Sufficient Dilated LMI Conditions for 𝐻_∞ Static Output Feedback Robust Stabilization of Linear Continuous-Time Systems

Abstract

New sufficient dilated linear matrix inequality (LMI) conditions for the 𝐻_∞ static output feedback control problem of linear continuous-time systems with no uncertainty are proposed. The used technique easily and successfully extends to systems with polytopic uncertainties, by means of parameter-dependent Lyapunov functions (PDLFs). In order to reduce the conservatism existing in early standard LMI methods, auxiliary slack variables with even more relaxed structure are employed. It is shown that these slack variables provide additional flexibility to the solution. It is also shown, in this paper, that the proposed dilated LMI-based conditions always encompass the standard LMI-based ones. Numerical examples are given to illustrate the merits of the proposed method.

1. Introduction

The interest in static output feedback controllers lies in their conceptual simplicity and ease in practical implementation. In addition, because in practice, only partial information through measurements is available, the most common, albeit often complex, dynamic output feedback design procedure can always be reformulated, via a simple plant augmentation, as a static output problem. Paradoxically, the design of static output feedback controllers remains an open question in control theory. For instance, it still cannot be entirely solved, for instance, through convex optimization. This tool is known to lead, in this case, to only bilinear matrix inequality (BMI) conditions that are necessary and sufficient, but nonconvex and NP-hard to solve [1]. Various approaches have been proposed to deal with this issue. While some authors presented Riccati-like conditions [2–4], others proposed rank-constrained LMI conditions [5, 6] or iterative linear matrix inequality (ILMI) conditions [7–12], which, albeit, can only be numerically solved with limited efficiency. Recently, other approaches offered a convex LMI formulation of the problem providing, however, only sufficient LMI conditions, for both continuous-time and discrete-time systems [13–19]. Since then, most of the research efforts have been deployed into the direction of relaxing the conservatism of those sufficient conditions. For instance, in [14, 19], the Lyapunov variable was forced to satisfy a linear matrix equality constraint. In [17], the Lyapunov matrix was restricted to a block diagonal structure. All these approaches used the standard analysis or synthesis conditions as a basis, which are known, in general, to lead to quite conservative results. Many efforts are now being made in the direction of further reducing conservatism of standard methods through a dilatation procedure as in [11, 20–30]. More specifically, the authors in [31] derived sufficient dilated LMI condition for the existence of a robust 𝐻₂ static output feedback controller for continuous- as well as discrete-time systems. In their approach, the authors used auxiliary slack variables with structure that provided extra degree of freedom and offered more flexibility for the controller design. Their result turns out to be significantly less conservative than the results reported in [14, 17, 19]. To our knowledge, the robust 𝐻_∞ static output feedback stabilization problem for linear continuous-time system has not been addressed in the literature. The main objective of this paper is to derive sufficient dilated LMI conditions for this problem. It will be shown that the obtained conditions are naturally less conservative than early standard LMI methods. The structural restriction imposed on a Lyapunov variable is bypassed by means of the auxiliary slack variables with structure and a scalar. This extra degree of freedom has provided additional flexibility that greatly helps in solving the robust 𝐻_∞ static output feedback control problem via the parameter-dependent Lyapunov functions (PDLF). It is also shown, in this paper, that the proposed dilated LMI-based solution always encompasses the standard LMI-based one. This means that the conservatism of our method can never be worse than the one of existing in LMI standard methods.

The paper is organized as follows. In the next section, the 𝐻_∞ static output feedback stabilization problem is formulated. In Section 3, new sufficient dilated LMI-based conditions, for this problem, are proposed. These results are extended to the case of systems with polytopic uncertainties in Section 4. Section 5 gives numerical examples illustrating the advantages of the proposed method and compares it, when applicable, to the standard approaches.

Notation 1. The notation used in this paper is standard. In particular, 𝑃>0 means that the matrix 𝑃 is symmetric and positive definite. Sym{𝐴} is used to denote the expression 𝐴+𝐴^𝑇, and * is used to denote symmetric matrix blocks.

