**5.2 Some specific results**

Example 5.1. For the HE4 system with


,

*C* ¼ ½ � *C*1*C*<sup>2</sup> , where

*DOI: http://dx.doi.org/10.5772/intechopen.89319*

*σ*ð Þ¼ *A*

we had the following results:

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

*K*ð Þ <sup>0</sup> ¼

*K*ð Þ <sup>0</sup> ¼

*σ*ð Þ <sup>0</sup>

by the RS Algorithm1

 ><

>:

*<sup>K</sup>*ð Þ <sup>0</sup> <sup>¼</sup> *<sup>K</sup>*ð Þ <sup>0</sup>

*<sup>σ</sup> <sup>A</sup>* � *BK*ð Þ <sup>0</sup> *<sup>C</sup>* � � <sup>¼</sup>

�0*:*00000185 0*:*00001067 �0*:*00071398 0*:*00083459 *:*99999992 0*:*00000007 0*:*00000769 0*:*00494278

*Algorithms for LQR via Static Output Feedback for Discrete-Time LTI Systems*

,

,

 >=

>; ,

 >>>>>=

>>>>>;

,

�0*:*00000041 0*:*99999963 0*:*00472357 �0*:*00007589 �0*:*00001393 �0*:*00000365 �0*:*00972548 �0*:*05509146 �0*:*00008138 �0*:*00007179 0*:*94529303 �0*:*01190104 �0*:*00001330 0*:*00001443 0*:*00205491 0*:*99008269

*:*00022183 0*:*05942943 0*:*05327854 �0*:*99534942 *:*00026594 0*:*00000044 0*:*00000044 �0*:*00000002 *:*00029502 0*:*00000254 �0*:*00000225 0*:*00000017 *:*99634129 0*:*00008613 �0*:*00002336 0*:*00001053 �0*:*00030142 0*:*00050357 �0*:*00044968 0*:*00003365 �0*:*00000027 0*:*00008353 0*:*00009039 �0*:*00000579

> *:*638773652186517, 0*:*847768449750652 *:*002501752901569, 0*:*960047795833900

�0*:*05281866 0*:*30558099 �0*:*04123125 �0*:*74370605 �0*:*07272045 0*:*16180699

�0*:*34989799 �0*:*70937255 �0*:*03438071 *:*36682921 �0*:*55329174 �0*:*42930790

�0*:*07894070 �0*:*83106320 0*:*63513665 �0*:*57049060 0*:*02944824 0*:*03985277

*:*01822942 �0*:*40097959 �0*:*32739026

*:*88529956, 0*:*98303140,

� ¼ 1*:*1052,

*:*99890558 � 0*:*00709027*i*, *:*99895917 � 0*:*00548801*i*, *:*99648905, 0*:*99228590

�0*:*32035712 0*:*23550532 0*:*55239497

*:*990353602254223

 >>>>><

>>>>>:

� �

max <sup>¼</sup> <sup>4</sup>*:*<sup>1660</sup> � <sup>10</sup>6, *<sup>K</sup>*ð Þ <sup>0</sup> �

by the RS algorithm for minimal-gain SOF (see [10])

 *<sup>K</sup>*ð Þ <sup>0</sup> h i, where

*C*<sup>1</sup> ¼

*C*<sup>2</sup> ¼

with

*Algorithms for LQR via Static Output Feedback for Discrete-Time LTI Systems DOI: http://dx.doi.org/10.5772/intechopen.89319*


with

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

2.The MC algorithm achieved a very close value of *σ*max *K*ð Þ *best* � � to the lower

As was expected, the MC algorithm seems to perform better locally, while the RS algorithm seems to perform better globally. Thus, practically, the best approach is to apply the RS algorithm in order to find a close neighborhood of a global minimum and then to apply the MC algorithm on the result, for the local optimization,

*:*99999985 0*:*00000014 0*:*00001538 0*:*00988556

*:*00053188 0*:*00000088 0*:*00000089 �0*:*00000005 *:*00059005 0*:*00000508 �0*:*00000451 0*:*00000034 �0*:*00060285 0*:*00100715 �0*:*00089937 0*:*00006730 �0*:*00000054 0*:*00016706 0*:*00018078 �0*:*00001158

*:*99268256 0*:*00018120 �0*:*00003706 0*:*00002045 �0*:*00008474 0*:*99978709 �0*:*00020742 0*:*00015757 *:*00841925 0*:*00020153 0*:*99963047 0*:*00000293

