Output feedback controller design for discrete LTI systems with polytopic uncertainty via direct searching of the design space

This paper concerns static output feedback stabilization of polytopic discrete linear time‐invariant (LTI) systems. The previous related studies were mainly based on linear matrix inequality (LMI) approaches which are naturally conservative. In this paper, a novel design algorithm is presented that iteratively partitions a primary design space to subspaces. Then, by assessing stabilizability status of each generated subspace, the algorithm determines the total stabilizable parts and removes the undesired parts of the design space. Mathematical theories are developed to determine the total de‐stabilizability or stabilizability of a given subspace. These subspaces' properties are detected through checking the existence of critical polynomials (which have roots on the unit circle of the complex plane) on their exposed edges. By omitting the undesired parts of the design space, the algorithm just searches the desired parts which are far smaller than the primary design space. This strategy improves the feasibility performance of the algorithm. Some illustrating examples are provided to show the steps of the design algorithm. Furthermore, 100 random models are generated to evaluate the feasibility performance of the proposed algorithm as compared to some existing methods. The results reveal the superiority of the proposed algorithm.


| INTRODUCTION
The polytopic uncertain model includes a type of parametric uncertainty which is exploited to model the existing uncertainty in practical models [1]. This type of uncertainty is frequently encountered in the analysis of various kinds of discrete-time systems [2,3]. In this type of model uncertainty, the system matrices are considered to be a convex combination of some known corner matrices [4][5][6].
One appropriate controller is the static output feedback (SOF) controller [7,8] due to its easy implementation. In the presence of polytopic uncertainties, robust SOF synthesis of LTI systems is widely investigated in the previous studies [9][10][11]. Although, some novel and valuable approaches are developed, this problem is still an open challenge [12]. It is mainly because, there is a lack of a suitable and acceptable approach that is able to solve this problem by a set of sufficient and necessary conditions [13,14]. Among various approaches in this area, the Lyapunov-based methods are the most widely popular and common approaches.
In the Lyapunov-based approaches, two groups of variables are considered as the design and Lyapunov variables [15,16]. The design variables are free variables in the control law which exist in the gain of the controller and the Lyapunov variables are used in the Lyapunov candidate functions. The main issue which arises in these approaches is the existence of coupling terms between the Lyapunov and design variables [10]. Indeed, the stability conditions are given in bilinear matrix inequality (BMI) form which cannot be efficiently solved by available linear matrix inequality (LMI) solvers. To cope with this issue, sufficient LMI conditions are developed to convexify these BMI conditions via utilizing LMI tricks or limiting assumptions [12,17]. Clearly, this approach is conservative due to the assumptions made during converting the problem into an LMI form. For instance, in Crusius and Trofino [17], the Lyapunov function is forced to have a special structure. Thus, the BMI conditions are converted to LMIs by inserting an LMI equality constraint on the Lyapunov variables. This change imposes conservativeness to the problem because the equality constraints result in a set of approximate conditions. The LMI tricks such as congruence transformation are also used to convexify the main nonconvex conditions [18].
Some approaches consider the parameter-dependent Lyapunov functions to decrease the conservativeness of the resulting stability criteria [19]. In such problems, the Lyapunov candidate function is considered to depend affinely on the uncertain parameters. Similar to the previous approaches, some LMI tricks are adopted to convexify the final conditions. These approaches significantly reduce the conservativeness which are also more feasible than the methods pertaining to the common Lyapunov functions. However, some conservatism still remains in their approach.
To overcome the disadvantages of the present approaches, this paper extends the direct searching idea which is recently developed to design robust stabilizing controllers for continuous LTI systems [20,21] to LTI discrete models. In other words, here, a design algorithm is proposed for robust SOF stabilization of discrete LTI systems in the presence of polytopic uncertainties. The controller may have some known and unknown variables in its structure where the unknown variables are noted by design variables. It is supposed that the design variables belong to a compact convex polygonal space which can be located anywhere in the whole possible real spaces, and its shape does not restrict the design methodology. A design algorithm tries to find a point inside the design space that is capable of stabilizing the uncertain model where the uncertain variables belong to the unit simplex.
The algorithm iteratively divides the primary design space to smaller parts and predicts the existence or nonexistence of feasible points in the given subspaces. If it detects that there is no feasible point in the design subspace, it marks it as total-destabilizability (TD) space. On the other hand, if it predicts that the whole subspace can robustly stabilize the model, it marks it as the total-stabilizability (TS) space. Investigation of the TD property of any subspaces is one of the most important novelties of this paper. The previous studies focus on the TS conditions, but this paper proposes a systematic way to predict the TD subspaces, and this achievement enables us to detect the undesired parts of the design space and delete them rapidly. Obviously, this tool may significantly improve the convergence rate of the design algorithm.
Briefly, the paper contributions are as follows: First, the present approaches for SOF design for discrete LTI systems with polytopic uncertainties are conservative. This paper presents a less-conservative approach for SOF design of such systems through direct searching of the design space. The method considers the whole design space and checks all parts for finding a feasible solution. Second, to increase the convergence rate of the method, the design space is partitioned to smaller subspaces, and they are searched for TS, TD, or undermined parts. Finding a TS subspace is satisfactory and ends the method, while finding a TD subspace means that no solution exists in this subspace. Thus, the algorithm omits this part and shrinks the design space. If the subspace is undetermined (neither TS nor TD), the method partitions that special subspace and continues the method. All the theories and the method for detecting TD and TS subspaces and the criteria for continuing the design space partitioning are expanded in the body of the paper. Third, the TS and TD determination of the subspaces depend on finding critical polynomials which are defined in the paper and the related theories are mentioned.
The paper is organized as follows. In Section 2, the uncertain plant model and its algebra are presented. In Section 3, the design algorithm which is the main contribution of this paper is mentioned. Extensive simulation examples are provided in Section 4 to assess the trueness of the proposed contributions of this paper. Finally, Section 5 concludes the paper.

| PLANT MODEL
Consider the following uncertain LTI model, in which, the uncertain parameters are assumed to have polytopic form: where k is the time index and x ℝ n , u ℝ n u , and y ℝ n y are the states, inputs, and outputs, respectively. Uncertain parameters are shown by vector θ ℝ n t assumed to be constant and belong to a unit simplex with the following definition: In (1), the system matrix C p ℝ n y Ân is supposed to be constant and known. However, the matrices A p (θ), B p (θ) affinely depend on the uncertain parameters: where A i f g n t i¼1 & ℝ nÂn and B i f g n t i¼1 & ℝ nÂn u are corner matrices of these system matrices. Now, consider the following control law, where f R n d is the vector of design parameters and F f ð Þ ℝ n u Ân y is an affine function of these design parameters.
According to (1) and (5), the closed loop system will be obtained as given below: The following assumptions are considered for the closed loop model (6): • The design parameters f should belong to a primary space Ω, called primary design space. • The characteristic polynomial of the closed loop system should affinely depend on f.

Definition 1.
Büyükköro glu et al. [22]. Let A ℝ n Â n be an arbitrary square matrix. This matrix is noted to be Schur stable if all its eigenvalues are inside the unit circle of the complex plane (in this paper, the unit circle is shown by C).
Definition 2. Let ψ be an arbitrary subspace inside the design space Ω. This subspace is said to be totally stabilizable (TS) if it satisfies the following condition: Definition 3. Let ψ be an arbitrary subspace in the design space Ω. It is noted to be total de-stabilizable (TD) if it holds the following condition: Definition 4. The real coefficient polynomial is critical if it has some roots on the boundary of the unit circle in the complex plane.

| MAIN RESULTS
This section presents the main contribution of the paper that is an algorithm to find a feasible point inside the primary design space Ω that stabilizes model (6). Assume the primary design space as a convex and compact space. At first, the whole space is considered, and its stability status is evaluated. The evaluation result is whether it is TS, TD, or none of them. If it is TS, one of the points inside it is returned as the feasible solution. If it is TD, there will be no feasible point inside this space, and the algorithm terminates by infeasibility. If the space is neither TS, nor TD, it is divided to two smaller subspaces through halving its largest edge. The method is repeated for the new subspaces, and it is continued until a feasible point is found or all the smaller parts are marked as TD. It is apparent that the sufficient conditions which are presented to exploit the TS or TD properties are more feasible in the smaller subspaces in comparison to larger ones.
Finding critical polynomials in a polynomial segment (a convex combination space of just two polynomials) plays a key role to develop TD conditions. For this purpose, an efficient method is proposed that can find all critical polynomials on a given polynomial segment. The critical polynomials are detected through finding roots of another polynomial that is defined based on the real and imaginary parts of corner polynomials.
The design algorithm has the following positive points: • The matrix gain in the control law may include some known and unknown variables that enable us to consider a free structure in the control law. • Since the algorithm directly searches the main design space, one can force the closed loop system to have specific structure such as minimum-phase and positive system. • The algorithm can be adopted to enhance the feasibility performance of previous sufficient approaches via supposing various control objectives. • The design algorithm is able to find all feasible simplexes of the primary simplex Ω whose largestedge-length is larger than a threshold value. • The design algorithm certainly terminates in a finite number of iterations that depends on the length of the largest-edge of the primary design simplex Ω.
This section consists of four subsections: fining critical polynomials on a polynomial segment, investigating the TS property of the design subspaces, investigating the TD property of the design subspaces, and proposing the design algorithm.

| Finding critical polynomials on a polynomial segment
It is known that a polynomial segment is a convex combination of two polynomials [23]. The proposed method detects the critical polynomials on a polynomial segment via finding possible roots on the unit circle of the complex plane. Before presenting the proposed methodology, the following theorems are needed. Theorem 1 mentions that the real coefficient polynomial in the complex plane can be decomposed to real and imaginary parts, in which, the real part only depends on the real complex variable, while the imaginary part depends on both real and imaginary parts of the complex variable. Theorem 2 derives two conditions to find critical polynomials on a polynomial segment based on Theorem 1. Theorem 1. Assume p is a real coefficient polynomial and v is an arbitrary complex number on the boundary of C. Then, p(v) can be decomposed as follows: where σ and ω are the real and imaginary parts of v and p r (σ) and p i (σ) are real coefficient polynomials of σ.
Proof. An induction technique can be used to successively prove the statement of the theorem for all polynomials with different degrees. Indeed, this technique shows the trueness of the following logical statement for each m {1,2,3…}: Q m ½ all polynomials with degree m that satisfy 9 ð Þ ð10Þ Note that each polynomial with degree 1 can be written as p(z) = a 0 +a 1 z where a 0 and a 1 are real numbers. Obviously, p(v) = a 0 +a 1 σ+ja 1 ω that implies According to (11) and (12), we can see that is true which means all polynomials with degree m satisfy (9). Let p z ð Þ ¼ P mþ1 i¼0 a i z i be an arbitrary real polynomial with degree m+1 and v = σ+jω be an arbitrary complex number on ∂ C ð Þ. Thus, one can obtain where q z ð Þ ¼ P m i¼0 a iþ1 z i is another polynomial with degree m. Since q has degree m and Q[m] is considered to satisfy the theorem hypothesis, q(v) can be decomposed as follows: Substituting (14) in (13) results in By expanding (15), the above equation can be expressed: Regarding v ∂ C ð Þ, one can easily conclude that σ 2 +ω 2 = 1 which implies that Then, by using (17), it can be easily concluded that Hence, p satisfies (9) which completes the proof.
To clarify the results of Theorem 1, the following numerical example is presented. Let p(z) be a real polynomial such as By substituting v = σ+jω in (20), one can get: Now, (21) can be factorized as follows: Since ω 2 = 1 À σ 2 , it can be easily concluded that Therefore, which approves the result of Theorem 1. In the following, Theorem 2 is presented for finding the critical polynomials in a given polynomial segment.
Theorem 2. Let p(z) and q(z) be two polynomials with similar degrees. Then, there are = 0 if and only if the following conditions hold: where p r (σ), q r (σ), p i (σ), and q i (σ) are defined in Theorem 1.
Proof. To prove the necessity part, let α [0,1] and v ∂C which leads to αp(v) Thus, the following results can be obtained due to Theorem 1 and the assumption ω ≠ 0: According to (28), (29), and α [0,1], q r (σ) and p r (σ) cannot have the same sign. The same result can be obtained for q i (σ) and p i (σ). Therefore, q r (σ)p r (σ) ≤ 0 and q i (σ) p i (σ) ≤ 0 which proves condition (27) [20]. By multiplying both sides of (28) by q i (σ) and (30) by q r (σ), it is easy to see that (26) is satisfied, as well. Finally, the necessity part of the theorem is proved.
To show the sufficiency part, let a v ∂C exist which satisfies (26) and (27). Therefore, there are real numbers α i ,α r [0,1] such that: Assume α ¼ max α i , α r ð Þin which α i and α r are defined, in the following: Clearly, α satisfies (28) and (29). On the other hand, αp(v)+(1 À α)q(v) can be decomposed as given below, according to Theorem 1: Since α satisfies (28) and (29), one can conclude that Therefore, it proves the sufficiency of the theorem.
Based on Theorem 2, the following method is presented to find the critical polynomials in a given polynomial segment which is noted by Method 1.

| Investigating the TS property of design subspaces
In this subsection, a theorem is presented to detect the TS property of a given subspace of the design space.
The theorem is mainly based on the Polya's approach which is frequently utilized in the control literature [24].
To completely explain the concepts of the main theorem of this subsection, the Polya's theorem which discusses the positive definiteness of homogenous polynomial matrices and Lemma 1 which discusses the robust stability of a point in the design space are presented in the following.
Theorem 3. (Polya's theorem) Sala and Arino [24]. Assume P(w) is a homogenous polynomial matrix in which w i f g n w i¼1 is the set of parameters and L is a large enough positive integer number. Then, the following statements are equivalent: i. 8w Δ n w : P w ð Þ> 0 ii. Matrix coefficients of all monomials in w 1 þ … þ w n w ð Þ L P w ð Þ are positive definite.
Polya's theorem can be very useful to check the positive definiteness of a given homogenous polynomial matrix. As stated in the theorem, the statements are equivalent only if L is sufficiently large. Indeed, if the second statement is not satisfied for a specific L, it cannot be concluded that the first statement satisfies because L may be considered larger. This fact is the main drawback of Polya's theorem because it is not known how much the parameter L should be large. Lemma 1. Lee et al. [25]. Assume f Ω is an arbitrary member of the design space. Then, the closed loop model 6 is robustly stable if and only if there exists symmetric homogenous polynomial matrix P(θ) that holds the following conditions: Note that the conditions (36) and (37) in Lemma 1 are not in LMI form which means the feasibility of these conditions cannot be investigated by convex optimization methods. The nonconvexity of these conditions can critically affect the feasibility region of the methods that apply to this problem. Thus, elementary LMI techniques are utilized to convexify the mentioned nonconvex conditions in Theorem 4.
Proof. Assume matrix G(θ, f ) is defined as follows: By congruence transformation of the matrix in (39) by G(θ, f ), the following result will be equivalently obtained: Using Schur theorem, the above inequality equals to: Therefore, the conditions are equivalent to conditions of Lemma 1 which conclude the robust stability of the model.
The feasibility of conditions (38) and (39) can be investigated by Polya's approach, the details of which are presented in Theorem 3. The degree of monomials of P(θ, f ) is considered to be finite and noted by d, in this paper.