2. Problem Statement and Preliminaries

Let us consider the linear continuous-time stationary system described by the state-space equations𝑧𝑥=̇=𝐶𝐴_𝑧𝑥+𝐵_𝑤𝑤+𝐵_𝑢𝑢,𝑥𝑦+=𝐷𝐶_𝑧𝑤𝑦𝑤+𝐷_𝑧𝑢𝑢,𝑥,(2.1) where 𝑥∈𝑅^𝑛 is the plant state vector, 𝑤∈𝑅^𝑚 is the exogenous input vector, 𝑢∈𝑅^𝑞 is the controller input vector, 𝑧∈𝑅^𝑝 is the controlled output vector, and 𝑦∈𝑅^𝑟 is the measured output vector. The static output feedback control problem is to find a constant matrix gain 𝐾∈𝑅^𝑞×𝑟 defined by the control law 𝑢=𝐾𝑦,(2.2) which stabilizes system (2.1) and guarantees that the 𝐻_∞ norm of the transfer matrix from 𝑤 to 𝑧 is less than a prescribed level 𝛾. The feedback connection of the system and the controller gives a linear system whose state-space representation is𝑥𝑧̇==𝐶𝐴_clcl𝑥+𝐵_cl𝑤,𝑥+𝐷_cl𝑤,(2.3) where the matrices 𝐴_cl, 𝐵_cl, 𝐶_cl, and 𝐷_cl are given by𝐶𝐴_clcl=𝐴+𝐵_𝑢𝐾𝐶_𝑦,𝐵_cl=𝐵_𝑤,=𝐶_𝑧+𝐷_𝑧𝑢𝐾𝐶_𝑦,𝐷_cl=𝐷_𝑧𝑤.(2.4) The closed-loop transfer matrix from 𝑤 to 𝑧, 𝐺_𝑤𝑧(𝑠), is then given by 𝐺_𝑤𝑧(𝑠)=𝐶_cl𝑠𝐼−𝐴_cl⁻¹𝐵_cl+𝐷_cl.(2.5) From the bounded real lemma [32, 33], system (2.1) is stabilizable via the static output feedback controller (2.2), and ‖𝐺_𝑤𝑧(𝑠)‖_∞<𝛾 if and only if there exist a symmetric and definite positive matrix 𝑋∈𝑅^𝑛×𝑛 and a general matrix 𝐾∈𝑅^𝑞×𝑟 such that⎡⎢⎢⎢⎢⎢⎣Sym𝐴_cl𝑋𝐵_cl𝑋𝐶^𝑇cl∗−𝛾𝐼_𝑚×𝑚𝐷^𝑇cl∗∗−𝛾𝐼_𝑝×𝑝⎤⎥⎥⎥⎥⎥⎦<0.(2.6) Note that despite the fact that 𝐴_cl and 𝐶_cl are both affine in 𝐾, as it is shown in (2.4), the matrix inequality (2.6) is a BMI in the variables 𝑋 and 𝐾, that is, non convex and NP-hard to solve [1]. In order to circumvent this difficulty, many authors searched, instead, LMI conditions which are only sufficient, hence gaining tractability at the expense of some more or less conservatism. For the incoming developments, a particular linear transformation is needed for system (2.1) by means of nonsingular state coordinate transformation matrix 𝑇∈𝑅^𝑛×𝑛 insuring that 𝐶_𝑦=𝐶_𝑦𝑇⁻¹=[𝐼_𝑟×𝑟0_{𝑟×(𝑛−𝑟)}] in case 𝐶_𝑦 is a full row-rank matrix, and thus leading to the transformed generalized plant representation⎡⎢⎢⎢⎢⎣𝐴𝐵_𝑤𝐵_𝑢𝐶_𝑧𝐷_𝑧𝑤𝐷_𝑧𝑢𝐶_𝑦⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣𝑇𝐴𝑇⁻¹𝑇𝐵_𝑤𝑇𝐵_𝑢𝐶_𝑧𝑇⁻¹𝐷_𝑧𝑤𝐷_𝑧𝑢𝐶_𝑦𝑇⁻¹⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣𝑇𝐴𝑇⁻¹𝑇𝐵_𝑤𝑇𝐵_𝑢𝐶_𝑧𝑇⁻¹𝐷_𝑧𝑤𝐷_𝑧𝑢𝐼_𝑟×𝑟0_{𝑟×(𝑛−𝑟)}⎤⎥⎥⎥⎥⎦.(2.7) The nonsingular state coordinate transformation matrix 𝑇 is usually taken as in [31] to be 𝑇=[𝐶^𝑇𝑦(𝐶_𝑦𝐶^𝑇𝑦)⁻¹𝐶^⟂𝑦]⁻¹, with 𝐶^⟂𝑦 denoting an orthogonal basis for the null space of 𝐶_𝑦.

The following lemma [17], for instance, proposed the following sufficient standard LMI conditions for the 𝐻_∞ static output feedback control problem.

Lemma 2.1. Assume that 𝐶_𝑦 is a full row-rank matrix. System (2.1) is stabilizable by the static output feedback controller (2.2) and ‖𝐺_𝑤𝑧(𝑠)‖_∞<𝛾 if there exist a symmetric, structured, and positive definite matrix 𝑋=diag(𝑋₁,𝑋₂) with 𝑋₁∈𝑅^𝑟×𝑟and 𝑋₂∈𝑅^{(𝑛−𝑟)×(𝑛−𝑟)} and a structured matrix 𝑈=⌊𝑈₁0_{𝑞×(𝑛−𝑟)}⌋ with 𝑈₁∈𝑅^𝑞×𝑟 such that ⎡⎢⎢⎢⎢⎣Sym𝐴𝑋+𝐵_𝑢𝑈𝐵_𝑤𝑋𝐶^𝑇𝑧+𝑈^𝑇𝐷^𝑇𝑧𝑢∗−𝛾𝐼_𝑚×𝑚𝐷^𝑇𝑧𝑤∗∗−𝛾𝐼_𝑝×𝑝⎤⎥⎥⎥⎥⎦<0.(2.8) Furthermore, a static output feedback controller gain is given by 𝐾=𝑈₁𝑋¹⁻¹.

