**1. Introduction**

The application of static-output-feedbacks (SOFs) for linear-quadratic regulators (LQR) is very attractive, since they are cheap and reliable and their implementation is simple and direct, because their components has direct physical interpretation in terms of sensors amplification rates and actuator activation power. Moreover, the long-term memory of dynamic feedbacks is useless for systems subject to random disturbances, to fast dynamic loadings or to random bursts and impulses, and the application of state feedbacks is not always possible due to unavailability of full-state measurements (see, e.g., [1]). Also, the use of SOF avoids the need to reconstruct the state by Kalman filter or by any other state reconstructor.

On the other hand, in practical applications, the entries of the needed SOFs are bounded, and since the problem of SOFs with interval constrained entries is NPhard (see [2, 3]), one cannot expect the existence of a deterministic efficient (i.e., polynomial-time) algorithm to solve the problem. Randomized algorithms are thus natural solutions to the problem. The probabilistic and randomized methods for the constrained SOF problem and robust stabilization via SOFs (among other hard problems) are discussed in [4–7]. For a survey of the SOF problem see [8], and for a recent survey of the robust SOF problem see [9].

The Ray-Shooting Method was recently introduced in [10], where it was used to derive the Ray-Shooting (RS) randomized algorithm for the minimal-gain SOF problem, with regional pole assignment, where the region can be non-convex and unconnected. The Ray-Shooting Method was successfully applied recently also to the following hard complexity control problems for continuous-time systems:

**2. Preliminaries**

defined by

Let a discrete-time system be given by

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

where *A* ∈ *<sup>p</sup>*�*p*, *B*∈ *<sup>p</sup>*�*<sup>q</sup>*

degree of stability. Let *<sup>K</sup>* <sup>∈</sup>S*<sup>q</sup>*�*<sup>r</sup>*

has a unique solution *P* >0, given by

*J x*ð Þ¼ 0,*<sup>K</sup>* <sup>X</sup><sup>∞</sup>

*P K*ð Þ¼ *mat Ip* <sup>⊗</sup> *Ip* � *Ac*ℓð Þ *<sup>K</sup> <sup>T</sup>* <sup>⊗</sup> *Ac*ℓð Þ *<sup>K</sup> <sup>T</sup>* � ��<sup>1</sup>

*k*¼0 *xT*

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *k*¼0 *xT*

<sup>¼</sup> *<sup>x</sup><sup>T</sup>*

Substitution of (4) into (3) and using *xk* <sup>¼</sup> *Ac*ℓð Þ *<sup>K</sup> <sup>k</sup>*

�*KCxk* into (2) yields:

stable leads to

**57**

�

*xk*þ<sup>1</sup> ¼ *Axk* þ *Buk*, *k* ¼ 0, 1, …

*xT*

where *Q* >0 and *R* >0. Let *uk* ¼ �*Kyk* be the SOF, and let *Ac*ℓð Þ *K* ≔ *A* � *BKC* denote the closed-loop matrix. Let denote the open unit disk, let 0 <*α* <1, and let *<sup>α</sup>* denote the set of all *z*∈ with j j *z* < 1 � *α* (where j j *z* is the absolute value of *z*). For a square matrix *Z*, we denote by *σ*ð Þ *Z* its spectrum. For any rectangular matrix *<sup>Z</sup>*, we denote by *<sup>Z</sup>*<sup>þ</sup> its Moore-Penrose pseudo-inverse. By k k*<sup>Z</sup> <sup>F</sup>* <sup>¼</sup> *trace ZTZ* � �<sup>1</sup>