As stated earlier in this subsection, Polya's approach cannot equivalently assess the feasibility conditions in practical applications. Furthermore, the monomials of P(θ, f) in Theorem 4 have no upper bound f. Although (38) and (39) are stated to be equivalent to the robust stability, the equivalency will be obtained if d goes to infinity. Since the monomials cannot have infinite degrees in practical application, these conditions will be conservative. Therefore, there are two sources of conservativeness in this approach which are related to the values of parameters L (in Theorem 3) and d (in Theorem 4).
Remark 1. It is apparent that the conservativeness of conditions (38) and (39) reduces whenever space φ & Ω becomes smaller. Indeed, the main idea behind the proposed design algorithm of this paper is to investigate the feasibility of conditions (38) and (39) in a small part of the design space. This fact can improve the feasibility of conditions which are used to check the TS property of each subspace of the design space.

| Investigating the TD property of design subspaces
The proposed design algorithm in this paper aims to find the feasible solution more quickly by removing the TD regions of the design space. In the following, the required theory for TD detection of subspaces is elaborated. First, the exposed edge theorem is mentioned which demonstrates that the root space of all polynomials in a given compact convex polynomial space is a subspace of the roots of its exposed edges. This result can be used to detect the TD property of a subspace of the design space.
Theorem 5. (Exposed edges) Bartlett et al. [26]. Assume p(z; β) is a parametric polynomial whose parameters belong to a compact space. Then, the boundary curve of its root space belongs to the root pace of its exposed edges.
A set of indicator points will be randomly selected (shown by N & Δ n t ) to explore the TD property of subspaces.
Proof. Assume θ is an arbitrary member of the evaluating set N satisfying conditions (43) and (44). All internal polynomials of {d (s; f,θ)j f φ} are not Schur stable regarding conditions (43) and (44) and exposed edges theorem. Hence, the existence of θ N that holds these conditions apparently completes the theorem proof.
In other words, the theorem mentions that a subspace of the design space is TD if all the polynomials on the exposed edges of the corresponding polynomial subspace are not Schur stable and there is no critical point on its exposed edges.

| The proposed algorithm
This subsection proposes the design algorithm which is the main contribution of this paper. The steps of the design algorithm can be briefly presented as follows: Design algorithm 1-Consider Ω to be a compact simplex space. To clarify the abovementioned steps, some explanatory notes are presented in the following: The design space is initialized in the first step. Notation S (i) stands for remained sub-simplexes of the primary simplex Ω that will be subsequently investigated after the i th iteration. Steps 5 and 6 determine the TS or TD properties of the current simplex s by utilizing Theorems 4 and 6, respectively. If s is TS, the algorithm returns it as the feasible solution and finalizes the iterations. If s is TD, the algorithm marks it as TD and goes to Step 3. To confirm when the algorithm should terminate, the length of the largest edge of s is used as a terminating condition for the algorithm (in step 7, lel(s) stands for the largest edge length of s). In Step 8, s is divided into two smaller simplexes via halving the largest edge of s, and these simplexes are inserted into S (i) , and the algorithm goes to Step 3. It is worth mentioning that the design algorithm will terminate based on a threshold value for the size of the obtained simplexes at each iteration.
In the following, some properties of the design algorithm are investigated involving the finite termination and special conservativeness. For this purpose, some notations should be defined involving S i ð Þ TD and S i ð Þ nonÀTD that contain the TD and non-TD simplexes of S (i) , respectively. Notice that TS and TD are not direct-opposite properties.
Theorem 7. The design algorithm will certainly terminate in up to l δ À Á n d AE Ç number of iterations where l is the largest-edge length of the primary simplex Ω, δ is the threshold value for the largest -edge length of the generated simplexes and n d is the number of design variables.
Three possible cases can happen about the termination of the algorithm: It may reach a TS simplex before the i th iteration, all generated simplexes can be detected to be TD before the i th iteration, or the generated simplexes can be divided into S nonÀTD is smaller than the threshold value δ which implies the termination of the design algorithm based on its seventh step.