Remarks 1. (i) Lemma 2.1 is only applicable when matrix 𝐶_𝑦 has a full row rank. A dual version of this lemma is readily obtained when matrix 𝐶_𝑦 is row-rank deficient, but instead, when matrix 𝐵_𝑢 has a full column rank.
(ii) Because of the structure imposed on the Lyapunov variable 𝑋, a great deal of conservatism is known to limit the usefulness of Lemma 2.1.

3. 𝐻_∞ Static Output Controller Synthesis

In order to reduce the conservatism encountered in the standard approach, in the following lemma, new sufficient dilated LMI-based conditions are now proposed.

Lemma 3.1. Assume that 𝐶_𝑦 is a full row-rank matrix. System (2.1) is stabilizable by the static output feedback controller (2.2) and ‖𝐺_𝑤𝑧(𝑠)‖_∞<𝛾 if for some positive scalar α, there exist a symmetric definite positive matrix 𝑌∈𝑅^𝑛×𝑛 and structured matrices 𝑍=^{𝑍10𝑟×(𝑛−𝑟)}_𝑍2𝑍3 with 𝑍₁∈𝑅^𝑟×𝑟, 𝑍₂∈𝑅^{(𝑛−𝑟)×𝑟}, and 𝑍₃∈𝑅^{(𝑛−𝑟)×(𝑛−𝑟)} and 𝐿=⌊𝐿₁0_{𝑞×(𝑛−𝑟)}⌋ with 𝐿₁∈𝑅^𝑞×𝑟 such that ⎡⎢⎢⎢⎢⎢⎢⎣𝛼Sym𝐴𝑍+𝐵_𝑢𝐿𝐵_𝑤𝛼𝑍^𝑇𝐶^𝑇𝑧+𝐿^𝑇𝐷^𝑇𝑧𝑢𝑌+𝐴𝑍+𝐵_𝑢𝐿−𝛼𝑍^𝑇∗−𝛾𝐼_𝑚×𝑚𝐷^𝑇𝑧𝑤0_𝑚×𝑛∗∗∗∗−𝛾∗𝐼_𝑝×𝑝𝐶−_𝑧𝑍+𝐷_𝑧𝑢𝐿Sym{𝑍}⎤⎥⎥⎥⎥⎥⎥⎦<0.(3.1) Furthermore, a static output feedback controller gain is given by 𝐾=𝐿₁𝑍¹⁻¹.

Proof. Assume that, for some positive scalar 𝛼, a solution with variables 𝑌, 𝑍=^{𝑍10𝑟×(𝑛−𝑟)}_𝑍2𝑍3 and 𝐿=⌊𝐿₁0_{𝑞×(𝑛−𝑟)}⌋ of inequality (3.1) exists. Substituting the expressions for 𝐴, 𝐵_𝑤, 𝐵_𝑢, 𝐶_𝑧, 𝐷_𝑧𝑤, and 𝐷_𝑧𝑢 into this inequality, defining 𝑍=𝑇⁻¹𝑍𝑇^−𝑇 and 𝑌=𝑇⁻¹𝑌𝑇^−𝑇, it comes that ⎡⎢⎢⎢⎢⎢⎢⎢⎣𝛼Sym𝑇𝐴𝑍+∗𝐵_𝑢𝐿𝑇^−𝑇𝑇^𝑇−𝑇𝛾𝐼𝐵^𝑤_𝑚×𝑚𝛼𝑇𝑍^𝑇𝐶^𝑇𝑧+𝑇⁻¹𝐿^𝑇𝐷^𝑇𝑧𝑢𝒜𝐷^𝑇𝑧𝑤0_𝑚×𝑛∗∗−𝛾𝐼_𝑝×𝑝𝐶^𝑧𝑍+𝐷_𝑧𝑢𝐿𝑇^−𝑇𝑇^𝑇∗∗∗−Sym𝑇𝑍𝑇^𝑇⎤⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.2) where 𝒜 denotes 𝑇(𝑌+𝐴𝑍+𝐵_𝑢𝐿𝑇^−𝑇−𝛼𝑍^𝑇)𝑇^𝑇. In view of the expressions of 𝐿 and 𝐾 in Lemma 3.1, the term𝐿𝑇^−𝑇, in this inequality, can be developed into 𝐿𝑇^−𝑇=𝐿₁0_{𝑞×(𝑛−𝑟)}𝑇^−𝑇=𝐾𝐼_𝑟×𝑟0_{𝑟×(𝑛−𝑟)}⎡⎢⎢⎣𝑍¹0_{𝑟×(𝑛−𝑟)}𝑍₂𝑍₃⎤⎥⎥⎦𝑇_−𝑇=𝐾𝐶_𝑦𝑇⁻¹⎡⎢⎢⎣𝑍₁0_{𝑟×(𝑛−𝑟)}𝑍₂𝑍₃⎤⎥⎥⎦𝑇_−𝑇=𝐾𝐶_𝑦𝑍.(3.3) Now, using (2.4) leads to ⎡⎢⎢⎢⎢⎢⎢⎢⎣𝛼Sym𝑇𝐴^cl𝑍𝑇^𝑇𝑇𝐵^cl𝛼𝑇𝑍^𝑇𝐶^𝑇cl𝑇𝑌+𝐴_cl𝑍−𝛼𝑍^𝑇𝑇^𝑇∗−𝛾𝐼_𝑚×𝑚𝐷^𝑇cl0_𝑚×𝑛∗∗−𝛾𝐼_𝑝×𝑝𝐶_cl𝑍𝑇^𝑇∗∗∗−Sym𝑇𝑍𝑇^𝑇⎤⎥⎥⎥⎥⎥⎥⎥⎦<0.(3.4) Now, pre- and postmultiplying this inequality by ^{𝑇−10𝑛×𝑚0𝑛×𝑝𝐴cl𝑇−10𝑚×𝑛𝐼𝑚×𝑚0𝑚×𝑝0𝑚×𝑛}_{0𝑝×𝑛0𝑝×𝑚𝐼𝑝×𝑝𝐶cl𝑇}⁻¹ and its transposed, respectively, lead to Inequality (2.6). This completes the proof.

In this lemma, the Lyapunov variable 𝑌 has been separated from the controller variable 𝐾. Hence, there is no need to impose any structure upon the Lyapunov variable as in Lemma 2.1. The structural restriction is, instead, imposed on the introduced auxiliary slack variables 𝑍. These slack variables, along the scalar 𝛼, also provide additional degrees of freedom, hence, possibly reducing conservativeness. The following theorem proves that the degree of conservatism can, indeed, be almost always reduced.

Theorem 3.2. If the standard LMI conditions of Lemma 2.1 are satisfied and achieve an upper bound 𝛾^𝑆, then the dilated inequality conditions of Lemma 3.1 are satisfied with an upper bound 𝛾^𝐷≤𝛾^𝑆.

Proof. Suppose that the matrix inequality conditions of Lemma 2.1 are satisfied and achieve an upper bound 𝛾^𝑆 with the variables 𝑋 and 𝑈. In the right-hand side of the matrix inequality conditions of Lemma 3.1, if we let 𝑌=𝑋, 𝑍=𝛼⁻¹𝑋, and 𝐿=𝛼⁻¹𝑈, with 𝛼>0, this right-hand side becomes ⎡⎢⎢⎢⎢⎢⎢⎢⎣Sym𝐴𝑋∗+𝐵_𝑢𝑈−𝛾𝐵𝐼_{𝑤𝑚×𝑚}𝑋𝐶^𝑇𝑧+𝑈^𝑇𝐷^𝑇𝑧𝑢𝛼⁻¹𝐴𝑋+𝐵_𝑢𝑈𝐷^𝑇𝑧𝑤0_𝑚×𝑛∗∗∗∗−𝛾∗𝐼_𝑝×𝑝𝛼⁻¹𝐶−_𝑧2𝑋𝛼+₋₁𝐷_𝑧𝑢𝑈𝑋⎤⎥⎥⎥⎥⎥⎥⎥⎦.(3.5) By virtue of the Schur complement, this matrix will be negative definite if and only if 𝑋>0 and ⎡⎢⎢⎢⎢⎣Sym𝐴𝑋+𝐵_𝑢𝑈𝐵_𝑤𝑋𝐶^𝑇𝑧+𝑈^𝑇𝐷^𝑇𝑧𝑢∗−𝛾𝐼_𝑚×𝑚𝐷^𝑇𝑧𝑤∗∗−𝛾𝐼_𝑝×𝑝+0.5×𝛼₋₁⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝐴𝑋0+𝐵_{𝑢𝑚×𝑛}𝑈𝐶_𝑧𝑋+𝐷_𝑧𝑢𝑈⎤⎥⎥⎥⎥⎦𝑋⁻¹⎡⎢⎢⎢⎢⎣𝐴𝑋0+𝐵_{𝑢𝑚×𝑛}𝑈𝐶_𝑧𝑋+𝐷_𝑧𝑢𝑈⎤⎥⎥⎥⎥⎦^𝑇<0.(3.6) As the first term in this matrix inequality is no other than the standard 𝐻_∞ conditions in Lemma 2.1 and is therefore negative definite, there always exists a sufficiently large scalar 𝛼>0 which achieves this condition. This proves that the dilated LMI 𝐻_∞ conditions in Lemma 3.1 always encompass the standard ones. Clearly, this means that the dilated-based approach yields upper bounds that are always such that 𝛾^𝐷≤𝛾^𝑆.

