On Parametrizations of State Feedbacks and Static Output Feedbacks and Their Applications

In this chapter, we provide an explicit free parametrization of all the stabilizing static state feedbacks for continuous-time Linear-Time-Invariant (LTI) systems, which are given in their state-space representation. The parametrization of the set of all the stabilizing static output feedbacks is next derived by imposing a linear constraint on the stabilizing static state feedbacks of a related system. The parametrizations are utilized for optimal control problems and for pole-placement and exact pole-assignment problems.


Introduction
The solution of the problem of stabilizing static output feedback (SOF) has a great practical importance, for several reasons: they are simple, cheap, reliable, and their implementation is simple and direct. Since in practical applications, full-state measurements are not always available, the application of stabilizing state feedback (SF) is not always possible. Obviously, in practical applications, the entries of the needed SOFs are bounded with bounds known in advance, but unfortunately, the problem of SOFs with interval constrained entries is NP-hard (see [1,2]). Exact pole assignment and simultaneous stabilization via SOF or stabilization via structured SOFs are also NP-hard problems (see [2,3] resp.). These problems become even harder when optimal SOFs are sought, when the optimality notions can be the sparsity (see [4]) of the controller (e.g., for reliability purposes of networked control systems (NCSs)), the cost or energy consumption of the controller (which are related to various norm-bounds on the controller), the H ∞ -norm, the H 2 -norm or the linear-quadratic regulator (LQR) functional of the closed loop. The practical meaning of the NP-hardness of the aforementioned problems is that the problems cannot be formulated as convex problems (e.g., through LMIs or SDPs) and cannot have any efficient algorithms (under the widespread belief that P 6 ¼ NP). Thus, one has to compromise the exactness (which might affect the feasibility of the solution) or the optimality of the solution. Therefore, one has to utilize the specific structure of the given problem, in order to describe effectively the set of all feasible solutions, by reducing the number of variables and constraints to the minimum, for the purpose of increasing the efficiency and accuracy of the available algorithms. This is the aim of the proposed method.
Several formulations and related algorithms were introduced in the literature for the constrained SOF and other control hard problems. The iterated linear matrix inequalities (ILMI), bilinear matrix inequalities (BMI), and semi-definite programming (SDP) approaches for the constrained SOF problem, for the simultaneous stabilizing SOF problem, and for the robust control via SOF (with related algorithms) were studied in: [5][6][7][8][9][10][11]. The problem of pole placement via SOF and the problem of robust pole placement via Static Feedback were studied in: [12,13]. In [14,15], the method of alternating projections was utilized to solve the problems of rank minimization and pole placement via SOFs, respectively. The probabilistic and randomized methods for the constrained SOF problem and robust stabilization via SOFs (among other hard problems) were discussed in [16][17][18][19]. In [20], the problem of minimal-gain SOF was solved efficiently by the randomized method. A nonsmooth analysis approach for H ∞ synthesis and for the SOF problem is given in [21,22], respectively. A MATLAB ® library for multiobjective robust control problems based on the non-smooth analysis approach was introduced in [23]. All these references (and many more references not brought here) show the significance of the constrained SOF problem to control applications.
Many problems can be reduced to the SOF constrained problem, including the reduction of the minimal-degree dynamic-feedback problem and robust or decentralized stability via static-feedback, reduced-order H ∞ filter problem, global minimization of LQR functional via SOF, and the design problem of optimal PID controllers (see [2,10,[24][25][26][27] respectively). It is worth mentioning [28], where the alternating direction method of multipliers was utilized to alternate between optimizing the sparsity of the state feedback matrix and optimizing the closed-loop H 2 -norm, where the sparsity measure was introduced as a penalty term, without any pre-assumed knowledge about the sparsity structure of the controller. The method of augmented Lagrangian for optimal structured static-feedbacks was considered in [29], where it is assumed that the structure is known in advance (otherwise, one should solve a combinatorial problem). The computation overhead of all the aforementioned methods can be reduced significantly, if good parametrization of all the SOFs of the given system could be found, where a parametrization can be called "good" if it takes into account the structure of the given specific system and if it well separates between free and dependent parameters, thus resulting in a minimal set of nonlinear nonconvex inequalities/equations needed to be solved.