�0*:*00000005 0*:*00013964 �0*:*00000914 0*:*99709909

�0*:*00000097 0*:*00002352 �0*:*00000082 �0*:*00000055 *:*00000679 0*:*00000357 �0*:*00013147 �0*:*00000146 *:*00117865 0*:*00072316 �0*:*02601681 �0*:*00016953 �0*:*00035692 0*:*00470509 0*:*00008442 �0*:*00000015 *:*00302236 0*:*00013040 �0*:*00468257 �0*:*00205817 *:*00286635 �0*:*00539604 �0*:*00009665 0*:*00000026 �0*:*00018970 0*:*00014386 �0*:*00517989 0*:*00234114 �0*:*04813606 �0*:*00000574 �0*:*00000060 �0*:*00000001

,

,

,

�0*:*00000082 0*:*99999927 0*:*00944714 �0*:*00015179 �0*:*00016276 �0*:*00014358 0*:*89058607 �0*:*02380208 �0*:*00002661 0*:*00002887 0*:*00410983 0*:*98016538 �0*:*00002930 �0*:*00000576 �0*:*01923117 �0*:*00428369 �0*:*32100350 �0*:*00003189 �0*:*00471073 0*:*02122199 *:*00098904 0*:*32051910 �0*:*02076100 �0*:*00474084 �0*:*01910442 0*:*01711196 0*:*00004146 �0*:*00066123

bound, for the systems AC6, HE3, DIS5, NN5, and NN16.

as is evidently seen from the performance of the RS + MC algorithm.

**5.2 Some specific results**

*Control Theory in Engineering*

*A*<sup>1</sup> ¼

*A*<sup>2</sup> ¼

*B* ¼

*A* ¼ ½ � *A*1*A*<sup>2</sup> , where

Example 5.1. For the HE4 system with

$$\sigma(A) = \begin{cases} 0.638773652186517, 0.847768449750652 \\ 1.002501752901569, 0.960047795833900 \\ 0.990353602254223 \end{cases} \Bigg{]} $$

,

,

we had the following results: by the RS algorithm for minimal-gain SOF (see [10])

$$\begin{aligned} K^{(0)} &= \left[ K\_1^{(0)} K\_2^{(0)} \right], \text{where} \\ K\_1^{(0)} &= \begin{bmatrix} -0.05281866 & 0.30558099 & -0.04123125 \\ -0.74370605 & -0.07272045 & 0.16180699 \\ -0.34989799 & -0.70937255 & -0.03438071 \\ 0.36682921 & -0.5329174 & -0.42930790 \\ 0.3668291 & -0.83106320 & 0.6513665 \end{bmatrix}, \\ K\_2^{(0)} &= \begin{bmatrix} -0.07894070 & -0.83106320 & 0.6513665 \\ -0.57049060 & 0.02944824 & 0.0398527 \\ 0.01822942 & -0.40097999 & -0.32739026 \\ -0.32035712 & 0.2350532 & 0.55239497 \end{bmatrix}, \\ \sigma(A - BK^{(0)}C) &= \begin{Bmatrix} 0.88529956, 0.98303140, \\ 0.99890558 \pm 0.00709027i, \\ 0.99895917 \pm 0.00548801i, \\ 0.99648905, 0.99228590 \end{Bmatrix}, \\ \sigma\_{\max}^{(0)} &= 4.1660 \cdot 10^6, \left| K^{(0)} \right| = 1.1052, \end{aligned}$$

by the RS Algorithm1

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

$$\begin{aligned} K^{(best)} &= \left[ K\_1^{(best)} K\_2^{(jet)} \right], \text{where} \\ K\_1^{(best)} &= \begin{bmatrix} 2.60115550 & 0.71670943 & 1.38518242 \\ -1.21472623 & 7.41955425 & -2.28748737 \\ 0.08032866 & -1.01678491 & -3.50944968 \end{bmatrix}, \\ &0.88639191 & -0.92895747 & 0.77017051 \\ 0.88639191 & -0.92895747 & 0.710952615 \\ -0.37181168 & -0.49434738 & -1.91992615 \\ -0.43167652 & 2.40298411 & -0.14274690 \\ -1.3462893 & -2.73288946 & -0.19682963 \\ -1.3462893 & -2.73288946 & -0.19682963 \end{bmatrix}, \\ \sigma(A - BK^{(best)}C) &= \left\{ \begin{bmatrix} 0.86726666, 0.9893915, \\ 0.99600356 \pm 0.01886288i, \\ 0.99670041 \pm 0.00199237i, \\ 0.97791267 \pm 0.01292326i, \end{bmatrix} \right\}, \\ \sigma^{(best)} &= 5.2247 \cdot 10^4, \|K^{(best)}\| = 8.1964, \end{aligned}$$

,

*<sup>K</sup>*ð Þ *best* <sup>¼</sup> *<sup>K</sup>*ð Þ *best*

*DOI: http://dx.doi.org/10.5772/intechopen.89319*

*<sup>σ</sup> <sup>A</sup>* � *BK*ð Þ *best <sup>C</sup>* � � <sup>¼</sup>

max <sup>¼</sup> <sup>3</sup>*:*<sup>1783</sup> � <sup>10</sup>4, *<sup>K</sup>*ð Þ *best* �

 >>>>>><

>>>>>>:

� �

The Ray-Shooting Method is a powerful tool, since it practically solves the problem of LQR via SOF, for real-life discrete-time LTI systems. The proposed hybrid algorithm RS + MC has good performance in terms of run-time, in terms of the quality of controllers (by reducing the starting point LQR functional value and by reducing the controller norm) and in terms of the success rate in finding a starting point feasible with respect to the needed *α*-stability. The RS + MC algorithm has a proof of convergence in probability to a global minimum (as is evidently seen from the experiments). This enlarges the practicality and scope of the Ray-Shooting Method in solving hard complexity control problems, and we expect to receive

*K*ð Þ *best* ¼

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

*K*ð Þ *best* ¼

*σ*ð Þ *best*

**6. Concluding remarks**

more results in this direction.

 *<sup>K</sup>*ð Þ *best* h i, where

*Algorithms for LQR via Static Output Feedback for Discrete-Time LTI Systems*

*:*16101573 0*:*52768736 �1*:*99112247

,

,

 >>>>>>=

>>>>>>;

�1*:*8453102 11*:*28218548 �3*:*45382160

*:*45368113 �0*:*97316671 �6*:*96467764

*:*57106266 �0*:*53606197 �0*:*45209883

�0*:*67588625 �0*:*18705486 �0*:*28941025

�0*:*69074912 �0*:*26655181 0*:*20004806

�0*:*83280356 3*:*20877857 �0*:*31860927

�3*:*09450298 �0*:*73194640 �0*:*07799223

*:*98195937 � 0*:*03551664*i*,

*:*99264852 � 0*:*01953917*i*,

*:*99354901 � 0*:*00357529*i*,

*:*99828371, 0*:*98490431

� ¼ 12*:*1029*:*

,

,

by the MC Algorithm 2

$$\begin{cases} & K^{(bar)} = \left[ K\_1^{(bar)} K\_2^{(bar)} \right], \text{where} \\ & K\_1^{(bar)} = \begin{bmatrix} 0.16219293 & 0.22685502 & -1.41842788 \\ 0.10558055 & 0.04299252 & -1.12924145 \\ -0.01638169 & -0.08184317 & -0.14133958 \\ -0.03859594 & 0.07947463 & 0.12534983 \end{bmatrix}, \\ & K\_2^{(bar)} = \begin{bmatrix} -0.23992446 & -0.17536269 & -0.44875450 \\ -0.14481130 & -0.18150136 & -0.27764220 \\ -0.04123859 & 0.04014410 & 0.05631322 \\ -0.06158264 & 0.05048672 & 0.16208075 \end{bmatrix}, \\ & \sigma(A - BK^{(bar)}C) = \begin{bmatrix} 0.53091472, 0.95364142, \\ 0.98741833, 0.99897830, \\ 0.99717581 \pm 0.00564301i, \\ 0.95033459 \pm 0.08956374i, \\ 0.95033459 \pm 0.08956374i, \end{cases}$$

and by RS + MC Algorithms 1 and 2:

*Algorithms for LQR via Static Output Feedback for Discrete-Time LTI Systems DOI: http://dx.doi.org/10.5772/intechopen.89319*

$$\begin{cases} & K^{(box)} = \left[K\_1^{(box)}K\_2^{(box)}\right], \text{where} \\ & K\_1^{(box)} = \begin{bmatrix} 0.16101573 & 0.52768736 & -1.99112247 \\ -1.8453102 & 11.28218548 & -3.4582160 \\ 0.45368113 & -0.97316671 & -6.96467764 \\ 0.97106266 & -0.53606197 & -0.45209833 \end{bmatrix}, \\ & \begin{aligned} K\_2^{(box)} = \begin{bmatrix} -0.67588625 & -0.18705486 & -0.28941025 \\ -0.69074912 & -0.26655181 & 0.20004806 \\ -0.83280356 & 3.20877857 & -0.31860927 \\ -3.09450298 & -0.73194640 & -0.07799223 \end{bmatrix}, \\ \end{cases} \\ \sigma(A - BK^{(box)}C) = \begin{cases} & 0.98195937 \pm 0.03551664i, \\ 0.99264852 \pm 0.01953917i, \\ 0.99354901 \pm 0.00357529i, \\ 0.99828371, 0.98490431 \end{cases} \end{cases}$$
 
$$\sigma\_{\max}^{(box)} = 3.1783 \cdot 10^4, \ \|K^{(box)}\| = 12.1029.$$

,