This theorem proves that the new dilated LMI conditions of Lemma 3.1 for the 𝐻_∞ static output controller synthesis do encompass the standard LMI-based ones in Lemma 1 in [15] when 𝑌=𝑋, 𝑍=𝛼⁻¹𝑋, 𝐿=𝛼⁻¹𝑈, and 𝛼∈[𝛼_min,+∞] (𝛼_min being the minimum value which satisfies the inequality condition (3.6)). When these parameters are set free in the new dilated LMI conditions of Lemma 3.1, it is likely that they actually reduce conservatism when compared to the standard LMI “undilated” counterparts. The best reduction in conservatism can be easily obtained through a simple line search of the scalar 𝛼.

4. Extension to Uncertain Systems

In this section, an extension to uncertain systems with polytopic uncertainties is made. To this end, let us consider the linear continuous-time uncertain system described by the following state space equations:𝑧𝑥=̇=𝐂𝐀_𝐳𝑥+𝐁_𝐰𝑤+𝐁_𝐮𝑢,𝑥𝑦+=𝐃𝐂_𝐳𝐰𝐲𝑤+𝐃_𝐳𝐮𝑢,𝑥,(4.1) where 𝐀, 𝐁_𝐰, 𝐁_𝐮, 𝐂_𝐳, 𝐂_𝐲, 𝐃_𝐳𝐰, and 𝐃_𝐳𝐮 are not precisely known but belong to a polytopic uncertainty domain Ω defined by Ω=⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎢⎢⎣𝐀𝐁_𝐰𝐁_𝐮𝐂_𝐳𝐃_𝐳𝐰𝐃_𝐳𝐮𝐂_𝐲⎤⎥⎥⎥⎥⎦=_𝑁_𝑖=1𝜃_𝑖⎡⎢⎢⎢⎢⎣𝐴_𝑖𝐵_𝑤𝑖𝐵_𝑢𝑖𝐶_𝑧𝑖𝐷_𝑧𝑤𝑖𝐷_𝑧𝑢𝑖𝐶_𝑦𝑖⎤⎥⎥⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭,(4.2) where 𝜃_𝑖≥0 and ∑^𝑁𝑖=1𝜃_𝑖=1.

With the static output feedback controller given by (2.2), the closed-loop transfer matrix from 𝑤 to 𝑧, 𝐆_𝐰𝐳(𝑠,𝜃), is given by𝐆_𝐰𝐳(𝑠,𝜃)=𝐂_𝐜𝐥𝑠𝐼−𝐀_𝐜𝐥⁻¹𝐁_𝐜𝐥+𝐃_𝐜𝐥,(4.3) where the matrices 𝐀_𝐜𝐥, 𝐁_𝐜𝐥, 𝐂_𝐜𝐥, and 𝐃_𝐜𝐥 are given by𝐂𝐀_{𝐜𝐥𝐜𝐥}=𝐀+𝐁_𝐮𝐊𝐂_𝐲,𝐁_𝐜𝐥=𝐁_𝐰,=𝐂_𝐳+𝐃_𝐳𝐮𝐊𝐂_𝐲,𝐃_𝐜𝐥=𝐃_𝐳𝐰.(4.4) In view of Lemma 2.1, it is straightforward to derive sufficient conditions for the existence of a static output feedback controller as in (2.2) which robustly stabilizes system (4.1) solely in case of 𝐂_𝑦 is certain and, in the same time, insures that a ‖𝐆_𝐰𝐳(𝑠,𝜃)‖_∞<𝛾.

Lemma 4.1. Assume that 𝐂_𝑦=𝐶_𝑦 is a full-row rank fixed matrix. The system (4.1) is robustly stabilizable by the static output feedback controller (2.2) and ‖𝐆_𝐰𝐳(𝑠,𝜃)‖_∞<𝛾 if there exist a symmetric structured matrix 𝑋=diag(𝑋₁,𝑋₂) with 𝑋₁∈𝑅^𝑟×𝑟and 𝑋₂∈𝑅^{(𝑛−𝑟)×(𝑛−𝑟)} and a structured matrix 𝑈=⌊𝑈₁0_{𝑞×(𝑛−𝑟)}⌋ with 𝑈₁∈𝑅^𝑞×𝑟 such that, for 𝑖=1,…,𝑁, ⎡⎢⎢⎢⎢⎣Sym𝐴_𝑖𝑋+𝐵_𝑢𝑖𝑈𝐵_𝑤𝑖𝑋𝐶^𝑇𝑧𝑖+𝑈^𝑇𝐷^{𝑇𝑧𝑢𝑖}∗−𝛾𝐼_𝑚×𝑚𝐷^{𝑇𝑧𝑤𝑖}∗∗−𝛾𝐼_𝑝×𝑝⎤⎥⎥⎥⎥⎦<0,(4.5) where a single nonsingular state coordinate transformation matrix 𝑇 is used for all the system vertices satisfying 𝐶_𝑦=𝐶_𝑦𝑇⁻¹=[𝐼_𝑟×𝑟0_{𝑟×(𝑛−𝑟)}], as 𝐂_𝑦 is certain. Furthermore, the static output feedback controller gain is given by 𝐾=𝑈₁𝑋¹⁻¹.

This lemma provides sufficient standard LMI-based conditions that are known to be, in general, overly conservative. This conservatism is due to the fact that not only common Lyapunov matrix is used for all the vertices, but, in addition, some structure was also imposed on this matrix. The following lemma provides, by means of parameter-dependent Lyapunov functions, new sufficient conditions for the existence of a robust static output feedback controller, as in (2.2), which robustly stabilizes system (4.1) with the matrix 𝐂_𝑦 allowed to be uncertain and, at the same time, insures that ‖𝐆_𝐰𝐳(𝑠,𝜃)‖_∞<𝛾.

Lemma 4.2. Assume that, for 𝑖=1,…,𝑁, matrix 𝐶_𝑦𝑖 has a full-row rank. The uncertain system (4.1) is robustly stable by a static output feedback controller and ‖𝐆_𝐰𝐳(𝑠,𝜃)‖_∞<𝛾 if, for some positive scalar 𝛼 and for 1≤𝑖, 𝑗≤𝑁, there exist symmetric definite positive matrices 𝑌_𝑗∈𝑅^𝑛×𝑛 and structured matrices 𝑍_𝑖=^{𝑍10𝑟×(𝑛−𝑟)}_{𝑍2𝑖𝑍3𝑖} with 𝑍₁∈𝑅^𝑟×𝑟, 𝑍_2𝑖∈𝑅^{(𝑛−𝑟)×𝑟}, and 𝑍_3𝑖∈𝑅^{(𝑛−𝑟)×(𝑛−𝑟)} and 𝐿=⌊𝐿₁0_{𝑞×(𝑛−𝑟)}⌋ with 𝐿₁∈𝑅^𝑞×𝑟 such that ⎡⎢⎢⎢⎢⎢⎢⎣𝛼Sym𝐴_𝑖𝑍_𝑖+𝐵_𝑢𝑖𝑗𝐿𝐵_𝑤𝑖𝛼𝑍^𝑇𝑖𝐶^𝑇𝑧𝑖+𝐿^𝑇𝐷^{𝑇𝑧𝑢𝑗}𝑇^𝑖𝑌_𝑗𝑇^𝑇𝑖+𝐴_𝑖𝑍_𝑖+𝐵_𝑢𝑖𝑗𝐿−𝛼𝑍^𝑇𝑖∗−𝛾𝐼_𝑚×𝑚𝐷^{𝑇𝑧𝑤𝑖}0_𝑚×𝑛∗∗∗∗−𝛾∗𝐼_𝑝×𝑝𝐶−_𝑧𝑖𝑍_𝑖+𝐷_𝑧𝑢𝑗𝐿Sym𝑍_𝑖⎤⎥⎥⎥⎥⎥⎥⎦<0,(4.6) where a different nonsingular state coordinate transformation matrix 𝑇_𝑖 is used for each of the system vertices satisfying 𝐶_𝑦𝑖=𝐶_𝑦𝑖𝑇^𝑖−1=[𝐼_𝑟×𝑟0_{𝑟×(𝑛−𝑟)}] and where 𝐵_𝑢𝑖𝑗=𝑇_𝑖𝐵_𝑢𝑗. Furthermore, a static output feedback controller gain is given by 𝐾=𝐿₁𝑍¹⁻¹.