In [30], a parametrization of all the SFs and SOFs of Linear-Time-Invariant (LTI) continuous-time systems is achieved by using a characterization of all the (marginally) stable matrices as dissipative Hamiltonian matrices, leading to a highly performance sequential semi-definite programming algorithm for the minimal-gain SOF problem. The proposed method there can be applied also to LTI discrete-time systems by adding semi-definite conditions for placing the closed-loop eigenvalues in the unit disk. A new parametrization for SOF control of linear parameter-varying (LPV) discrete-time systems, with guaranteed ℓ 2 -gain performance, is provided in [31]. The parametrization there is given in terms of an infinite set of LMIs that becomes finite, if some structure on the parameter-dependent matrices is assumed (e.g., an affine dependency). The H 2 -norm guaranteed-performance SOF control for hidden Markov jump linear systems (HMJLS) is studied in [32], where the SOFs are parameterized via convex optimization with LMI constraints, under the assumptions of full-rank sensor matrices and an efficient and accurate Markov chain state estimator. In [33], an iterative LMI algorithm is proposed for the SOF problem for LTI continuous-time negative-imaginary (NI) systems with given H ∞ norm-bound on the closed loop, based on decoupling the dependencies between the SOF and the Lyapunov certificate matrix.
When solving an optimization problem, it is important to have a convenient parametrization for the set of feasible solutions. Otherwise, one needs to use the probability method (i.e., the "generate and check" method), which is seriously doomed to the "curse of dimensionality" (see [16]). In [13], a closed form of all the stabilizing state feedbacks is proved (up to a set of measure 0), for the purpose of exact pole assignment, when the location errors are optimized by lowering the condition number of the similarity matrix, and the controller performance is optimized by minimizing its Frobenius norm. The parametrization in [13] is based on the assumptions that the input-to-state matrix B has a full rank and at least one real state feedback leading to diagonalizable closed-loop matrix exists, where a necessary condition for the existence of such feedback is that the multiplicity of any assigned eigenvalue is less than or equal to rank B ð Þ. In this context, it is worth mentioning [34] in which a parametrization of all the exact pole-assignment state feedbacks is given, under the assumption that the set of needed closed-loop poles should contain sufficient number of real eigenvalues (which make no problem if the problem of pole placement is of concern, where it is generally assumed that the region is symmetric with respect to the real axis and contains a real-axis segment with its neighborhood). The results of [34] and of the current chapter are based on a controllability recursive structure that was discovered in [35].
In this chapter, using the aforementioned controllability recursive structure, we introduce a parametrization of the set of all stabilizing SOFs for continuous-time LTI systems with no other assumptions on the given system (for discrete-time LTI systems, the parametrization is much more involved and will be treated in a future work). As opposed to the notable works [36][37][38], where for the parametrization one still needs to solve some LMIs in order to get the Lyapunov matrix, here we give an explicit recursive formula for the Lyapunov matrix and for the feedback in the case of SF, and a constrained form for the Lyapunov matrix and for the feedback in the case of SOF.
The rest of the chapter goes as follows: In Section 2, we set notions and give some basic useful lemmas, and in Section 3, we introduce the parametrization of the set of all stabilizing static-state feedbacks for LTI continuous-time systems. In Section 4, we introduce the constrained parametrization of the set of all stabilizing SOFs for LTI continuous-time systems. The effectiveness of the method is shown on a real-life system. Section 5 is based on [34] and is devoted to the problem of exact pole assignment by SF, for LTI continuous-time or discrete-time systems. The effectiveness of the method is shown on a real-life system. Finally, in Section 6, we conclude with some remarks and intentions for a future work.

Preliminaries
By  we denote the complex field and by  À the open left half-plane. For z ∈  we denote by R z ð Þ its real part, while by I z ð Þ we denote its imaginary part. For a square matrix Z, we denote by σ Z ð Þ the spectrum of Z. For a  pÂq matrix Z, we denote by Z T its transpose, and by z i,j or by Z i,j , its i, j ð Þ'th element or block element. A square matrix Z in the continuous-time context (in the discrete-time context) is said to be (asymptotically) stable, if any eigenvalue λ ∈ σ Z ð Þ satisfies R λ ð Þ < 0, i.e., Consider a continuous-time system in the form: where A ∈  nÂn , B ∈  nÂm , C ∈  rÂn , and x, u, y are the state, the input, and the measurement, respectively. Assuming that the state x is fully accessible and fully available for feedback, we define u ¼ ÀK 0 ð Þ x to be the state feedback (SF). When the state is not fully accessible or not fully available for feedback but the measurement y is available for feedback, we define u ¼ ÀKy to be the static output feedback (SOF). The problems that we consider here are the following: • (SF-PAR): How can one parameterize the set of all K 0 ð Þ ∈  mÂn such that the closed-loop A À BK 0 ð Þ is stable, and what is the best parametrization (in terms of minimal number of parameters and minimal set of constraints)?
• (SOF-PAR): How can one parameterize the set of all K ∈  mÂr such that the closed-loop A À BKC is stable, and what is the best parametrization?
The parameterizations will be used for achieving other goals and performance keys for the system, other than stability, which is the feasibility defining basic key.
