New Techniques for Optimizing the Norm of Robust Controllers of Polytopic Uncertain Linear Systems

The history of linear matrix inequalities (LMIs) in the analysis of dynamical systems dates from over 100 years. The story begins around 1890 when Lyapunov published his work introducing what is now called the Lyapunov’s theory (Boyd et al., 1994). The researches and publications involving the Lyapunov’s theory have grown up a lot in recent decades (Chen, 1999), opening a very wide range for various approaches such as robust stability analysis of linear systems (Montagner et al., 2009), LMI optimization approach (Wang et al., 2008), H2 (Apkarian et al., 2001; Assuncao et al., 2007a; Ma & Chen, 2006) or H∞ (Assuncao et al., 2007b; Chilali & Gahinet, 1996; Lee et al., 2004) robust control, design of controllers for systems with state feedback (Montagner et al., 2005), and design of controllers for systems with state-derivative feedback (Cardim et al., 2009). The design of robust controllers can also be applied to nonlinear systems.


Introduction
The history of linear matrix inequalities (LMIs) in the analysis of dynamical systems dates from over 100 years. The story begins around 1890 when Lyapunov published his work introducing what is now called the Lyapunov's theory (Boyd et al., 1994). The researches and publications involving the Lyapunov's theory have grown up a lot in recent decades (Chen, 1999), opening a very wide range for various approaches such as robust stability analysis of linear systems (Montagner et al., 2009), LMI optimization approach (Wang et al., 2008), H 2 (Apkarian et al., 2001;Assunção et al., 2007a;Ma & Chen, 2006) or H ∞ (Assunção et al., 2007b;Chilali & Gahinet, 1996;Lee et al., 2004) robust control, design of controllers for systems with state feedback (Montagner et al., 2005), and design of controllers for systems with state-derivative feedback . The design of robust controllers can also be applied to nonlinear systems.
In addition to the various current controllers design techniques, the design of robust controllers (or controller design by quadratic stability) using LMI stands out for solving problems that previously had no known solution. These designs use specialized computer packages (Gahinet et al., 1995), which made the LMIs important tools in control theory.
extended stability point, allowing extra matrices through a relaxation in the case of extended stability, making feasible a relaxation in the stability analysis (here called projective stability) through extra matrices. The synthesis of the controller K is depending now on an auxiliary matrix V, not necessarily symmetrical, and in this situation it becomes completely decoupled from Lyapunov's matrix P, leaving it free.
Two critical points in the design of robust controllers are explored here. One of them is the magnitude of the designed controllers that are often high, affect their practical implementation and therefore require a minimization of the gains of the controller to facilitate its implementation (optimization of the norm of K).The other one is the fact that the system settling time can be larger than the required specifications of the project, thus demanding restrictions on LMIs to limit the decay rate, formulated with the inclusion of the parameter γ in LMIs.
The main focus of this work is to propose new methods for optimizing the controller's norm, through a different approach from that found in (Chilali & Gahinet, 1996), and compare it with the optimization method presented in (Assunção et al., 2007c) considering the different criteria of stability, aiming at the advantages and disadvantages of each method, as well as the inclusion of a decay rate (Boyd et al., 1994) in LMIs formulation.
In (Siljak & Stipanovic, 2000) an optimization of the controllers's norm was proposed for decentralized control, but without the decay rate, so no comparisons were made with this work due to the necessity to insert this parameter to improve the performance of the system response.
The LMIs of optimization that will be used for new design techniques, had to be reformulated because the matrix controller synthesis does not depend more on a symmetric matrix, a necessary condition for the formulation of the existing LMI optimization. Comparisons will be made through a practical implementation in the Quanser's 3-DOF helicopter (Quanser, 2002) and a general analysis involving 1000 randomly generated polytopic uncertain systems.