Proof. Suppose that, for some scalar 𝛼>0, a solution to inequalities (4.6) exists with variables 𝑍_𝑖=^{𝑍10𝑟×(𝑛−𝑟)}_{𝑍2𝑖𝑍3𝑖}, 𝑌_𝑗, and 𝐿=⌊𝐿₁0_{𝑞×(𝑛−𝑟)}⌋. Substituting 𝐴_𝑖, 𝐵_𝑤𝑖, 𝐵_𝑢𝑖𝑗, 𝐶_𝑧𝑖, 𝐷_𝑧𝑤𝑖, and 𝐷_𝑧𝑢𝑗 into these inequalities, defining 𝑍_𝑖=𝑇^𝑖−1𝑍_𝑖𝑇^𝑖−𝑇 and using the fact that the term 𝐿𝑇^𝑖−𝑇, in this inequality, can be developed into 𝐿𝑇^𝑖−𝑇=𝐿₁0_{𝑞×(𝑛−𝑟)}𝑇^𝑖−𝑇=𝐾𝐼_𝑟×𝑟0_{𝑟×(𝑛−𝑟)}⎡⎢⎢⎣𝑍¹0_{𝑟×(𝑛−𝑟)}𝑍_2𝑗𝑍_3𝑗⎤⎥⎥⎦𝑇_𝑖−𝑇=𝐾𝐶_𝑦𝑖𝑇^𝑖−1⎡⎢⎢⎣𝑍₁0_{𝑟×(𝑛−𝑟)}𝑍_2𝑗𝑍_3𝑗⎤⎥⎥⎦𝑇_𝑖−𝑇=𝐾𝐶_𝑦𝑖𝑍_𝑖,(4.7) it comes that ⎡⎢⎢⎢⎢⎢⎢⎢⎣𝛼Sym𝑇^𝑖𝐴_𝑖+∗𝐵_𝑢𝑗𝐾𝐶_𝑦𝑖𝑍_𝑖𝑇^𝑇𝑖−𝑇𝛾_𝑖𝐼𝐵_{𝑤𝑖𝑚×𝑚}𝛼𝑇_𝑖𝑍^𝑇𝑖𝐶^𝑇𝑧𝑖+𝐶^𝑇𝑦𝑖𝐾^𝑇𝐷^{𝑇𝑧𝑢𝑗}ℬ𝐷^{𝑇𝑧𝑤𝑖}0_𝑚×𝑛∗∗∗∗−𝛾∗𝐼_𝑝×𝑝−𝒞Sym𝑇^𝑖𝑍_𝑖𝑇^𝑇𝑖⎤⎥⎥⎥⎥⎥⎥⎥⎦<0,(4.8)where ℬ denotes 𝑇_𝑖(𝑌_𝑗+(𝐴_𝑖+𝐵_𝑢𝑗𝐾𝐶_𝑦𝑖)𝑍_𝑖−𝛼𝑍^𝑇𝑖)𝑇^𝑇𝑖 and 𝒞 denotes (𝐶_𝑧𝑖+𝐷_𝑧𝑢𝑗𝐾𝐶_𝑦𝑖)𝑍_𝑖𝑇^𝑇𝑖.
Now, pre- and postmultiplying this inequality by ⎡⎢⎢⎣^{𝑇𝑖−10𝑛×𝑚0𝑛×𝑝−𝛼𝑇𝑖−1}_{0𝑚×𝑛𝐼𝑚×𝑚0𝑚×𝑝0𝑚×𝑛0𝑝×𝑛0𝑝×𝑚𝐼𝑝×𝑝0𝑝×𝑛0𝑛×𝑛0𝑛×𝑚0𝑛×𝑝𝑍𝑖−𝑇𝑇𝑖−1}⎤⎥⎥⎦ and its transposed, respectively, it becomes equivalent to ⎡⎢⎢⎢⎢⎢⎢⎣−2𝛼𝑌_𝑗𝐵_𝑤𝑖0_𝑛×𝑝𝑌_𝑗𝑍^𝑖−1+𝐴_𝑖+𝐵_𝑢𝑗𝐾𝐶_𝑦𝑖+𝛼𝐼_𝑛×𝑛∗−𝛾𝐼_𝑚×𝑚𝐷^{𝑇𝑧𝑤𝑖}0_𝑚×𝑛∗∗−𝛾𝐼_𝑝×𝑝𝐶_𝑧𝑖+𝐷_𝑧𝑢𝑗𝐾𝐶_𝑦𝑖∗∗∗−Sym𝑍^𝑖−1⎤⎥⎥⎥⎥⎥⎥⎦<0.(4.9) Multiplying this inequality by 𝜃_𝑖𝜃_𝑗 and summing up lead to ⎡⎢⎢⎢⎢⎢⎢⎣−2𝛼𝐘𝐁_𝐰0_𝑛×𝑝𝐘𝐙⁻¹+𝐀+𝐁_𝐮𝐾𝐂_𝑦+𝛼𝐼_𝑛×𝑛∗−𝛾𝐼_𝑚×𝑚𝐃^𝑇𝐳𝐰0_𝑚×𝑛∗∗−𝛾𝐼_𝑝×𝑝𝐂_𝐳+𝐃_𝐳𝐮𝐾𝐂_𝑦∗∗∗−Sym𝐙⁻¹⎤⎥⎥⎥⎥⎥⎥⎦<0,(4.10) where 𝐘=∑^𝑁𝑗=1𝜃_𝑗𝑌_𝑗 and 𝐙⁻¹=∑^𝑁𝑖=1𝜃_𝑖𝑍^𝑖−1. This obviously implies that 𝐘>0. On the other hand, pre- and postmultiplying this inequality by ^{𝐼𝑛×𝑛0𝑛×𝑚0𝑛×𝑝(𝐀𝐜𝐥+𝛼𝐼)𝐙𝑇0𝑚×𝑛𝐼𝑚×𝑚0𝑚×𝑝0𝑚×𝑛}_{0𝑝×𝑛0𝑝×𝑚𝐼𝑝×𝑝𝐂𝐜𝐥𝐙}^𝑇 and its transposed, respectively, lead to ^{Sym{𝐀𝐜𝐥𝐘}𝐁𝐜𝐥𝐘𝐂𝐓𝐜𝐥∗−𝛾𝐼𝑚×𝑚𝐃𝐓𝐜𝐥}_{∗∗−𝛾𝐼𝑝×𝑝}, which completes the proof.

Similarly to the certain case, in this lemma, the Lyapunov variable has been separated from the controller parameter. This has permitted the use of parameter-dependent Lyapunov functions (PDLFs) and, hence, provided the opportunity to possibly lessen conservativeness. In addition, contrarily to many results reported in the literature (as in, for instance, [14, 17]), the output matrix 𝐂_𝑦 is allowed to be uncertain. This constitutes an important contribution of this paper. It is also important to note that when the output matrix remains certain (fixed), our proposed LMI conditions always recover the standard LMI conditions. A numerical example is given to support this claim.

