General Output Feedback Stabilization for Fractional Order Systems : An LMI Approach

and Applied Analysis 3 3. Main Results 3.1. Static Output Feedback Control Theorem 4. Design the controller (4) for the system (1) with 0 < α < 1; the corresponding closed-loop control system is asymptotically stable, if there exist matrices Z ∈ P, G ∈ R, andH ∈ R, such that [ Ξ Z T B + C T H T − K T


Introduction
In recent years, fractional order systems (FOSs) have attracted considerable attention from control community, since many engineering plants and processes cannot be described concisely and precisely without the introduction of fractional order calculus [1][2][3][4][5][6].Due to the tremendous efforts devoted by researchers, a number of valuable results on stability analysis [7][8][9] and controller synthesis [10][11][12][13] of FOSs have been reported in the literature.
Since it is usually not possible or practical to sense all the states and feed them back, it is practically important and theoretically appealing to stabilize systems by output feedback controller (OFC) [14][15][16][17][18]. Linear matrix inequality (LMI) is one of the most effective and efficient tools in controller design and a great deal of LMI-based methods of OFC design have been proposed over the last decade.Generally, these methods can be broadly classified into three categories: iterative algorithm [19,20], singular value decomposition (SVD) method [21][22][23][24], and slack variable method [25,26].
Paradoxically, only few studies deal with OFC design for FOSs.One finds some existing results as presented in [27][28][29][30] only.Based on SVD method, [27] designs the OFC for a type of FOS with time delay.Nevertheless, the SVD method is inherently conservative, particularly for such a large number of decision variables.Reference [28] designs an  ∞ static OFC which is also based on SVD method, but the stable region reduced repeatedly in the transformation process.Reference [29] studies the FOS stabilization problem based on its approximation model, which does not face the original fractional order systems directly.Reference [30] gives sufficient conditions for static OFC which can stabilize the FOS with the order 1 ≤  < 2. Nonetheless, there is no discussion about 0 <  < 1 and dynamic OFC.Furthermore, the original free decision matrix variables  11 ,  12 ,  21 , and  22 are limited to  11 =  22 =  and  12 =  21 = 0, which shall cause an increase in conservatism.
This motivates us to adopt our previous stability criteria [9] in output feedback controller synthesis for FO-LTI systems, which shall make the resulting controller less conservative and more applicable in practice.Preferably, the new method is applicable for all the fractional order 0 <  < 2. In addition, the output feedback stabilization of uncertain FOS is discussed in the paper.

Problem Formulation and Preliminaries
Consider the following FO-LTI system: where the order 0 <  < 2; () ∈ R  , () ∈ R  , and () ∈ R  are the system state, the control input, and the measurable output, respectively; the system matrices , , and  are the constant real matrices with appropriate dimensions.We make the usual assumptions that the pair {, } is controllable and the pair {, } is observable, which guarantee the existence of real matrices  0 and  0 such that  +  0 and  +  0  are stable.Thus one may find a real matrix  satisfying the stability of  + .
The following Caputo definition is adopted for fractional derivatives of order  for function (): where the fractional order  − 1 <  ≤ ,  ∈ N, and the Gamma function Γ() = ∫ ∞ 0  −  −1 d.In this paper, the following general OFC is considered: where () ∈ R  is the controller state variable.
Set   = 0,   = 0,   = 0, and   = ; the static OFC is derived: Set   = ,   = ,   = , and   = 0; one can obtain the following dynamic OFC: This paper aims at finding the proper condition so that the resulting closed-loop system is asymptotically stable with the three types of OFCs.For this purpose, the following lemmas are first introduced.
Lemma 1 (see [25]).Let Φ, , and  be given matrices with appropriate dimensions; then hold if and only if there exists an appropriate dimension matrix  which satisfies where the operator sym() represents  +   and ⋆ stands for the symmetrical part matrix; for example, Proof.Without loss of generality, we assume that Φ ∈ R × , ,  ∈ R × , and  ∈ R × .Set  = [ Φ  ⋆ 0 ],  = ,  = [0 ×   ], and  = [  −   ]; then (7) can be transformed into Select ], using the projection lemma in [31], and one gets the equivalent LMIs: This establishes the proof.
Proof.Since  and   have the same set of eigenvalues, the systems D  () = () and D  () =   () have the same stability.
As a result, ∃ ∈ P × , sym() < 0 is equivalent to Because one completes the proof of Remark 3.
Abstract and Applied Analysis 3

Static Output Feedback Control
Theorem 4. Design the controller (4) for the system (1) with 0 <  < 1; the corresponding closed-loop control system is asymptotically stable, if there exist matrices  ∈ P × ,  ∈ R × , and  ∈ R × , such that is feasible, and the controller gain is given by where Ξ = sym(  +   0 ), and  0 is an additional initialization matrix, which is derived from  0 =  −1 .The matrices  ∈ P × and  ∈ R × satisfy the following LMI: Proof.First, we design a virtual state feedback controller () =  0 () for the system in (1), which yields By using Lemma 2, one obtains that the system ( 19) is asymptotically stable if and only if there exists  ∈ P × , such that sym( Define  =  0 ; one can easily get ( 18) from (20).The use of Remark 3 with Hereafter, considering the actual output feedback controller () = (), then the resulting closed-loop dynamic system can be described as From Remark 3, one can obtain that the system ( 22) is asymptotically stable if and only if there exists a matrix  2 ∈ P × , such that Owing to the existence of the nonlinear terms   2 , inequality (23) is not an LMI.For the purpose of using the MATLAB LMI toolbox to solve the matrix inequality, we need to linearize the matrix inequality.

Dynamic Output Feedback Control I.
Under the control of ( 5), if we define the augmented state () = [  ()   ()]  , then the related closed-loop control system can be rewritten as where As a result, the controllability matrix   related to {, } can be described as which implies that rank By virtue of All of these stated above lead up to the following: In other words, {, } is controllable.The corresponding observability matrix   satisfies which implies that rank Since   ∈ R (+)×(+) , one has rank (  ) ≤ min ( + , ( + ) ) =  + . ( Proceeding forward, one has rank ( Consequently, {, } is observable.Thus, Theorem 5 has been proved completely. Theorem 6. Design the controller (5) for system (1) with 0 <  < 1; if there exist matrices  ∈ P (+)×(+) ,  ∈ R × , and  ∈ R ×(+) , such that is feasible, then the resulting closed-loop control system in (25) is asymptotically stabilizable by the output feedback controller where Ξ = sym(  +   0 ), and  0 is an additional initialization matrix, which is derived from  0 =  −1 .The matrices  ∈ P (+)×(+) and  ∈ R × satisfy the following LMI:  , then the related closed-loop control system can be rewritten as where (39) Remark 7. In a manner similar to the proof of Theorem 5, one gets that {, } is controllable and {, } is observable.
Remark 10.Theorems 4, 6, and 8 consider the stability problem of the systems in ( 22), (25), and (38), respectively.One can observe that there are more decision variables in the LMIs in Theorem 6 or 8 than in Theorem 4, which shall increase the computational complexity to solve those LMIs in the former.Of course, since those theorems are just solving convex feasibility problems, and meanwhile the dimensions , , and  are of limited practical magnitude, there shall not be a computational burden.
Theorem 12. Design the controller (4) for system (1) with 1 ≤  < 2; the related closed-loop control system is asymptotically stable, if there exist matrices  ∈ P 2×2 ,  ∈ R × , and  ∈ R ×2 , such that is feasible, and the controller gain is given by where Ξ = sym(  Ã+  B 0 ), and  0 is an additional initialization matrix, which is derived from  0 =  −1 .The matrices  ∈ P 2×2 and  ∈ R ×2 satisfy the following LMI: Remark 13.Because the way of how to get the controller in Theorem 4 is very similar to that of Theorems 6, 8, and 12, therefore the proof of the latter three has been omitted.
There are  2 + 2 + 2 + 8 2 decision variables and  + 4 inequalities in Theorem 12 that need to be solved.At the same time, Theorem 2.3 needs  2 +  +  + 7 2 + 2 decision variables and +6 inequalities.Our approach may need more decision variables since some decision variables in Theorem 2.3 are set to be equal or zeros by force, which may become more conservative.In addition, our approach needs less inequalities than that in Theorem 2.3, which shall reduce computational burden.
Remark 15.In analogy to the above mentioned case with 0 <  < 1, we can easily get the other stabilization criterion controlled by ( 3) or (5).Considering expanding the th order system to 0.5th order system before or after substituting the controller (3) or ( 5) into it, we can get th or 0.5th controller design criterion.