spectral norm of *Z*. By *LZ* and *RZ*, we denote the (left and right) orthogonal projections *I* � *Z*þ*Z* and *I* � *ZZ*<sup>þ</sup> on the spaces *Ker Z*ð Þ and *Ker Z*<sup>þ</sup> ð Þ, respectively. For a topological space X and a subset U ⊂ X, we denote by *int* ð Þ U the interior of U, i.e., the largest open set included in U. By U we denote the closure of U, i.e., the smallest closed set containing <sup>U</sup>, and by *<sup>∂</sup>*U ¼ U � *int* ð Þ <sup>U</sup> we denote the boundary of <sup>U</sup>. Let <sup>S</sup>*<sup>q</sup>*�*<sup>r</sup>* denote the set of all matrices *<sup>K</sup>* <sup>∈</sup> *<sup>q</sup>*�*<sup>r</sup>* such that *<sup>σ</sup>*ð Þ *Ac*<sup>ℓ</sup> <sup>⊂</sup> (i.e., stable in the discrete-time sense), and let <sup>S</sup>*<sup>q</sup>*�*<sup>r</sup>* denote the set of all matrices *<sup>K</sup>* <sup>∈</sup> *<sup>q</sup>*�*<sup>r</sup>* such that *σ*ð Þ *Ac*<sup>ℓ</sup> ⊂ *α*. If the last is nonempty, we say that *Ac*<sup>ℓ</sup> is *α*-stable and we call *α* the

*<sup>k</sup> Qxk* <sup>þ</sup> *uT*

, *C* ∈ *<sup>r</sup>*�*p*, and *x*<sup>0</sup> ∈ *<sup>p</sup>*. Let the LQR cost functional be

� �, (2)

*<sup>k</sup> Ruk*

*<sup>α</sup>* be given. Substitution of the SOF *uk* ¼ �*Kyk* ¼

*<sup>P</sup>* � *Ac*ℓð Þ *<sup>K</sup> TPAc*ℓð Þ¼ *<sup>K</sup> <sup>Q</sup>* <sup>þ</sup> *CTKTRKC* (4)

� �*Ac*ℓð Þ *<sup>K</sup> <sup>k</sup>*

� *vec Q* <sup>þ</sup> *<sup>C</sup>TKTRKC* � � � �*:* (5)

*<sup>k</sup> <sup>Q</sup>* <sup>þ</sup> *<sup>C</sup>TKTRKC* � �*xk:* (3)

*x*<sup>0</sup> with the fact that *Ac*ℓð Þ *K* is

*x*0

(1)

<sup>2</sup> we

<sup>2</sup> we denote the

*yk* ¼ *Cxk*

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

*J x*ð Þ 0, *<sup>u</sup>* <sup>≔</sup> <sup>X</sup><sup>∞</sup>

denote the Frobenius norm of *<sup>Z</sup>*, and by k k*<sup>Z</sup>* <sup>¼</sup> max *<sup>σ</sup> <sup>Z</sup>TZ* � � � � � � <sup>1</sup>

*J x*ð Þ¼ 0, *<sup>K</sup>* <sup>X</sup><sup>∞</sup>

*k*¼0 *xT*

Since *<sup>Q</sup>* <sup>þ</sup> *CTKTRKC* <sup>&</sup>gt;0 and *Ac*ℓð Þ *<sup>K</sup>* is stable, it follows that the Stein equation

*<sup>k</sup> <sup>P</sup>* � *Ac*ℓð Þ *<sup>K</sup> TPAc*ℓð Þ *<sup>K</sup>* � �*xk*

<sup>0</sup> *P K*ð Þ*x*<sup>0</sup> <sup>¼</sup> *P K*ð Þ<sup>1</sup>

� � �

<sup>0</sup>*Ac*ℓð Þ *<sup>K</sup> Tk <sup>P</sup>* � *Ac*ℓð Þ *<sup>K</sup> TPAc*ℓð Þ *<sup>K</sup>*

<sup>2</sup>*x*<sup>0</sup>

� � � 2 *:*

*k*¼0


The contribution of the research presented in the current chapter is as follows:


The reminder of the chapter is organized as follows:

In Section 2 we formulate the problem and give some useful lemmas (without a proof). In Section 3, we introduce the randomized algorithm for the problem of LQR via SOF for discrete-time LTI systems. Section 4 is devoted to the deterministic algorithm for the problem. In Section 5, we give the results of the algorithms for some real-life systems. Finally, in Section 6 we conclude with some remarks.

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