5. Numerical Examples

In this section, numerical examples are presented to illustrate the merit of the proposed 𝐻_∞ static output feedback synthesis method, both in the cases of with or without uncertainties. When applicable, a comparison with the standard conditions, for instance the ones reported in [17], is then made. In all the following examples, we need to design an 𝐻_∞ static output feedback controller which not only stabilizes the considered system (certain or uncertain), but also minimizes the 𝐻_∞ norm of the closed-loop transfer function.

Example 5.1. Take the continuous-time, unstable, and certain system ⎡⎢⎢⎢⎢⎣𝐴𝐵_𝑤𝐵_𝑢𝐶_𝑧𝐷_𝑧𝑤𝐷_𝑧𝑢𝐶_𝑦⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣0.1020−0.2111200.3−1200001100110110⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(5.1) The simulation results for an 𝐻_∞ static output feedback controller synthesis are given in Table 1. Clearly, for this example, the proposed dilated LMI conditions of Lemma 3.1 give a very significant improvement (around 30%), when compared with the classical standard conditions of Lemma 1 in [17].

Table 1 

Controller gain and the optimal 𝐻_∞ guaranteed cost.

Figure 1 depicts the effect of varying the scalar parameter 𝛼, involved in Lemma 3.1, upon the performance level 𝛾. Clearly, from this figure, it appears that for small values of this parameter 𝛼, no recovery is achieved, and the performance level remains above the standard level. When 𝛼 is greater than a certain threshold, a better than standard performance level is achieved culminating at a unique minimum, before tending to the standard level for greater values of 𝛼. This seems to be a general trend as the coming examples will show.

Figure 1 

The relation between the performance level 𝛾 and the scalar 𝛼.

Example 5.2. Consider the continuous-time system with polytopic uncertainties of two vertices defined by ⎡⎢⎢⎢⎢⎣𝐴₁𝐵_𝑤1𝐵_𝑢1𝐶_𝑧1𝐷_𝑧𝑤1𝐷_𝑧𝑢1𝐶_𝑦1⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎣210−41110114121⎤⎥⎥⎥⎥⎥⎥⎦,⎡⎢⎢⎢⎢⎣𝐴₂𝐵_𝑤2𝐵_𝑢2𝐶_𝑧2𝐷_𝑧𝑤2𝐷_𝑧𝑢2𝐶_𝑦2⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎣1−10−51110121321⎤⎥⎥⎥⎥⎥⎥⎦.(5.2) In this case, the output matrix is fixed, that is, supposed to be certain. The simulation results are given in Table 2. Clearly, for this example, the proposed dilated LMI condition gives, for a given realization, a very significant improvement (around 25%), when compared with the classical standard conditions of Lemma 3 in [17].

Table 2 

Controller gain and the optimal 𝐻_∞ guaranteed cost.

Figure 2 depicts the effect of varying the scalar parameter 𝛼 upon the performance level 𝛾. The general trend suspected in Example 5.1 (Figure 1) seems to be confirmed.

Figure 2 

The relation between the performance level 𝛾 and the scalar 𝛼.

Example 5.3. Consider the continuous-time system with polytopic uncertainties defined by ⎡⎢⎢⎢⎢⎣𝐴₁𝐵_𝑤1𝐵_𝑢1𝐶_𝑧1𝐷_𝑧𝑤1𝐷_𝑧𝑢1𝐶_𝑦1⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎣210−11110122110⎤⎥⎥⎥⎥⎥⎥⎦,⎡⎢⎢⎢⎢⎣𝐴₂𝐵_𝑤2𝐵_𝑢2𝐶_𝑧2𝐷_𝑧𝑤2𝐷_𝑧𝑢2𝐶_𝑦2⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎣2−10−311100.511221⎤⎥⎥⎥⎥⎥⎥⎦,(5.3) in which the output matrix is uncertain. The application of Lemma 4.2 and a line search of the scalar 𝛼 yielded, for the realization (𝜃₁,𝜃₂)=(0.4,0.6), the guaranteed 𝐻_∞ performance 𝛾=5.08, for 𝛼=4.2, a controller gain 𝐾=−4.4266 and an actual performance level ‖𝐆_𝐰𝐳(𝑠,𝜃)‖_∞=1.52.

6. Conclusion

In this paper, we have presented new sufficient dilated LMI conditions for solving 𝐻_∞ static output feedback control problems for continuous-time systems. Auxiliary slack variables with even more relaxed structure are employed in order to provide additional flexibility in the design. The method easily and successfully extends to the case of systems with polytopic uncertainties by means of parameter-dependent Lyapunov functions. It is shown that the proposed dilated LMI-based conditions always encompass the standard LMI-based ones. The consequence is a significant reduction of conservatism. The numerical examples have supported all these claims.