A square matrix Z is said to be non-negative (denoted as . For a matrix Z ∈  pÂq , we denote by Z þ the Moore-Penrose pseudo-inverse (see [39,40] for definition and properties). By L Z , R Z we denote the orthogonal projections I q À Z þ Z and I p À ZZ þ , respectively, where I s denotes the identity matrix of size s Â s. Note that Z þ Z and ZZ þ (as well as L Z and R Z ) are symmetric and orthogonally diagonalizable with eigenvalues from 0, 1 f g. By diag and bdiag we denote diagonal and block-diagonal matrices, respectively.
A system triplet A, B, C ð Þis SOF stabilizable (or just stabilizable) if and only if there exist K and P > 0 such that for some given R > 0 (R ¼ I can always be chosen), where E ¼ A À BKC. For the "if" direction, note that (2) implies the negativity of the real part of any eigenvalue of E, implying that the closed-loop E is stable. For the "only-if" direction, under the assumption that E ¼ A À BKC is stable for some given K, one can show that P ≔ Ð ∞ 0 exp Et ð ÞR exp E T t À Á dt is well defined, satisfies P T ¼ P and P > 0, and is the unique solution for (2).
Note that the set of all SOFs is given by K ¼ B þ XC þ þ L B S þ TR C where S, T are any m Â r matrices and X is any n Â n matrix such that E ¼ A À BB þ XC þ C is stable. Thus, one can optimize K by utilizing the freeness in S, T without changing the closed-loop performance achieved by X. This characterization of the feasibility space shows its effectiveness in proving theorems, as will be seen along the chapter (see also [20,35]). We also conclude that A, B, C ð Þis stabilizable if and only if A, BB þ , C þ C ð Þis stabilizable. In the sequel, we make use of the following lemma (see [39]): When the condition is satisfied, the set of all solutions is given by where Z is arbitrary matrix. Moreover, we have: Similarly, the equation YA ¼ B has solutions if and only if BA þ A ¼ B (equivalently, BL A ¼ 0). When the condition is satisfied, the set of all solutions is given by where W is arbitrary matrix. Moreover, we have:

Parametrization of all the static state feedbacks
We start with the following lemma known as the projection lemma (see [41], Theorem 3. When (5) is satisfied then, X is a stabilizing SF if and only if X is a solution for Moreover, one specific solution for (6) is given by Similarly, A T , C þ C À Á is stabilizable if and only if there exists Q > 0 such that When (8) is satisfied then, Y T is a stabilizing SF (i.e., A T À C þ CY T or, equivalently, A À YC þ C is stable) if and only if Y is a solution for: One specific solution for (9) is given by Remark 3.1 The explicit formulas (7) and (10) are our little contribution to the projection lemma, Lemma 3.1. Unfortunately, we do not have such an explicit formulas for LTI discrete-time systems.
In order to describe the set of all solutions for (6) and (9), we need the following lemma that can be proved easily: Lemma 3.2 Let P > 0, Q > 0. Then, the set of all solutions for: is given by Similarly, the set of all solutions for: is given by The following theorem describes the set of all solutions for (6) and (9), using the controllers (7) and (10): Theorem 3.1 Let P > 0 satisfy (5) and let X 0 be given by (7). Then, X is a solution for (6) if and only if: where W satisfies W T ¼ ÀW, R B W ¼ 0 and L is arbitrary. Similarly, let Q > 0 satisfy (8) and let Y 0 be given by (10). Then, Y is a solution for (9) if and only if: where V satisfies V T ¼ ÀV, VL C ¼ 0 and M is arbitrary.

Proof:
Assume that X is a solution for (6). Since X 0 is also a solution for (6), it follows Conversely, let X be given by (13) and from which we conclude that X satisfies (6), since X 0 satisfies (6). The second claim is proved similarly. ■ In the following we describe the set P of all matrices P > 0 satisfying (5). Note that in Theorem 3.1 the existence of P > 0 satisfying (5) is guaranteed by the assumption that A, BB þ ð Þis stabilizable and as a result of Lemma 3.1. Let P ∈ P and let W arbitrary such that W T ¼ ÀW, R B W ¼ 0 Let X P ð Þ denote the set of all matrices X satisfying (15) for a fixed P ∈ P, and let K P ð Þ denote the set of all matrices K satisfying (15) for a fixed P ∈ P. Note that for a fixed P ∈ P, the set X P ð Þ is convex (actually affine) and ∪ P ∈ P X P ð Þ contains all the stabilizing X parameters of the stabilizable pair A, BB þ ð Þ . Finally, ∪ P ∈ P K P ð Þ contains all the stabilizing SF's K of the stabilizable pair A, B ð Þ. For a stabilizable pair A T , C T À Á , let Q be the set of all matrices Q > 0 satisfying (8), and let Let Y Q ð Þ denote the set of all matrices Y satisfying (16) for a fixed Q ∈ Q, and let K Q ð Þ denote the set of all matrices K satisfying (16) for a fixed Q ∈ Q. Then, ∪ Q ∈ Q K Q ð Þ contains all the stabilizing SFs K of the stabilizable pair A T , C T À Á . In the following we assume (without loss of generality, see Remark 4.2) that A, BB þ ð Þis controllable. Under this assumption, we recursively (go downwards and) define a sequence of sub-systems of the given system A, BB þ ð Þ . Since BB þ is symmetric matrix (with simple eigenvalues from the set 0, 1 f g), it is diagonalizable by an orthogonal matrix. Let U denote an orthogonal matrix such that [35] and see Lemma 5.1 in the following).