Quadratic stability of continuous time linear systems
Consider (1) an autonomous linear dynamic system without state feedback. Lyapunov proved that the systemẋ (t) = Ax(t) with x(t) ∈ R n e A ∈ R n×n a known matrix, is asymptotically stable (i.e., all trajectories converge to zero) if and only if there exists a matrix P = P ∈ R nxn such that the LMIs (2) and (3) are met (Boyd et al., 1994).
Consider in equation (2) that A is not precisely known, but belongs to a politopic bounded uncertainty domain A. In this case, the matrix A within the domain of uncertainty can be written as convex combination of vertexes A j , j = 1, ..., N, of the convex bounded uncertainty domain (Boyd et al., 1994), i.e. A(α) ∈ A and A shown in (4).
A sufficient condition for stability of the convex bounded uncertainty domain A (now on called polytope) is given by the existence of a Lyapunov's matrix P = P ∈ R n×n such that the LMIs (5) and (6) A(α) P + PA(α) < 0 ( 5 ) are checked for every A(α) ∈ A (Boyd et al., 1994). This stability condition is known as quadratic stability and can be easily verified in practice thanks to the convexity of Lyapunov's inequality that turns the conditions (5) and (6) equivalent to checking the existence of P = P ∈ n×n such that conditions (7) and (8) are met with j = 1, ..., N.
It can be observed that (5) can be obtained multiplying by α j ≥ 0 and adding in j of j = 1 to j = N.
Due to being a sufficient condition for stability of the polytope A, conservative results are generated, nevertheless this quadratic stability has been widely used for robust controllers's synthesis.

Decay rate restriction for closed-loop systems
Consider a linear time invariant controllable system described in (9) .
with A ∈ R n×n , B ∈ R n×m the matrix of system input, x(t) ∈ R n the state vector and u(t) ∈ R m the input vector. Assuming that all state are available for feedback, the control law for the same feedback is given by (10) being K ∈ R m×n a constant elements matrix. Often the norm of the controller K can be high, leading to saturation of amplifiers and making the implementation in analogic systems difficult. Thus it is necessary to reduce the norm of the controllers elements to facilitate its implementation.
From the quadratic Lyapunov's function (12), to establish a lower limit on the decay rate of (9), with (13) .
Adding the restriction on the decay rate (13) in the equation (14) and making the appropriate simplifications, (15) and (16) are met.
As the inequality (15) became a bilinear matrix inequality (BMI) it is necessary to perform manipulations to fit them back into the condition of LMIs. Multiplying the inequalities (17) and (18) on the left and on the right by P −1 , making X = P −1 and G = KX results: If the LMIs (17) and (18) are feasible, a controller that stabilizes the closed-loop system can be given by K = GX −1 .
This system can be described as convex combination of the polytope's vertexes shown in (20). .
with A and B belonging to the uncertainty polytope (21) being r the number of vertexes (Boyd et al., 1994).
Knowing the existing theory for uncertain systems, Theorem 3.1 theorem can be enunciated (Boyd et al., 1994): Theorem 3.1. A sufficient condition which guarantees the stability of the uncertain system (20) subject to decay rate γ is the existence of matrices X = X ∈ R n×n and G ∈ R m×n , such that (22) and (23) are met.
When the LMIs (22) and (23) are feasible, a state feedback matrix which stabilizes the system can be given by (24).
Proof. The proof can be found at (Boyd et al., 1994).
Thus, it can be feedback into the uncertain system shown in (19) being (22) and (23) sufficient conditions for the polytope asymptotic stability, now for a closed-loop system subject to decay rate.

Optimization of the K matrix norm of the closed-loop system
In many situations the norm of the state feedback matrix is high, precluding its practical implementation. Thus Theorem 4.1 was proposed in order to limit the norm of K (Assunção et al., 2007c;Faria et al., 2010).
Theorem 4.1. Given an fixed constant μ 0 > 0, that enables to find feasible results, it can be obtained a constraint for the K ∈ R m×n matrix norm from the state feedback, with K = GX −1 , X = X > 0 ∈ R n×n and G ∈ R m×n finding the minimum value β, β > 0 such that KK < β μ 0 2 I m . The optimum value for β can be found solving the optimization problem with the LMIs (25), (26) and (27).
where I m and I n are the identity matrices of m and n order respectively.
Proof. The proof can be found at (Assunção et al., 2007c).

