Less Conservative Optimal Robust Control of a 3-DOF Helicopter

Thiswork proposes an improved technique for design andoptimization of robust controllers norm for uncertain linear systems,with state feedback, including the possibility of time-varying the uncertainty. The synthesis techniques used are based on LMIs (linear matrix inequalities) formulated on the basis of Lyapunov’s stability theory, using Finsler’s lemma.The design has used the addition of the decay rate restriction, including a parameter γ in the LMIs, responsible for decreasing the settling time of the feedback system. Qualitative and quantitative comparisonsweremade betweenmethods of synthesis and optimization of the robust controllers norm, seeking alternatives with lower cost and better performance that meet the design restrictions. A practical application illustrates the efficiency of the proposed method with a failure purposely inserted in the system.


Introduction
The history of linear matrix inequalities (LMIs) in the analysis of dynamical systems dates back to over 100 years.The story begins around 1890 when Lyapunov published his work introducing what is now called Lyapunov's theory [1].The researches and publications involving Lyapunov's theory have grown up a lot in recent decades [2], opening a very wide range for various approaches such as robust stability analysis of linear systems [3].
In addition to the various current controllers design techniques, the design of optimal robust controllers (or controller design by quadratic stability) using LMI stands out for solving problems that previously had no known solution [4].These designs use specialized computer packages [5].
Recent publications have found a certain conservatism inserted in the analysis of quadratic stability, which led to a search for solutions to eliminate this conservatism.Finsler's lemma [6] has been widely used in control theory for the stability analysis by LMIs, with results similar to the LMI quadratic stability, but with extra matrices, allowing a certain relaxation in the stability analysis (referred as "extended" stability), by obtaining a larger region of feasibility.However, these methods only guarantee stability for uncertain timeinvariant systems or with very small variation rate [7].
The main focus of this paper is to propose an improved method of design and optimization technique than the one presented in [8], including the possibility of time-varying the uncertainty, searching for lower gains of controllers that have the same dynamic performance when the system is forced to have a faster transient with the inclusion of decay rate restriction (a D-stability performance index [9] which ensures that the eigenvalues of the uncertain system are on the left of  scalar) in the LMIs formulation.
Comparisons will be made through a practical application in a Quanser's 3-DOF helicopter [10], considering a failure during the landing treated as a time variant uncertainty, and a generic analysis involving 1000 randomly generated polytopic uncertain systems.

Robust Stability Using Decay Rate as a Performance Index
Consider the controllable uncertain linear time-invariant system described in the state space form: ẋ () =  ()  () +  ()  () . (1 This system can be described as convex combination of the polytope vertexes: with   > 0,  = 1, . . ., , where  is the number of the polytope vertexes [1].Considering the uncertain system (2) and exiting Lyapunov theory for designing controllers, the following theorem is stated [1].Theorem 1.A sufficient condition for the uncertain system (2) stability guarantee subject to decay rate greater than or equal to  is the existence of matrices  =   ∈ R × and  ∈ R × , such that with  = 1, . . ., .
When the LMIs ( 4) and ( 5) are feasible, a state feedback matrix that stabilizes the system can be found as Proof.See [1].
Thus, the feedback of the uncertain system presented in (1) can be done, with (4) and (5) being sufficient conditions for asymptotic stability of the polytope for a state feedback system with decay rate restriction.If the solution of LMIs is feasible, the uncertain system's stability is guaranteed.
In many situations the norm of the state feedback matrix is high, precluding its practical application.In [11] an optimization method that minimizes the gains of the controller designed via LMIs (4) and ( 5) was proposed, but this does not eliminate the inherent conservatism.
Thus, in [8] a new way to optimize the norm of  controller was presented eliminating the conservatism of the LMIs by Finsler's lemma.
We can use the Finsler's lemma to express the stability in terms of LMIs, with advantages over the existing Lyapunov's theory [1], once it introduces new variables (, X) under conditions which involve only L, B, and B ⊥ .Lemma 2 (Finsler).Consider  ∈ R   , L ∈ R   ×  , and B ∈ R   ×  with rank(B) <   and B ⊥ a basis for the null space of B (i.e., BB ⊥ = 0).Then the following conditions are equivalent: (1)   L < 0, ∀ ̸ = 0 : B = 0 Proof.See [6] or [12].