Theorem 8. The design algorithm converges to a feasible solution if the primary design simplex Ω includes a feasible simplex such as s that satisfies the following conditions: where c s is the center of s.
Proof. In the worst situation, the algorithm continues up to i ¼ l δ À Á n d AE Ç number of iterations without finding any TS simplex based on Theorem 7 (l = lel(Ω)). According to Theorem 7, the largest-edge length of all simplexes of S i ð Þ nonÀTD is certainly smaller than δ. It is clear that the intersection of simplex s with all simplexes of S i ð Þ TD is empty due to the definition of TD and TS simplexes. This fact can be mathematically mentioned as follows: Using (47), simplex s entirely belongs to the union space of simplexes of S i ð Þ nonÀTD . Since the smallest-edge length of s is larger than δ and lel s ð Þ < δ for all s S i ð Þ nonÀTD , no vertex pair of s can be inside one simplex of S i ð Þ nonÀTD that results in the following relation: Assume c s lies within simplex s S i ð Þ nonÀTD that differs with simplexes s v f g v ∂ c s ð Þ . It is obvious that the distance between center and each vertex of s is surely larger than δ based on the assumption of this theorem. Due to lel s ð Þ < δ and condition (45), simplex s cannot include any boundary point of simplex s which concludes s fully belongs to s (i.e., s & s). This fact contradicts the first assumption of the proof about the failure of the design algorithm to find any TS simplex because the generated simplex s completely belongs to the feasible simplex s that should be detected to be TS. Theorem 8 states that the design algorithm will surely find at least one TS simplex if and only if there is a feasible simplex in the primary design space satisfying conditions (45) and (46). It is worth mentioning that these conditions directly depend on the threshold value δ such that the size of the feasible simplex can be forced to have an appropriate volume dependent on δ. In fact, the design algorithm finds all feasible parts of the primary space that are appropriately small while the LMI-based approaches ideally explore the largest convex subspace of the design space. This fact is one of the most important advantages of the design algorithm as compared to the previous ones.

| SIMULATION RESULTS
In this section, four simulation examples are provided to evaluate the performance of the proposed algorithm. The first two examples analyze a specific model, while in Example 3, the results of the proposed approach are compared to the results of a group of existing approaches. The final example elaborates the ability of the design algorithm to be applied to a real application. Example 1. Consider the discrete model in (1) with the following corner matrices: The proposed algorithm is used to find a proper controller. The algorithm proceeds sequentially in 21 iterations to reach a feasible solution in the design space which is co{[À10 À 10] T , [10 0] T , [0 10] T }. Figure 1 shows meshes obtained in the algorithm's procedure. The center of the feasible region shown in iteration 21 ( Figure 1C) is one of the solutions of this problem given by the following: Figure 1D shows the TD simplexes which are detected in the whole iterations of the algorithm. TD detection of the parts of the primary space will improve the convergence rate of the algorithm because by detecting them, and there is no need to search the TD simplexes at all. In this example, a set of large sub-simplexes of the design space are detected to be TD, and deleting them enhances the convergence rate of the algorithm. Example 2. This example is provided to compare the conservativeness of the proposed algorithm to a group of previous methods in a specific model. The previous methods are previous works [17,18,27].
Consider a discrete LTI system 1 with the following corner matrices:   Figure 3 shows the internal states of the closed loop model which shows that the closed loop model is stable in the mentioned situation.
Example 3. The feasibility performance of the proposed design algorithm in this paper is compared to some existing approaches. For this purpose, 100 polytopic uncertain models are randomly generated that are supposed to have different dimensions.