Robust Output Feedback Control. Consider the system (1)
where 0 <  < 1 is uncertain, and the system matrices  and  can be described as where  0 and  0 are the constant matrices and the uncertain terms Δ and Δ are given by herein ,   , and   are constant matrices; the unknown variable matrix () ∈ R × satisfies   ()() ≤   .
Theorem 16.Design the controller (4) for the system (1) with conditions in (48); the corresponding closed-loop control system is asymptotically stable for all admissible uncertainties, if there exist matrices  ∈ P × ,  ∈ R × ,  ∈ R × , and a set of positive scalars  1 ,  2 ,  3 , such that is feasible, and the controller gain is given by where 0 is an additional initialization matrix, which is derived from  0 =  −1 .The matrices  ∈ P × ,  ∈ R × , and positive scalars  1 ,  2 satisfy the following LMI: where Remark 17.By using the similar approach in Theorem 4, the theorem can be easily derived, wherefore the proof is omitted.

Illustrative Examples
All the numerical examples illustrated in this paper are implemented via the piecewise numerical approximation algorithm.For more information about the algorithm one can refer to [29].
Example 1.Consider the system as follows: It is completely controllable and observable.One gets that the eigenvalues of the system matrix are 1.7500 + 0.6614 and 1.7500 − 0.6614, which are denoted by EV0.Thereby, the original system with  = 0.6 is unstable.If applying the method in Theorems 4, 6, and 8 to design OFCs, using the MATLAB LMI toolbox, one can get the following feasible OFCs: If one uses EV1, EV2, and EV3 to represent the eigenvalues of the closed-loop control system matrices which are controlled by the three aforementioned controller, respectively, then the distribution of those eigenvalues in the complex plane is shown in Figure 1.
According to the method in (43), one gets the equivalent system with 0.7th order.Based on Remark 15, applying The same as Example 1, one obtains the eigenvalues distribution of those equivalent closed-loop control systems in the complex plane as shown in Figure 2. The responses of open-loop system 0 1 t (s)  Designing 1.4th order OFC as (3)-( 5) and using the method in (43), one gets the equivalent system with 0.7th order.By using the method for the case of 0 <  < 1, one gets the following feasible OFCs:  () = −12.6764() , Under the control of the obtained OFCs, one gives the eigenvalues distribution area of the equivalent closed-loop control system as shown in Figure 3.
One can observe Figures 1, 2, and 3 that if the system (1) can be stabilized by OFC, no matter 0 <  < 1 or 1 ≤  < 2, we can get the needed controller via our approach.
Example 4. Consider the system as follows: If we design the static OFC as then one can get the eigenvalues of the closed-loop system matrix  +  as To be obvious, when the order  > 1, no matter how we choose ,  +  always has eigenvalue in unstable region.
That is why we discuss the design of the dynamic OFCs in ( 3) and ( 5).Based on the method in this paper, we get the feasible dynamic OFCs as From the results in Figure 4, we can obtain that the system in (59) can be stabilized by the obtained dynamic OFCs.
Figure 5 shows the output signal of the open-loop system and the closed-loop system, respectively.The corresponding control input is given in Figure 6.From the simulation results, one can conclude that the proposed method can easily obtain OFC which is able to stabilize such uncertain fractional order system.

Conclusions
In this paper, the methods of designing general OFC for FOSs with the order 0 <  < 2 have been investigated.For the case of 0 <  < 1, LMI-based sufficient conditions for static/dynamic OFC design are proposed.Based on the equivalence transformation, the related results are generalized to the systems with 1 ≤  < 2. Compared with existing results, the new proposed approaches require fewer decision variables and have less restrictions conditions which are helpful for reducing the conservatism of the obtained results.
The numerical examples have shown the effectiveness of the proposed design methods.It is believed that the approaches provide a new avenue to solve such problem.
[1]of.According to[1], {, } is controllable, if and only if the controllability matrix   is full row rank, where   = [  ⋅ ⋅ ⋅  −1 ].{, } is observable, if and only if the observability matrix   is full column rank, where