Recursively, assume that the pair A i ð Þ , B i ð Þ was defined and is controllable. Let the base case, in which also k b ¼ n b ). Now, we go upward and define the Lyapunov matrices and the related SFs of the sub-systems. For the base case i ¼ b, let P b ð Þ > 0 be arbitrary (note that it is a free parameter!). Let and note that R B b ð Þ ¼ 0 in the base case. Now, it can be checked that ð Þ > 0 satisfying: and assume that K iþ1 ð Þ , as is defined by (19). Similarly, let P i ð Þ denote the set of all P i ð Þ > 0 satisfying: and assume that K i ð Þ P i ð Þ À Á was parameterized through P i ð Þ > 0 ranging in the set P i ð Þ , as is defined by (20). Now, we need to characterize the matrices P i ð Þ > 0 belonging to the set P i ð Þ .
Multiplying (20) from the left by U i ð ÞT and from the right by U i ð Þ we get: (20) is therefore equivalent to: which is equivalent to: (22) implies that the system: is stable. Now, and we need to define c P i ð Þ 1,1 in order to complete c P i ð Þ to a strictly non-negative matrix. Since: 1,1 is arbitrary strictly non-negative matrix (a free parameter!).
Conversely, if P i ð Þ > 0 satisfies (20) then (23) is stable and thus: if and only if: since the last equation has unique strictly non-negative solution and since 1,1 > 0 is arbitrary (a free parameter!) and Þ is given by (26). We thus have a parametrization of all P i ð Þ > 0 satisfying (20). Specifically, P 0 ð Þ is the set of all P 0 ð Þ > 0 satisfying (5).
Then, it can be checked that We conclude the discussion above with the following: Þbe a controllable pair. Then, in the above notations, for (27). Similarly to the discussion above, relating to A T , C þ C À Á and defining sub- the related sub-system and specifically, Q 0 ð Þ is the set of all Q 0 ð Þ > 0 satisfying (8).
The parametrizations of all the stabilizing SF's of A, BB þ ð Þand A T , C þ C À Á are given in the following: Þbe a given controllable pair. Then, the set of all stabilizing SF's of A, BB þ ð Þis given by Similarly, let A T , C þ C À Á be a given controllable pair. Then, the set of all stabilizing SF's of where M is arbitrary, V satisfies V T ¼ ÀV, VL C ¼ 0, and Q > 0 satisfies

Parametrizations of all the static output feedbacks
In this section, we give two parametrizations for the set of all the stabilizing SOFs. We start with the following lemma, which was extensively used in [20]: are stabilizable and there exist matrices X, Y ∈  nÂn such that A À BB þ X and A À YC þ C are stable and BB þ X ¼ YC þ C. When the conditions hold, the set of all stabilizing SOFs related to the chosen matrices X, Y is given by Moreover, this condition can be simplified to (meaning that it does not include matrix inverses): We can state now the first parametrization for the set of all the stabilizing SOFs: are controllable. Then, the system has a stabilizing static output feedback if and only if there exist P, Q > 0 and W, V such that In this case, A À BKC is stable if and only if where S, T, L are arbitrary.
where F, H, M are arbitrary. We conclude this section with a second SOF parametrization: In this case, the set of all K's such that A À BKC is stable, is given by is controllable (see [35], Lemma 3.1, p. 536). Thus, we may assume without loss of generality that the given pair is controllable. If A, B, C ð Þis a system triplet such that A, BB þ ð Þand A T , C þ C À Á are stabilizable then, there exists an orthogonal matrix V such that b is partitioned accordingly (see [35], Theorem 4.1 and Remark 4.1, p. 539). Thus, the assumption in Corollary 4.2 that the given pairs are controllable does not make any loss of generality of the results. The effectiveness of the method is shown in the following example, but first, for the convenience of the reader, we summarize the whole method in Algorithm 1 (with its continuation in Algorithm 2). Let f K ð Þ denote a target function of the SOF K, to be minimized (e.g., K k k F , the LQR functional, the H ∞ -norm or the H 2 -norm of the closed loop, the pole-placement errors of the closed loop, or any other key performance that depends on K).
Regarding the LQR problem, let the LQR functional be defined by: where Q > 0 and R ≥ 0 are given. We need to find u t ð Þ that minimizes the functional value for any initial disturbance x 0 from the equilibrium point 0. Assuming that u t ð Þ is realized by a stabilizing SOF, let u t ð Þ ¼ ÀKy t ð Þ ¼ ÀKCx t ð Þ. Then, by substitution of the last into (29), we get: Now, since Q þ C T K T RKC > 0 and since E ≔ A À BKC is stable, the Lyapunov equation: has unique solution P LQR K ð Þ> 0 given by: By substitution of (31) into (30), we get:

Algorithm 1. An Algorithm For Optimal SOF's.
Require: An algorithm for optimizing f K ð Þ under LMI and linear constraints, an algorithm for computing the Moore-Penrose pseudo-inverse and an algorithm for orthogonal diagonalization.
19. let F b ð Þ be a symbol for arbitrary matrix Algorithm 2. An Algorithm For Optimal SOF's, Continued.
Now, if x 0 is known then we can minimize J x 0 , K ð Þby minimizing x T 0 P LQR K ð Þx 0 . Otherwise, and if we design for the worst-case, we need to minimize σ max P LQR K ð Þ À Á .
In the following examples, we have executed the algorithm on Processor: Intel (R) Core(TM) i5-2400 CPU @ 3.10GHz 3.10 GHz, RAM: 8.00 GB, Operating System: Windows 10, System Type: 64-bit Operating System, x64-based processor, Platform: MATLAB ® , Version: R2018b, Function: fmincon. Example 4.1 A system of Boeing B-747 aircraft (the "AC5" system in [42] and see also [43]) is given by the general model (given here with slight changes): where x is the state, w is the noise, u is the control input, z is the regulated output, and y is the measurement, where: , Note that A, B ð Þand A T , C T À Á here are controllable. Let u ¼ u r À Ky, where u r is a reference input. Then, u ¼ u r À KCx À KD 2,1 w and substitution the last into the system yields the closed-loop system: where the behavior of z is of our interest. Note that we actually have: For the stabilization via SOF with minimal Frobenius-norm, we need to minimize f K ð Þ ¼ K k k F . For the LQR problem we need to minimize f K ð Þ ¼ x T 0 P LQR K ð Þx 0 when x 0 is known and to minimize f K ð Þ ¼ σ max P LQR K ð Þ À Á when x 0 is unknown, where P LQR K ð Þ is given by (32). For the H ∞ and the H 2 problems, we need to These problems needed to be solved under the constraint that A À BKC stable, i.e., that Applying the algorithm we had: Now, we parameterize all the matrices Let w 1 be arbitrary and let   For a comparison, in this (small) example we had seven scalar indeterminate, four scalar equations and four scalar inequalities while by the BMI method indeterminate and eight scalar inequalities. This shows the potential of the method in reducing the number of variables and inequalities/equations, thus enabling to deal efficiently with larger problems. Moreover, the method removes the decoupling of P and K, in the sense that now K depends on P and the dependence of P on K has removed, thus making the problem more relaxed. Figures 1 and 2 show the impulse response and the step response of the closedloop system, in terms of the regulated output z ¼ y, where w ¼ 0 and u r is the delta Dirac function or the unit-step function, respectively. While the amplitudes seem to be reasonable, the settling time of order 10 5 seems unreasonable. This happens because lowering the SOF-norm results in pushing the closed-loop eigenvalues toward the imaginary axes, as can be seen from the dense oscillations. We therefore must set a barrier on the abscissa of the closed-loop eigenvalues as a constraint. Note however that as a starting point for other optimization keys where we need any stabilizing SOF that we can get, the above SOF might be sufficient. Step response of the closed loop with the minimal-norm SOF.
The entries of z ¼ y under w ¼ 0 and u r ¼ 0, when the closed-loop system is derived by the initial condition x 0 ¼ 1 1 1 1 ½ T are depicted in Figure 3. The results might not be satisfactory regarding the amplitudes or the settling time; however, as a starting point for other optimization keys where we need any stabilizing SOF that we can get, the above SOF might be sufficient.
For the problem of pole placement via SOF, assume that the target is to place the closed-loop eigenvalue as close as possible to À10 AE i, À 1 AE 0:1i. Then, starting from: with K k k F ¼ 1:256029021159584 Á 10 5 . The resulting closed-loop eigenvalues are:  Figures 4 and 5 depict the impulse response and the step response of the closed loop with the pole-placement SOF. The amplitudes look reasonable but the settling time might be unsatisfactory.
Regarding the H ∞ -norm of the closed loop, starting from: in CPU À Time ¼ 0:703125 sec ½ the fmincon function has converged to the optimal point The simulation results of the closed-loop system are given in Figure 6, where w is normally distributed random disturbance, where each entry is N 0, 10 6 À Á The results here are good.