Robust Stability of Systems Using Finsler's Lemma with
Decay Rate Restriction.
and L = [ 2   0 ], note that B = 0 corresponds to the feedback system with  and   L < 0 represents the stability restriction with decay rate formulated from the quadratic Lyapunov function [1].In this case the dimensions of Lemma's 2 variables are   = 2 and   = .
Thus, it is possible to characterize stability through the quadratic Lyapunov function ((()) := ()  ()), generating new degrees of freedom for the synthesis of controllers.
From existing proof of Finsler's lemma it can be concluded that the properties of one to four are equivalent.Thus, we can rewrite fourth property as follows: conveniently choosing the matrix variables X = [   ], with  ∈ R × nonsymmetric and  a relaxation constant that has the function of flexible matrix X in the LMI [13].This constant can be defined by making a one-dimensional search.Applying the congruence transformation at the right, in the fourth property, and making  = (  ) −1 ,  = , and  =   , the following LMIs were found: with  ∈ R × ,  ̸ =   ,  ∈ R × , and  ∈ R × .These LMIs meet the restrictions for the asymptotic stability of the system with state feedback.The stability resulting of the LMIs derived from Finsler's lemma referred to as extended stability [14].The advantage of using Finsler's lemma formulation for robust stability analysis is the freedom of Lyapunov's function, now defined as () = ∑  =1     , ∑  =1   = 1,   ≥ 0 and  = 1, . . ., .As () depends on , the Lyapunov matrix use fits to time-invariant polytopic uncertainties, with permitted rate of variation being sufficiently small.Thus in [8] the following theorem was presented.
When LMIs ( 9) and ( 10) are feasible, a state feedback matrix that stabilizes the system can be given by Proof.See [8].
Thus, it can be feedback into the uncertain system with ( 9) and (10) sufficient conditions for asymptotic stability of the polytope.

Optimization of 𝐾 Norm Using
Finsler's Lemma.In [8] there was a difficulty in applying the existing theory for the optimization of  matrix norm [11] for the new structure of LMIs.This was due to the nonsymmetry of  matrix for the controller synthesis, condition that was necessary to the LMI development when the controller synthesis matrix was  =  −1 .The solution found was using the idea of the optimization procedure for redesign presented in [15].Thus, in [8] the new optimization method adequacy was proposed, with the minimization of a scalar  obeying the relation    <   with   the Lyapunov function, to the new relaxed parameters through the following theorem.Theorem 4. A constraint for the  ∈ R × matrix norm of state feedback can be obtained, with  =  −1 and   =     , being  ∈ R × ,  ∈ R × , and   ∈ R × ,   =    > 0 finding the minimum ,  > 0, such that    <   ,  = 1, . . ., .One can get the optimal value of  solving the optimization problem with the LMIs: where   denotes the identity matrix of  order.
This way of optimizing the norm of  showed better results than the one presented in [11].However, the optimization LMI, in order to be bound by the Lyapunov's matrix   , still does not have the minimal gains that would be found to meet the design requirements.
In order to improve the optimization performance, an alternative is presented in Section 2.4 to optimize the norm of  controller reducing the conservatism of the LMIs design with a convenient manipulation.

Robust Stability of Systems Using Reciprocal Projection
Lemma with the Decay Rate Restriction.In order to verify the advantages of the new formulation proposed in Section 2.4, one of the best examples of optimal design of  was borrowed from [8] for comparison purposes.As in the extended stability case, the advantage of using the reciprocal projection lemma for robust stability analysis is Lyapunov's function degree of freedom, now defined as () = ∑  =1     , ∑  =1   = 1,   ≥ 0 and  = 1, . . ., .As described before Theorem 3, the use of () fits to time-invariant polytopic uncertainties, with permitted rate of variation being sufficiently small.To verify this, Theorem 5 follows.2) is the existence of matrices  ∈ R × ,   =    ∈ R × , and  ∈ R × , such that LMIs ( 13) are met as follows:

Theorem 5. A sufficient condition which guarantees the stability of the uncertain system (
with  = 1, . . ., . When the LMIs ( 13) are feasible, a state feedback matrix which stabilizes the system can be given by Proof.See [8].
Theorem 6 shows the optimization of  for LMIs (13).
Theorem 6.A constraint for the  ∈ R × matrix norm of state feedback is obtained, with  =  −1 ,  ∈ R × , and  ∈ R × finding the minimum ,  > 0, such that    < , being  =  −1  −1 and therefore  =   > 0. We can get the optimal value of  solving the optimization problem with the LMIs as follows: where   and   denote the identity matrices of  and  order, respectively.