The proposed algorithm and some of the previous methods are evaluated to compare their feasibilities. Table 1 shows the number of feasible models for the methods and model's dimensions (50 random models are generated for each system's orders). Table 1 reveals that the design algorithm results in a less-conservative Notice that some of the previous methods focus on the optimal robust controller design problem. To fairly adopt these methods, they are evaluated by considering sufficiently large predefined optimal gains which causes their optimality conditions do not affect their results. Example 4. This example reveals the ability of the design algorithm to be applied to real applications. For this purpose, the simplified model of vertical dynamics of a helicopter is considered which is borrowed from [28,29]: where x(k) ℝ 4 consists of four internal states, namely, horizontal velocity, vertical velocity, pitch rate, and pitch angle, respectively [28]. Also, notation θ Δ 4 stands for the uncertain vector of the system. System matrices A p , B p , and C p are given from da Silva and Tarbouriech [29], as follows: , In model (53), matrix e A plays the uncertainty role of the system with the following definition [28]: It is apparent that these system matrices can be equivalently rewritten as given in Equations (3) and (4).
The primary simplex Ω & ℝ 8 is considered as follows (please note that the control problem includes 8 design variables): where I 8 is the eighth dimensional identity matrix and e 8 ℝ 8 is a column vector whose entries are 1. The design algorithm reaches a TS simplex in the above primary design space whose center is given in the following as a robust stabilizer point: T A B L E 1 Number of feasible models out of 100 in Example 3 n ¼ 2, n u ¼ 2 The obtained feasible solution is implemented in the closed loop system to evaluate its performance. The simulation results are given in Figure 4 T and all corners of the uncertain space Δ 4 : Figure 4 demonstrates that the point in (57) robustly stabilizes the closed loop model while the convergence rate is relevantly small. To increase the convergence rate of the closed loop model, the design algorithm is exploited to solve the other design problem in which the where γ > 1 imposes the convergence rate of the model. The design algorithm reaches the following feasible point for γ = 1.05.
f ¼ 0 2:5 2:5 2:5 0 0 2:5 0 ½ T ð58Þ Figure 5 shows the internal states of the closed loop model that is controlled by the controller whose parameters are given in (58). Figure 5 depicts the faster convergence of the closed loop which is controlled by (58) as compared to the previous closed loop response in Figure 4.

| CONCLUSION
This paper studied the problem of robust SOF controller for discrete LTI systems with polytopic uncertainty. The existing methods try to find the controller parameters based on LMI aspects which are conservative. This paper presents a design algorithm which is able to search any compact convex design space. The algorithm iteratively finds the feasible solution by partitioning the design space into TD, TS, or undetermined subspaces. A novel method is presented to explore the existence of critical polynomials on a polynomial segment. Then, this concept is utilized in determining the TD subspaces in the design space. These subspaces are instantly eliminated from the design space, and the algorithm changes the way to other subspaces. Through determination of TS subspaces, the algorithm is able to find a feasible solution to the design problem. If the subspace is neither TS nor TD, the algorithm halves it and continues the same procedure in each simplex.
The numerical examples show the less conservativeness and efficiency of the design parameters in comparison to the previous methods.
For the future researches, development of the direct search algorithm to solve the robust and optimal stabilization problems of polytopic time-varying uncertain discrete LTI systems is suggested. The design algorithm can be extended to the more complicated models such as time-varying uncertain systems with some minor modifications. It is also applicable for solving optimal problems such as optimal H ∞ performance. Heidar Ali Talebi received the B.Sc. degree in electrical engineering from Ferdowsi University, Mashhad, Iran, in 1988, the M.Sc. degree in electrical engineering from Tarbiat Modarres University, Tehran, Iran, in 1991, and the Ph.D. degree in electrical engineering from Concordia University, Montreal, QC, Canada, in 1997. He was a Postdoctoral Fellow and a Research Fellow at Concordia University and The University of Western Ontario. In 1999, he joined Amirkabir University of Technology, Tehran, Iran. He is currently a professor and chair in the Department of Electrical Engineering, and his research interests include control, robotics, medical robotics, fault diagnosis and recovery, intelligent systems, adaptive control, nonlinear control, and real-time systems.