New optimization of the K matrix norm of the closed-loop system
It can be verified that the LMIs given in Theorem 4.1 can produce conservative results, so in order to find better results, new methodologies are proposed.
Using the theory presented in (Assunção et al., 2007c) for the optimization of the norm of robust controllers subject to failures, it is proposed an alternative approach for the same problem grounded in Lemma (5.1).
The approach of the optimum norm used was modified to fit to the new structures of LMIs that will be given in sequence. At first, this new approach has produced better results comparing to the existing ones for the optimization stated in Theorem 4.1 using the set of LMIs (22) and (23). Note that in the first statement of the lemma, (28) is met for all x ∈ R m .
Knowing that (29) is true and λ max (L L) the maximum eigenvalue of L L, which is real (every symmetric matrix has only real eigenvalues). Besides, when x is equal to the eigenvector of L L associated to the eigenvalue λ max (L L), and x (L L)x = λ max (L L)x x. Thus, from (28) and (29), β ≥ λ max (L L).
Similarly, for every z ∈ R n , the second assertion of the lemma results in (30).
Consequently, every non-zero eigenvalue of L L is also an eigenvalue of LL . Therefore, λ max (L L) = λ max (LL ), and from (29) and (30) the lemma is proved .
Knowing that P = X −1 is the matrix used to define Lyapunov's quadratic function, Theorem 5.1 is proposed.
Theorem 5.1. Given a constant μ 0 > 0, a constraint for the state feedback K ∈ R m×n matrix norm is obtained, with K = GX −1 , X = X > 0, X ∈ R n×n and G ∈ R m×n by finding the minimum of β, β > 0 such that K K < β μ 0 I n . You can get the minimum β solving the optimization problem with the LMIs (32), (33) and (34).
where I m and I n are the identity matrices of m and n order respectively.
Proof. Applying the Schur complement for the first inequality of (32) results in (35).
on which K is the optimal controller associated with (22).
It follows that minimizing the norm of a matrix is equivalent to the minimization of a β > 0 variable such that K K < β μ 0 I n , with μ 0 > 0. Note that the position of the transposed matrix was replaced in this condition, comparing to that used in Theorem 4.1.
A comparison will be shown between the optimization methods, using the robust LMIs with decay rate (22) and (23) in the results section. Since the new method may suit the relaxed LMIs listed below, it was used in the comparative analysis for the control design for extended stability and projective stability.
Finsler's lemma shown in Lemma (5.2) can be used to express stability conditions referring to matrix inequalities, with advantages over existing Lyapunov's theory (Boyd et al., 1994), because it introduces new variables and generate new degrees of freedom in the analysis of uncertain systems with the possibility of nonlinearities elimination.
Lemma 5.2 (Finsler). Consider w ∈ R n x , L ∈ R n x ×n x and B ∈ R m x ×n x with rank(B) < n x e B ⊥ a basis for the null space of B (i.e., BB ⊥ = 0). Then the following conditions are equivalent: Proof. Finsler's lemma proof can be found at (Oliveira & Skelton, 2001;Skelton et al., 1997).

Stability of systems using Finsler's lemma restricted by the decay rate
Consider the closed-loop system (9). Defining w = x and L = 2γP P P 0 . Note that Bw = 0 corresponds to (9) and w Lw < 0 corresponds to stability constraint with decay rate given by (12) and (13). In this case the dimensions of the lemma's variables (5.2) are: n x = 2n and m x = n. Considering that P is the matrix used to define the quadratic Lyapunov's function (12), the properties 1 and 2 of Finsler's lemma can be written as: which results in the equations of stability, according to Lyapunov, including decay rate: Thus, it is possible to characterize stability through Lyapunov's quadratic function (V(x(t)) = x(t) Px(t)), generating new degrees of freedom for the synthesis of controllers.
From Finsler's lemma proof follows that if the properties 1 and 2 are true, then properties 3 and 4 will also be true. Thus, the fourth propriety can be written as (39).
Choosing conveniently the matrix of variables X = Z aZ , with Z ∈ R n×n invertible and not necessarily symmetric and a > 0 a fixed relaxation constant of the LMI (Pipeleers et al., 2009).
These LMIs meet the restrictions for the asymptotic stability (Feron et al., 1996) of the system described in (9) with state feedback given by (10). It can be checked that the first principal minor of the LMI (40) has the structure of the result found in the theorem of quadratic stability with decay rate . Nevertheless, there is also, as stated in the Finsler's lemma, a greater degree of freedom because the matrix of variables Y, responsible for the synthesis of the controller, doesn't need to be symmetric and the Lyapunov's matrix now turned into Q, which remains restricted to positive definite, is partially detached from the controller synthesis, since that Q = Y PY.
The stability of the LMIs derived from Finsler's lemma stability is commonly called extended stability and it will be designated this way now on.