Regarding the H 2 -norm of the closed loop, starting from: The simulation results of the closed-loop system is given in Figure 6, where w is normally distributed random disturbance, where each entry is N 0, 10 6 À Á distributed. The maximum absolute values of the entries of z ¼ y are The results here are excellent. We conclude that the best performance of the closed-loop system is achieved with the optimal H 2 -norm SOF; however, since the Frobenius-norm of the SOF controller is high, the cost of construction and of operation of the SOF controller might be high, and there are no "free meals." Note also that by minimizing the SOF Frobenius-norm, the eigenvalues of the closed loop tend to get closer to the imaginary axes (to the region of lower degree of stability), while by minimizing the H 2 -norm, the eigenvalues of the closed loop tend to escape from the imaginary axes (to the region of higher degree of stability). These are conflicting demands, and therefore, one should use some combination of the related key functions or to use some multiobjective optimization algorithm in order to get the best SOF in some or all of the needed key performance measures.
The following counterintuitive example shows that the SOF problem can be unsolvable (or hard to solve) even for small systems. In the example we show how nonexistence of SOF can be detected by the method: Response of the closed loop with the optimal H 2 -norm SOF to a 10 6 variance zero-mean normally distributed random disturbance.
Applying the algorithm we have The "while-loop" stops because B 1 ð Þ B 1 ð Þþ ¼ 1. Let P 1 ð Þ ¼ p 1 where p 1 > 0. Then, We therefore have: as the free parametrization of all the state feedbacks for which A 0 ð Þ À B 0 ð Þ K 0 ð Þ is stable-for any choice of p 1 , p 2 > 0. Now L C 0 ð Þ ¼ 1 2 1 À1 À1 1 ! and the equations K 0 ð Þ L C 0 ð Þ ¼ 0 are equivalent to the single equation: Assuming p 1 , p 2 > 0, the last equation implies that leading to a contradiction with p 2 being real positive number.

A parametrization for exact pole assignment via SFs
This section is based on the results reported in [34], where the proofs of the following lemma and theorem can be found. The aim of this section is to introduce a parametrization of all the SF's for the exact pole-assignment problem, when the set of eigenvalues can be given as free parameters (under some reasonable assumptions). This is done as part of the research of the problem of parametrization of all the SOFs for pole assignment. Note that the problem of exact pole assignment by SOFs is NP-hard (see [3]), meaning that an efficient algorithm for the problem probably does not exist, and therefore an effective description of the set of all solutions might not exist too. Also note that with SOFs, the feasible set Ω might exclude some open set from being a feasible set for the closed-loop spectrum (see [12]). These make the full aim very hard (if not impossible) to achieve. We therefore focus here on the problem of exact pole assignment via SFs. Let the control system be given by: where Þin the discrete-time context. We assume without loss of generality that A, B ð Þis controllable. The problem of exact pole assignment by SF is defined as follows: • (SF-EPA) Given a set Ω ⊆  À , Ω j j ¼ n (in the discrete-time context Ω ⊆  À ), symmetric with respect to the x-axis, find a state feedback F ∈  mÂn such that the closed-loop state-to-state matrix E ¼ A À BF has Ω as its complete set of eigenvalues, with their given multiplicities.
In [13], a closed form of all the exact pole-placement SFs is proved (up to a set of measure 0), based on Moore's method. In order to minimize the inaccuracy of the eigenvalues final placement and in order to minimize the Frobenius-norm of the feedback, a convex combination of the condition number of the similarity matrix and of the feedback norm was minimized. The parametrization proposed in [13] is based on the assumptions that there exists at least one real state feedback that leads to a diagonalizable state-to-state closed-loop matrix and that B is full rank. A necessary condition for such SF to exist is that the final multiplicity of any eigenvalue is less than or equal to rank B ð Þ. Here, we do not assume that B is full rank and we only assume that Ω contains sufficient number of real eigenvalues. A survey of most of the methods for robust pole assignment via SFs or by SOFs and the formulation of these methods as optimization problems with optimality necessary conditions is given in [44]. In [45] a performance comparison of most of the algorithmic methods for robust pole placement is given. A formulation of the general problem of robust exact pole assignment via SFs as an SDP problem and LMI-based linearization is introduced in [46], where the robustness is with respect to the condition number of the similarity matrix, which is made in order to hopefully minimize the inaccuracy of the eigenvalues final placement. Unfortunately, one probably cannot gain a parametric closed form of the SFs from such formulations. Moreover, the following proposed method is exact and therefore enables the use of the parametrization free parameters for other (and maybe more important) optimization purposes. Note that since the proposed method is exact, the closed-loop eigenvalues thyself can be inserted to the problem as parameters.