New Formulation for Robust Stability of Systems
Using Finsler's Lemma with Decay Rate Restriction.
. Check that the lemma variables (1) dimensions are   = 2 and   = .Considering that  is the matrix used to define the quadratic Lyapunov function, we will have the second propriety of Finsler's lemma written as follows: (2) ∃ =   > 0 such that which results in the necessary and sufficient condition for the system's stabilizability, including decay rate: (2)  −  +   −     + 2 < 0.
Thus, it is possible to characterize stability through the quadratic Lyapunov function ((()) = ()  ()), generating further degrees of freedom for the controllers synthesis.
From existing Finsler's lemma proof (Lemma 2), it can be concluded that the second and fourth properties are equivalent.Thus, we can rewrite the fourth property as follows: (17) Conveniently choose the matrix variables Y = [  1 2 ], with  1 ,  2 ∈ R × .Developing the fourth property, we have Thus the following LMIs subject to decay rate greater than or equal to  were found: These LMIs meet the restrictions for the asymptotic stability of the system with state feedback.It can be seen that the first principal minor of LMI (19) has the structure of the results found with the theorem of stability with decay rate.However, there is also, as stated in Finsler's lemma, a high degree of freedom, due to the relaxation variable matrices  1 and  2 , without being symmetric, and for a robust stability approach, they may be polytopic:  1 () = ∑  =1    1 and  2 () = ∑  =1    2 , ∑  =1   = 1,   ≥ 0 and  = 1, . . ., .Therefore the following theorem is proposed.
Theorem 7. In order to guarantee the stability of the uncertain system (2) subject to decay rate greater than or equal to  a sufficient condition is the existence of matrices  1 and  2 ∈ R × ,  =   ∈ R × , and  ∈ R × , such that When LMIs ( 21), ( 22), and ( 23) are feasible, a state feedback matrix that stabilizes the system can be given by  =  −1 . (24) Proof.Assume that LMIs (21), (22), and (23) are feasible.
There is a great advantage in using the new formulations (21) and ( 22) using Finsler's lemma compared with the old one (9) due to insertion of two polytopic matrices  1 and  2 , relaxing more the LMIs when compared to the old formulation where there was only the polytopic Lyapunov matrix   ; moreover as described before Theorem 3, the use of () fits to time-invariant polytopic uncertainties, with permitted rate of variation being sufficiently small, unlike this new formulation where the Lyapunov matrix  is not polytopic, allowing variations in .
Below we have an alternative optimization of the norm of , which together with LMIs (21), ( 22), and (23) shows better results for the controller gains, as will be seen in Section 4.

3-DOF Helicopter
Consider the schematic model in Figure 2 of the 3-DOF helicopter [10] shown in Figure 1.This equipment is a patrimony of Control Research Laboratory at FEIS, UNESP.Two DC motors are mounted at the two ends of a rectangular frame and drive two propellers.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 (pitch angle ()).A positive voltage to either motor also causes an elevation of the body (elevation angle () of the arm).If the body pitches, the thrust vectors result in a travel of the body (travel angle () of the arm) as well.The objective of this experiment is to design a control system to track and regulate the elevation and travel of the 3-DOF helicopter [10].The state space model that describes the helicopter is [10] [ The variables  and  represent the integrals of the angles  of yaw and  of travel, respectively.Matrices  and  are presented in sequence: Details of the model constants are given in Table 1.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 a polytope of two vertexes is constituted with an uncertainty in the input matrix of the system acting on the helicopter voltage between 0.7  and   .The polytope is described as follows.