Robust stability of systems using Finsler's lemma restricted by the decay rate
As discussed for the condition of quadratic stability, the stability analysis can be performed for a robust stability condition considering the continuous time linear system as a convex combination of r vertexes of the polytope described in (20). The advantage of using the Finsler's lemma for robust stability analysis is the freedom of Lyapunov's function, now , it can be defined a Lyapunov's function Q j for each vertex j. As Q(α) depends on α, the Lyapunov matrix use fits to time-invariant polytopic uncertainties, being permitted rate of variation sufficiently small. To verify this, Theorem 5.2 is proposed. (20) is the existence of matricesY ∈ R n×n , Q ∈ R n×n , Q j = Q j > 0 e G ∈ R m×n , decay rate greater than γ and a fixed constant a > 0 such that the LMIs (42) and (43) are met.

Theorem 5.2. A sufficient condition which guarantees the stability of the uncertain system
with j = 1, ..., r. When the LMIs (42) and (43) are feasible, a state feedback matrix which stabilizes the system can be given by (44).
Proof. Multiplying (42) and (43) by α j ≥ 0, and adding in j, for j = 1 to j = N, LMIs (45) and (46) Thus, the uncertain system shown can be fed back in (19) with (45) and (46) sufficient conditions for asymptotic stability of the polytope.
Observation 1. In the LMIs (42) and (43), the constant "a" has to be fixed for all vertexes and to satisfy the LMIs and it can be found through a one-dimensional search.

Optimization of the K matrix norm using Finsler's lemma
The motivation for the study of an alternative optimization of the K matrix norm of state feedback control was due to less conservative results obtained with Finsler's lemma. This way expecting to find, for some situations, controllers with lower gains, thus being easier to implement than those designed using the existing quadratic stability theory (Faria et al., 2010), avoiding the signal control saturations.
Some difficulty in applying the existing theorem (Faria et al., 2010) was found to the new structure of LMIs, as the controller synthesis matrix Y is not symmetric, a condition that was necessary for the development of Theorem (4.1) when the controller synthesis matrix was X = P −1 . Thus, Theorem 5.3 is proposed.
Theorem 5.3. A constraint for the K ∈ R m×n matrix norm of state feedback can be obtained, with K = GY −1 and Q j = Y P j Y, being Y ∈ R n×n , G ∈ R m×n and P ∈ R n×n , P j = P j > 0 finding the minimum β, β > 0, such that K K < βP j , j = 1...N. You can get the optimal value of β solving the optimization problem with the LMIs (47) and (48).
where I m denotes the identity matrix of m order.
Replacing G = KY and Q j = Y P j Y in (50), (51) is met.
on which K is the optimal controller associated with (42) and (43).
Thus it was possible the adequacy of the proposed optimization method with the minimization of a scalar β, using the inequality of minimization K K < βP j with P j the Lyapunov's matrix, to the new relaxed parameters.

Stability of systems using reciprocal projection lemma restricted by the decay rate
Another tool that can be used for stability analysis using LMIs is the reciprocal projection lemma (Apkarian et al., 2001) set out in Lemma 5.3.
Lemma 5.3 (reciprocal projection lemma). Consider Y = Y > 0 a given matrix. The following statements are equivalent Proof. Reciprocal projection lemma proof can be found at (Apkarian et al., 2001).
Consider the Lyapunov's inequality subject to a decay rate given by (15) and (16), which can be rewritten as (52) and (53).
where X P −1 e P is the Lyapunov's matrix. The original Lyapunov's inequality (15) can be recovered by multiplying the inequality (52) on the left and on the right by P.
Assuming ψ 0 e S = (A − BK)X + γX, it will be verified that the first claim of the reciprocal projection lemma will be exactly Lyapunov's inequality subject to the decay rate described in (52): From the reciprocal projection lemma, if the first statement is true, then the second one will also be true as (54) shows.
Multiplying (54) on the left and on the right by I 0 0 X −1 with P X −1 results in (55).
Multiplying (55) on the left and on the right by Applying the Schur complement in V YV, (57) is found. ⎡ performing the linearizing variable change Y P −1 results in (58).
In literature it can be found a formulation close to the insertion of the decay rate but with different positioning of the parameter of decay rate (Shen et al., 2006). It is easy to verify that some conservatism was introduced with the choice of Y P −1 , but the state feedback matrix is unrelated to the Lyapunov's matrix P, which results in relaxation of Lyapunov's LMI. Using the dual form (A − BK) → (A − BK) (Apkarian et al., 2001) results in inequality (59).
Performing the change of variable Z KV and inserting the constraint P > 0, the LMIs (60) and (60) that guarantee system stability can be found. ⎡ The inequalities (60) and (61) are LMIs, and being feasible, it is deduced a state feedback matrix that can stabilize the system (9) -(10) given by (62).
The result of relaxation of LMIs is interesting in the design of robust controllers, proposed below.

Robust stability of systems using reciprocal projection lemma restricted by the decay rate
A stability analysis for a robust stability condition can be performed considering the continuous time linear system an convex combination of r vertexes of the polytope described in (20). As in the extended stability case, the advantage of using the reciprocal projection lemma for robust stability analysis is the Lyapunov's function degree of freedom, now defined as P(α) = r ∑ j=1 α j P j , r ∑ j=1 α j = 1, α j ≥ 0 e j = 1...r, i.e., it is defined a Lyapunov's function P j for each vertex j. As described before Theorem 5.2, the use of P(α) fits to time-invariant polytopic uncertainties, being permitted rate of variation sufficiently small. To verify this, Theorem 5.4 is proposed.
Theorem 5.4. A sufficient condition which guarantees the stability of the uncertain system (20) is the existence of matrices V ∈ R n×n , P j = P j ∈ R n×n and Z ∈ R m×n , such that LMIs (63) and (64) with j = 1, ..., r.
When the LMIs (63) and (64) are feasible, a state feedback matrix which stabilizes the system can be given by (65).
It appears that K is unique and there are r Lyapunov's matrices P j , generating a relaxation in the LMIs. The same trend was observed in the formulation via Finsler's lemma in which variables were the Lyapunov's matrices Q j , but in (65) and (66) there is a greater degree of freedom with the inclusion of V in the design of the control matrix K, V being totally disconnected from P j , j = 1, ..., n.

Optimization of the K matrix norm using reciprocal projection lemma
A study was carried out to fit the LMIs to the new relaxed parameters once the state feedback matrix K is completely detached from the Lyapunov's matrix P(α). Therefore, relevant changes took place in the optimization proposed in this study to suit the reciprocal projection lemma. This optimization has provided interesting results in practice.
Due to the lack of relations to assemble LMI able to optimize the module of K it was proposed a minimization procedure similar to the optimization procedure for redesign presented in (Chang et al., 2002) inserting an extra restriction to the LMIs (63) and (64).
Thus Theorem 5.5 was proposed.
Theorem 5.5. A constraint for the K ∈ R m×n matrix norm of state feedback is obtained, with K = ZV −1 , V ∈ R n×n and Z ∈ R m×n finding the minimum β, β > 0, such that K K < βM, being M = V −1 V −1 and therefore M = M > 0. You can get the optimal value of β solving the optimization problem with the LMIs (67) and (68) which I m and I n denote the identity matrices of m and n order respectively.
Multiplying on the left and on the right (71) for V −1 e V −1 respectively and naming where K is the optimal controller associated with (63) and (64).
Due to M being defined as M = V −1 V −1 and so M = M > 0, it is possible to find a relationship that optimizes the matrix K minimizing a scalar β, with the relation of minimizing K K < βM.

Practical application in the 3-DOF helicopter
Consider the schematic model in Figure (2) of the 3-DOF helicopter (Quanser, 2002) shown in Figure (1). Two DC motors are mounted at the two ends of a rectangular frame and drive two propellers. The motors axis are parallel and the thrust vector is normal to the frame. The helicopter frame is suspended from the instrumented joint mounted at the end of a long arm and is free to pitch about its center (Quanser, 2002).
The arm is gimbaled on a 2-DOF instrumented joint and is free to pitch and yaw. The other end of the arm carries a counterweight such that the effective mass of the helicopter is light enough for it to be lifted using the thrust from the motors. A positive voltage applied to the front motor causes a positive pitch while a positive voltage applied to the back motor causes a negative pitch (angle pitch (ρ)). A positive voltage to either motor also causes an elevation of the body (i.e., pitch of the arm). If the body pitches, the thrust vectors result in a travel of the body (i.e., yaw (ε) of the arm) as well. If the body pitches, the impulsion vector results in the displacement of the system (i.e., travel (λ) of the system). The objective of this experiment is to design a control system to track and regulate the elevation and travel of the 3-DOF Helicopter.
The 3-DOF Helicopter can also be fitted with an active mass disturbance system that will not be used in this work. The state space model that describes the helicopter is (Quanser, 2002) shown in (73).
The variables ξ and γ represent the integrals of the angles ε of yaw and λ of travel, respectively. The matrices A and B are presented in sequence.
The values used in the project were those that appear in the MATLAB programs for implementing the original design manufacturer, to maintain fidelity to the parameters. The constants used are described in Table ( Gravitational constant (m/s 2 ) g 9.81 Table 1. Helicopter parameters Practical implementations of the controllers were carried out in order to view the controller acting in real physical systems subject to failures.
The trajectory of the helicopter was divided into three stages. The first stage is to elevate the helicopter 27.5 o reaching the yaw angle ε = 0 o . In the second stage the helicopter travels 120 o , keeping the same elevation i.e., the helicopter reaches λ = 120 o with reference to the launch point. In the third stage the helicopter performs the landing recovering the initial angle ε = −27.5 o .
During the landing stage, more precisely in the instant 22 s, the helicopter loses 30% of the power back motor. The robust controller should maintain the stability of the helicopter and have small oscilation in the occurrence of this failure.
To add robustness to the system without any physical change, a 30% drop in power of the back motor is forced by inserting a timer switch connected to an amplifier with a gain of 0.7 in tension acting directly on engine, and thus being constituted a polytope of two vertexes with an uncertainty in the input matrix of the system acting on the helicopter voltage between 0, 7V b and V b . The polytope described as follows.
Vertex 1 (100% of V b ): Fixing the decay rate equal to 0.8, there were designed: a controller with quadratic stability using the existing optimization (Assunção et al., 2007c), a controller with quadratic stability with the proposed optimization and controllers with extended stability and projective stability also with the proposed optimization to perform the practical implementation.
This controller was implemented in helicopter and the results are shown in Figure 3.
In (75) follows the quadratic stability controller design with the proposed optimization follows (Theorem 5.1).
This controller was implemented in the helicopter and the results are shown in Figure 4.
This controller was implemented in the helicopter and the results are shown in Figure (5).
This controller was implemented in the helicopter and the results are shown in Figure 6.  Fig. 3. Practical implementation of the designed K by quadratic stability with the optimization method presented in (Assunção et al., 2007c).
The graphics of Figures 3, 4, 5 and 6, refer to the actual data of the angles and voltages on the front motor (V f ) and back motor (V b ) measured with the designed controllers acting on the plant during the trajectory described as a failure in the instant 22 s. Tensions (V f ) and (V b ) on the motors were multiplied by 10 to match the scales of the two graphics.
Note that the variations of the amplitudes of (V f ) and (V b ) using optimized controllers proposed (75) and (76)     controller in the literature (74) shown in Figure 3. This is due to the fact that our proposed controllers (75) and (76) have lower gains then (74). For this implementation the projective stability designed controllers with proposed optimization (77) obtained the worst results as Figure 6.
It was checked that the γ used in the implementation of robust controllers, if higher, forces the system to have a quick and efficient recovery, with small fluctuations.

General comparison of the two optimization methods
In order to obtain more satisfactory results on which would be the best way to optimize the norm of K, a more general comparison has been made between the two methods as Theorems 4.1 and 5.1.
There were randomly generated 1000 uncertain polytopes of second order systems, with only one uncertain parameter (two vertexes) and after that, 1000 uncertain polytopes of fourth order uncertain systems, with two uncertain parameter (four vertexes). The 1000 uncertain polytopes were generated feasible in at least one case of optimization for γ = 0.5, and the consequences of γ increase were analyzed and plotted in a bar charts showing the number of controllers with lower norm due to the increase of γ, shown in Figure 7 for second-order systems and in Figure 8 to fourth-order systems.
The controllers designed with elevated values of γ do not have much practical application due to the fact that the increase of γ affect the increasing of the norm and make higher peaks of the transient oscillation, used here only for the purpose of analyzing feasibility and better results for the norm of K, so comparisons were closed in γ = 100.5, because this γ is already considered high.
In Figure 8 can be seen that the proposed optimization method produces better results for all cases analyzed. Due to the complexity of the polytopes used in this case (fourth-order uncertain systems with two uncertainties (four vertexes)), is natural a loss of feasibility with the increase of γ, and yet the proposed method shows very good results.

General comparison of the new design and optimization methods
A generic comparison between the three methods of design and optimization of K was also carried out: design by quadratic stability with proposed optimization shown in Theorem 5.1, design and proposed optimization with extended stability shown in Theorem 5.3 (using the parameter a = 10 −6 in the LMIs) and projective stability design with proposed optimization shown in Theorem 5.5.
Initially 1000 polytopes of second order uncertain systems were randomly generated, with only one uncertain parameter (two vertexes) and after that, fourth order uncertain systems, with two uncertain parameter (four vertexes). The 1000 polytopes were generated feasible in at least one case of optimization for γ = 0.5 and the consequences of γ increase were analyzed. In fourth-order uncertain systems, the 1000 polytopes were generated feasible in at least one case of optimization for γ = 0.2 and then, the consequences of γ of 0.2 in 0.2 increase were analyzed. This comparison was carried out with the intention of examining feasibility and better results for the norm of K. So, a bar graphics showing the number of controllers with lower norm with the increase of γ was plotted, and is shown in Figures 9 and 10.   Both figures 9 and 10 show that the proposed optimization method using quadratic stability showed better results for the controller norm with the increase of γ, due to optimization this method no longer depend on the matrices that guarantee system stability as it can be seen in equation (22). In contrast, using the proposed optimizations with extended stability and projective stability, they still depend on the matrices that guarantee system stability as seen in equations (51) and (72) and this is the obstacle to finding better results for these methods.

Conclusions
At the 3-DOF helicopter practical application, the controllers designed with the proposed optimization showed lower values of the controller's norm designed by the existing optimization with quadratic stability, except the design for projective stability which had the worst value of the norm for this case, thus showing the advantage of the proposed method regarding implementation cost and required effort on the motors. These characteristics of optimality and robustness make our design methodology attractive from the standpoint of practical applications for systems subject to structural failure, guaranteeing robust stability and small oscillations in the occurrence of faults.
It is clear that the design of K via the optimization proposed here achieved better results than the existing optimizing K (Assunção et al., 2007c), using the LMI quadratic stability for second order polytopes with one uncertainty. The proposed optimization project continued to show better results even when the existing optimization has become totally infeasible for fourth order polytopes with two uncertainties.
By comparing the three optimal design methods proposed here (quadratic stability, extended stability, and projective stability) it can be concluded that the design using quadratic stability had a better performance for both analysis: 1000 second order polytopes with one uncertainty and for the 1000 fourth order polytopes with two uncertainties, showing so that the proposed optimization ensures best results when used with the quadratic stability.