A completely different notion of robustness with respect to pole placement is considered in the following works: Robust pole placement in LMI regions and H ∞ design with pole placement in LMI regions are considered in [47,48], respectively. An algorithm based on alternating projections is introduced in [15], which aims to solve efficiently the problem of pole placement via SOFs. A randomized algorithm for pole placement via SOFs with minimal norm, in nonconvex or unconnected regions, is considered in [20]. , be the intended closed-loop eigenvalues, where the αs denote the paired complex-conjugate eigenvalues (with nonzero imaginary part), the βs denote the real eigenvalues, and 2c 1 , … , 2c m , r 1 , … , r ℓ denote their respective multiplicities, where 2 P m i¼1 c i þ P ℓ j¼1 r j ¼ n. In the following we would say that the size of the set (actually, the multiset) Ω is n (counting multiplicities) and we would write Ω j j ¼ n. Note that A, B ð Þis controllable if and only if A, BB þ ð Þis controllable, and also note that BB þ is a real symmetric matrix with simple eigenvalues in the set 0, 1 f g and thus is orthogonally diagonalizable matrix. Let U denote an orthogonal matrix such that: " # be partitioned accordingly. We cite here the following lemma taken from [34] connecting between the controllability of the given system and the controllability of its sub-system: Lemma 5.1 In the notations above, A, BB þ ð Þis controllable if and only if b Again, we use the recursive controllable structure.
5 be partitioned accordingly, with sizes ð Þ is controllable. The recursion stops when B i ð Þ B i ð Þþ ¼ I k i for some i ¼ b (which we call the base case). Note that in the worst case, the recursion stops when the rank k b ¼ 1. Theorem 5.1 In the above notations, assume that P ℓ j¼1 r j ≥ a, where a is the number of parity alternations in the sequence n 0 , n 1 , … , n b h i . Let Ω 0 ¼ Ω. Then, there exist a sequence Ω 0 ⊇ Ω 1 ⊇ ⋯ ⊇ Ω b of symmetric sets with size Ω i j j ¼ n i (counting multiplicities) and there exist a real state feedback Moreover, an explicit (recursive) formula for Þis given by: 1 Consider the problem of exact pole assignment via SF for the same system from Example 4.1. We therefore assume here that the full state is available for feedback control. Now, using the calculations from Example 4.1, we have: n 0 , n 1 h i¼ 4, 2 h i implying that the number of parity alternations is a ¼ 0. We therefore can assign by the method any symmetric set of eigenvalues to the closed loop. Let Ω 0 ¼ α, α, β, β È É be the eigenvalues to be assigned, and let Ω 1 ¼ β, β È É . Now, We have completed the pole-assignment SF parametrization. As an application, assume that α ¼ À10 þ i, β ¼ À1 þ 0:1i. Then, resulting with the closed-loop eigenvalues: In our calculations, we used MATLAB ® , which has a general precession of 5 À 7 significant digits in computing eigenvalues. Thus, we have almost no loss of digits by the method. For a comparison, see the last case in Example 4.1 and note that while exact pole assignment can be achieved by SF, in general, it cannot be achieved by SOF because the last is a NP-hard problem (see the introduction of this section). Even regional pole placement is hard to achieve by SOF because of the nonconvexity of the SOF feasibility domain.
Remark 5.1 Note that the indices k 0 , k 1 , … , k b h ias well as the indices n 0 , n 1 , … , n b h i , can be calculated from A, B ð Þin advance. After calculating these indices and the number a of parity alternations in the sequence n 0 , n 1 , … , n b h i , the designer can define Ω as to satisfy the assumption of Theorem 5.1, i.e., being symmetric with at least a real eigenvalues, in a parametric way, and get a parametrization of all the real SF leading to Ω as the set of closed-loop eigenvalues. Next, the designer can play with the specific values of these and of other free parameters, in order to gain the needed closed-loop performance requirements. This is in contrast with other methods where the parametrization is calculated ad-hoc for a specific set of eigenvalues, where any change in the set of eigenvalues necessitates new execution of the method.
Remark 5.2 Note that F iþ1 can be replaced by where H iþ1 is any real matrix, without changing the closed-loop eigenvalues (if one seek feedbacks with minimal Frobenius-norm, then he should take H iþ1 ¼ 0, otherwise he should leave H iþ1 as another free parameter). Thus, the freeness in H b , … , H 1 , H 0 h i and in G bþ1 , G b , … , G 1 h i makes the freeness in F 0 (e.g. in order to globally optimize the H ∞ -norm of the closed loop, the H 2 -norm or the LQR functional of the closed loop or any other performance key thereof). Note also that the sequences F b , … , F 1 , F 0 h iand G bþ1 , G b , … , G 1 h i can be calculated for Ω as in Theorem 5.1, where the eigenvalues in Ω are given as free parameters. In that case, it can be easily proved by induction that the state feedbacks F b , … , F 1 , F 0 h idepend polynomially on the eigenvalues parameters and on the other free parameters mentioned above (for complex eigenvalue α they depend polynomially on R α ð Þ, I α ð Þ). Finally, it is wort mentioning that the complementary theorem of Theorem 5.1 was also proved in [34], meaning that under the assumptions of Theorem 5.1, any SF that solves the problem has the form given in the theorem (up to a factor of the form given in Remark 5.2).

Concluding remarks
In this chapter, we have introduced an explicit free parametrization of all the stabilizing SF's of a controllable pair A, B ð Þ. This enables global optimization over the set of all the stabilizing SF's of such pair, because the parametrization is free. For a system triplet A, B, C ð Þ, we have shown how to get the parametrization of all the SOFs of the system by parameterizing all the SFs of A, B ð Þand all the SFs of A T , C T À Á and then imposing the compatibility constraint (28). We have also shown a parametrization of all the SOFs of the system triplet A, B, C ð Þby imposing the linear constraint K 0 ð Þ L C 0 ð Þ ¼ 0 on the SF K 0 ð Þ of the pair A, B ð Þ, where K 0 ð Þ was defined recursively and parameterizes the set of all SFs of A, B ð Þ. This leads to a set of polynomial equations (after multiplying by the l.c.m. of the denominators of the rational entries of K 0 ð Þ ) and inequalities that can be brought to polynomial equations. The resulting polynomial set of equations can be solved (parametrically) by using the Gröbner basis method (see e.g., [49][50][51][52]). By applying the Gröbner basis method, one would get an indication to the existence of solutions and in case that solutions do exist, it would tell what are the free parameters and how other parameters depend on the free parameters. It seems that the proposed method makes the Gröbner basis computations overhead (or other methods thereof) reduced significantly, thus enabling SOF global optimization for larger systems.
In view of Theorem 5.1 (with its complementary theorem proved in [34]), we have introduced a sound and complete parametrization of all the state feedbacks F which make the matrix b E ¼ U T A À BF ð Þ U k-complementary n À k ð Þ-invariant with respect to Ω 1 (see [34] for the definition and properties), where U is orthogonal such that where Ω is symmetric and has at least a (being the parity alternations in the sequence n 0 , … , n b h i ) real eigenvalues, where Ω 1 ⊆ Ω is symmetric with maximum real eigenvalues with size Ω 1 j j ¼ n À k. Assuming Ω as above, we have generalized the results of [13] in the sense that we do not assume the existence of real state feedback F that brings the closed-loop E ¼ A À BF to diagonalizable matrix, which actually means that the geometric and algebraic multiplicity coincides for any eigenvalue of the closed loop, and we do not assume the restriction on the multiplicity of each eigenvalue to be less than or equal to rank B ð Þ. However, in cases where the number of real eigenvalues in Ω is less than a, one should use the parametrizations given in [13], in [45] or in the references there. Note that in communication systems, where complex SFs and SOFs are sought, the introduced method is complete (with no restrictions) since the number of parity alternations and the restriction on Ω to contain as much real eigenvalues, were needed only to guarantee that F i for i ¼ b, … , 0 is real in each stage, which is needless in communication systems.
In view of Example 5.1, one can see that the accuracy of the final location of the closed-loop eigenvalues given by the proposed method depends only on the accuracy of computing B i ð Þþ and U i ð Þ for i ¼ 0, … , b and in the algorithm that we have to compute the closed-loop eigenvalues (see [53], for example) in order to validate their final location, and it has nothing to do with the specific values of the specific eigenvalues given in Ω. Therefore, by the proposed method, once that B i ð Þþ and U i ð Þ for i ¼ 0, … , b were computed as accurate as possible, the location of the closedloop eigenvalues will be accurate accordingly. Thus, by the proposed method the designer can save time since he can do it parametrically only once, and afterward he only needs to play with the specific values of the eigenvalues until he gets a satisfactory closed-loop performance, where he can be sure that the accuracy of the final placement will be the same for all of his trials independently on the specific values of the chosen eigenvalues. Also, the given parametrization of F 0 is polynomially dependent on the free parameters and thus is very convenient for applying automatic differentiation and optimization methods.
To conclude, we have introduced parametrizations of SFs and SOFs that are based on the recursive controllable structure that was discovered in [35]. The results has powerful implications for real-life systems, and we expect for more results in this direction. Unfortunately, for uncertain systems, the method cannot work directly because of the dependencies of A, B, C ð Þin uncertain parameters, for which we cannot compute U i ð Þ for i ¼ 0, … , b. However, if a nominal systemÃ,B,C À Á is known accurately then, the method can be applied to that system and the free parameters of the parametrization can be used to "catch" the uncertainty of the whole system, together with the closed-loop performance requirements. The research of this method will be left for a future work.