Comparison of Techniques for Controller 𝐾 Design and Optimization
To verify the efficiency of the new optimization technique practical applications of the controllers were carried out, in order to view the controllers acting in real systems subject to failures.The trajectory of the helicopter was divided into three stages as shown in Figure 3.The first stage is to elevate the helicopter 27.5 ∘ reaching the yaw angle  = 0 ∘ .In the second stage the helicopter travels 120 ∘ , keeping the same elevation; that is, the helicopter reaches  = 120 ∘ with reference to the launch point.In the third stage the helicopter performs the landing recovering the initial angle  = −27.5 ∘ .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 oscillation in the occurrence of this failure.
Fixing the decay rate grater or equal to 0.8, the following was designed: a controller with extended stability (Theorem 3) using the optimization (Theorem 4), controllers with projective stability (Theorem 5) using the optimization (Theorem 6), and the new proposed formulation for extended stability (Theorem 7) also with the improved optimization (Theorem 8) to perform the practical application.The value of the relaxation constant  in the LMIs of Theorem 3 can be adequately specified by the designer.In this example we did not find difficulty in specifying  = 10 −6 .The designer has an alternative to perform a one-dimensional search to set this constant.Theorems 3 and 5 hypothesis establishes a sufficiently low time variation of .However, for comparison purposes of Theorems 3 and 5 with Theorem 7, the same abrupt loss of power test made with controller (48) was done with controllers (46) and (44).
Respectively, the controllers norm was 56.47 for the controller designed with extended stability (Theorem 3) and optimization (Theorem 4), 110.46 for the controller designed with projective stability (Theorem 5) and optimization (Theorem 6), and 44.84 for the controller designed with the new proposed formulation for extended stability (Theorem 7) and optimization (Theorem 8).
Below we have the controller designed with the LMIs from Theorem 3 and optimization from Theorem 4 for the application illustrated in Figure 4   In Figures 4, 5, and 6 the states are shown: elevation (), pitch (), and travel () in degrees for the trajectory previously stated with the application of the controllers (44), (46), and (48), respectively.All three figures also show, respectively, the control signals (voltage) in the front (  ) and back (  ) engines, in which it is possible to verify that the control signals in Figure 6 are smoother compared with those of Figures 5 and 4.This smoothness is due to the fact that the norm of controller (48) is smaller than the others.
Note that even if the proposed method has smaller norm than the ones presented in [8], the transients before and after failure are almost the same with small differences in amplitude.

General Comparison of the Design and Optimization
Methods.A generic comparison between the three methods of design and optimization was carried out to obtain more satisfactory results on what would be the best way to optimize the norm of .One thousand polytopes of second order uncertain systems were randomly generated, with only one uncertain parameter (two vertexes).The one thousand polytopes were generated feasible in at least one case of design and optimization for  = 0.5 and the consequences of the  increase were analyzed as Figure 7 shows.
As we can see in Figure 7, the proposed formulation for extended stability with the optimization method showed

𝛾
Projective stability [8] Extended stability [8] Proposed formulation better results for the norm of the one thousand polytopes of second order uncertain systems randomly generated.This happened for all situations of  increase, that is, from 0.5 to 100.5 due to the insertion of the polytopic matrices  1 and  2 , relaxing more the LMIs when compared to the existing formulation where there were only the polytopic Lyapunov matrices   , thus allowing more efficient optimization using LMIs (31) and (32).Results for extended stability [8] are so small that they hardly appear in the bar chart due to the magnitude values of the other two methods.

Conclusions
The new proposed formulation for extended stability with the optimization method showed better results compared with those presented in [8].Controllers were implemented for all methods in 3-DOF helicopter and for the performance of controllers in the system; it was found that the proposed method showed better results requiring lower energy expenditure.The controller designed by the proposed technique presented in this paper showed the norm reasonably smaller than the existing techniques; in addition to this for the existing techniques the abrupt variation of the uncertainty

Figure 3 :
Figure 3: Tracking curves for  and .

Figure 6 :
Figure 6: Practical application of the  designed by the proposed formulation for extended stability with the optimization method (Theorems 7 and 8).

Figure 7 :
Figure 7: Number of controllers with lower norm for the design methods considering the increase in .
subject to decay rate greater than or equal to , a sufficient condition is the existence of matrices  ∈ R × ,   =    ∈ R × , and  ∈ R × , such that
and its norm: