## 5 Applications in Control Theory
### Control of Chaos
#### 5.2.2 Adaptive and Predictive Control  
In [114], Chen and Dong presented interesting analysis and simulations on the control of chaotic dynamic systems using conventional feedback control strategies. The trajectories of the chaotic Lozi system are controlled to its equilibrium points using conventional feedback controls. The proposed controlled Lozi map is  
\[\left\{ \begin{array}{l}x_{k+1}=1-a|x_{k}|+y_{k},\\ y_{k+1}=bx_{k}-Kv_{k}.\end{array} \right. \tag{75}\]  
For this system, Chen and Dong proved the following. **Proposition 2** ([114]). : _A necessary and sufficient condition for the local controllability of the chaotic Lozi system (3) to its equilibrium point \((x^{*},y^{*})=\left(\frac{1}{a-(b-1)},\frac{b}{a-(b-1)}\right)\) is \(v_{k}\longrightarrow 0\), as \(k\longrightarrow\infty\), where \(u_{k}=-Kv_{k}\) is the feedback controller applied to the system shown in (75) and \(v_{k}\) is the executive control input in the feedback \(u_{k}=-Kv_{k}\)._Computer simulations have been performed for \(a=1.8\), \(b=-1\), \(K=-0.195\); \(a=1.8\), \(b=0.4\), \(K=1.205\); \(a=1.8\), \(b=0.997\), \(K=1.802\); and \(a=1.8\), \(b=1.0\), \(K=1.805\). Improving this method, Gonzalez [115] considered the Lozi map and showed that the trajectories can be driven to the fixed point as well as to other fixed reference points \((x^{*},bx^{*})\). The controlled the Lozi map defined as  
\[\left\{ \begin{array}{l}x_{k+1}=1-a|x_{k}|+y_{k}+u_{k},\\ y_{k+1}=bx_{k}.\end{array} \right. \tag{76}\]  
The following two control algorithms were proposed. * Case 1: the value of \(a\) is unknown and the value of \(b\) is precisely known. In this case, the chosen control law is \[u_{k}=\bar{a}_{k}|x_{k}|-bx_{k-1}-1+x^{*},\] with \[\bar{a}_{k+1}=\bar{a}_{k}+\frac{\alpha|x_{k}|}{\gamma+x_{k}^{2}}(x_{k+1}-(- \bar{a}|x_{k}|+bx_{k-1}+1+u_{k}))\] and \(\alpha>0\), \(\gamma\geq 0\). With \(a_{k}=\bar{a}_{k}-a\), the closed loop system emerges: \[\left\{ \begin{array}{l}x_{k+1}=a_{k}|x_{k}|+x^{*},\\ a_{k+1}=a_{k}-\frac{\alpha|x_{k}|^{2}}{\gamma+x_{k}^{2}}a_{k}.\end{array} \right.\]
* Case 2: both values of \(a\) and \(b\) are known approximately. In this second case, the chosen control law is \[u_{k}=\bar{a}_{k}|x_{k}|-1+x^{*},\] (77) where the parameter estimator is provided by \[\bar{a}_{k+1}=\bar{a}_{k}+\frac{\alpha|x_{k}|}{\gamma+x_{k}^{2}}(x_{k+1}-(- \bar{a}|x_{k}|+1+u_{k}))\] (78) and \(\alpha>0\), \(\gamma\geq 0\). With \(a_{k}=\bar{a}_{k}-a\) and \(z_{k}=x_{k-1}\), the closed loop systems emerges: \[\left\{ \begin{array}{l}x_{k+1}=a_{k}|x_{k}|+bz_{k}+x^{*},\\ z_{k+1}=x_{k-1},\\ a_{k+1}=a_{k}-\frac{\alpha|x_{k}|^{2}}{\gamma+x_{k}^{2}}(\bar{a}_{k}|x_{k}|+bz_ {k}).\end{array} \right.\] (79)  
In this case, there is now the law \(b\). It was proved that for every real \(a\) and \(-1<b<1/2\), \(b\
eq 0\) there exist \(\alpha\) and \(\beta\) such that the fixed point is asymptotically stable for (79). This means that by applying the adaptive strategy (77) and (78) to the control system (76) and taking \((x_{1}(0),x_{2}(0),\bar{a}_{0})\) in the neighborhood of \((x^{*},gx^{*},a-sgn(x^{*})b)\), we obtain \((x_{1}(k),x_{2}(k))\longrightarrow(x^{*},gx^{*})\) as \(k\longrightarrow+\infty\); at the same time, we obtain a better approximation to the values of \(a\) and \(b\). These two proposed control laws can be applied when \(0<b<1\), \(a>b+1\), and \(2a+b<4\), that is, for the regions in the parameter space where chaos can occur (see Figure 3). It has been shown that when using these control laws \((x,y)\) converges towards \((x^{*},bx^{*})\). In [116], Park et al. investigated the possibility of controlling discrete-time chaotic dynamical systems through a generalized predictive control (GPC) method which does not require a precise mathematical model of the chaotic dynamics. GPC methods may be considered as a kind of adaptive control strategy. By combining the ARMAX (AutoRegressive Moving Average model with eXogenous noise input) modeling with the GPCtechnique, they developed a strategy for adaptive control of uncertain discrete-time chaotic systems.