cm−2 (200 MeV Xe) and 0.475 × 1014 cm−2 (60 MeV Ar). The results imply stoichiometric sputtering, due to the collection efficiency of N in the carbon foil collector of 0.35 [55], which is close to that of Ti. Thus, the total sputtering yield (Ti + N) is obtained by doubling Ysp (Ti) in Table 5. The sputtering yields of TiN (YEC) due to elastic collisions can be estimated assuming that YEC is proportional to the nuclear stopping power. Here, the proportional constant is obtained to be ~1.6 nm/keV using the experimental yields of 0.527 (0.6 keV Ar) and 0.427 (0.6 keV N) [94] and 0.7 (0.5 keV Cd) [88]. Ysp(TiN)/YEC ranges from 2.5 × 103 to 6 × 103 . The XRD intensity degradations YXD and Ysp(Ti + N) are plotted as a function of the electronic stopping power Se in Figure 10. It appears that both fit to the power-law: YXD = (0.0224Se) 1.26 and Ysp = (1.17Se) 1.95. The exponents are comparable for XRD intensity degradation and sputtering.

**Figure 9.** Areal density of sputtered Ti from TiN on SiO2 substrate collected in carbon foil vs. ion fluence for 60 MeV Ar (), 89 MeV Ni (∇), 99 MeV Xe (o) and 198 MeV Xe (∆) ions. An estimated error of areal density is 20%.

**Figure 10.** XRD intensity degradation YXD (10−12 cm2 ) (o, +) and sputtering yields Ysp (Ti + N) (, x) vs. electronic stopping power Se (keV/nm). Se is calculated by TRIM1997 (o, ) and by SRIM2013 (+, x). Power-law fits are indicated by dotted lines: YXD = (0.0224Se) 1.26 and Ysp = (1.17Se) 1.95 .


*4.3. Particle Swarm Optimization (PSO)*

Particle swarm optimization is inspired by the behavior of bird flocking and fish schooling [36]. Initially, the particles are generated with random positions and zero velocity, and the objective function is then evaluated. In the next step, the best positions of each particle, which is called the personal best position (*PPbest*(*i*)), are updated and the best position among all particles, which is called the global best position, is updated (*PGbest*). The velocity and position of each particle can be updated according to the *PPbest*(*i*) and the *PGbest* by the following equations.

$$V\_{P(i)} = wP(i) + c\_1r\_1(P\_{\text{Pbest}}(i) - P(i)) + c\_2r\_2(P\_{\text{Gbest}} - P(i))\tag{20}$$

$$(P(i)\_{n \le \upsilon} = P(i) + V\_{P(i)}) \tag{21}$$

where *VP*(*i*) is the velocity of the *i*th particle; *P*(*i*) is the current position of the *i*th particle; *PPbest*(*i*) is the personal best position of the *i*th particle; *PGbest* is the global best position among all particles; *P*(*i*)*new* is the updated position of the *i*th particle, *w* = 0.9 − (((0.9 − 0.4) ∗*it*)/*itermax*); *it* is the *it*th iteration; *c*1 = *c*2 = 2 and *r*1, *r*2 are random numbers randomly generated between 0 and 1.