Design of Switching Multiobjective Controller: A New Approach

Abstract

Design of switching 𝐻₂/𝐻_∞ output-feedback controller for discrete-time LTI systems with state-multiplicative noise is considered. The closed loop system achieves a minimum bound on the stochastic 𝐻₂ performance level, while satisfying the 𝐻_∞ performance. The proposed controller is based on a fuzzy supervisor which manages the combination of two separate 𝐻₂ and 𝐻_∞ controllers. A convex formulation of the two controllers leads to a structure which benefits from the advantages of both controllers to ensure a good performance in both the transient phase (𝐻₂ controller) and the steady phase (𝐻_∞ controller). The stability analysis uses the Lyapunov technique, inspired from switching system theory, to prove that the closed loop system with the proposed controller structure remains globally stable despite the configuration changing.

1. Introduction

Systems with stochastic nature have received much attention in the last decade, mainly in the 𝐻_∞ control theory framework. Solutions to various control and estimation problems that ensure the worst case performance bound in the 𝐻_∞ sense have been derived, in both, the continuous-time framework and the discrete-time counterpart. The modeling of parameter system uncertainties as white-noise processes in a linear setting is encountered in many areas of applications such as nuclear fission, heat transfer, population models, and immunology. In control theory, such models are encountered in gain scheduling when the scheduling parameters are corrupted with measurement noise [1–3]. Following the researches done in 1960s and 1970s, where the main issues were stability and control of continuous-time state multiplicative systems in the stochastic 𝐻₂ framework (see [4] and the references therein), researches in the last decade have focused on the 𝐻_∞ control setting. Thus, the continuous time stochastic state-multiplicative bounded real lemma (BRL) and the discrete-time counterpart were obtained. In [5], a discrete-time stochastic estimation for a guidance-motivated tracking problem was solved, and its results were shown to achieve better results than those achieved by the Kalman-filter. A parameter dependent approach for designing static output-feedback controller for linear time-invariant systems with state-multiplicative noise is introduced in [2], which achieve a minimum bound on either the stochastic 𝐻₂ or the 𝐻_∞ performance level. But previous studies paid less attention to mixed 𝐻₂/𝐻_∞ control of these systems because of their complexity [6].

Combination of different techniques to obtain the different performances is widely used today [4, 7, 8]. This method results in hybrid dynamical systems which include continuous and discrete dynamics and a mechanics (supervisor) managing the interaction between these dynamics [9].

In an actual engineering control problem, various conflicting requirements such as disturbance rejection and robustness to changing conditions and plant uncertainties have to be satisfied. General multiobjective control problems are difficult and remain mostly open up to now. By multiobjective control, we mean synthesis problems with a mix of performances. The mixed 𝐻₂/𝐻_∞ control is an important robust control method and has been studied by many researchers. The mixed 𝐻₂/𝐻_∞ control is concerned with the design of a controller that minimizes the 𝐻₂ performance of the system with respect to some input noises while it guarantees certain worst case performance with respect to other external disturbances. Compared with the sole 𝐻_∞ control, the mixed 𝐻₂/𝐻_∞ control is more attractive in engineering. Since the former is a worst-case design which tends to be conservative, whereas the later minimizes the average performance with a guaranteed worse-case performance. For the general multiobjective control problem, the usual approach is to design one controller and to force all Lyapunov matrices used to test the several design specifications to be the same. These constraints offer a computationally efficient solution to the control problems with multiple objectives and are thoroughly investigated through the previous works [10]. Designing one controller and specifying the closed-loop objectives in terms of a common Lyapunov function is the core of the Lyapunov-shaping paradigm and constitutes an important source of conservatism [10, 11].

In order to release these constraints and let several Lyapunov matrices be simultaneously considered in the Multiobjective control synthesis problem, the switching approach is advocated in this paper. Extensive benefits are obtained from this feature in order to offer a less conservative controller parameterization in the multiobjective control synthesis problems. While the new switching controllers release the constraints on the Lyapunov instrumental variable, this is obtained at the expense of designing a controller for each specification and adding a simple constraint to ensure the stability. The number of free parameters is significantly increased as compared with the available techniques. Finally, it can be easily verified that the results obtained with the standard parameterization are always encompassed by the presented switching formulations. In [12, 13], a new framework of switching controller structure for Multiobjective control (mix𝐻₂/𝐻_∞) of singular perturbation systems has been developed that has a better performance rather than conventional controller structure. In the present paper, the switching 𝐻₂ and the 𝐻_∞ static control problems for discrete-time linear systems are solved that contain stochastic white-noise parameter uncertainties in the matrices of the state-space model that describe the system. The simple design methods of [2] are applied to derive the static output-feedback gains that satisfy the prescribed 𝐻₂ and 𝐻_∞ performance criteria, separately. A fuzzy supervisor is proposed for hybrid combination of 𝐻₂ and 𝐻_∞ controllers to use their advantages and to ensure the required performances and the stability of the closed loop system.

The main contribution of the presented work is combining 𝐻₂ and 𝐻_∞ controllers using a supervisor, which manages the gradual transition from one controller to another. This method is applied to use the advantages of each controller. The gradual transition attenuates the uncontrollability and instability problems related to the abrupt switch. The control signal is obtained via a weighted sum of the two signals given by the 𝐻₂ and the 𝐻_∞ controllers. This weighted sum is managed thanks to a fuzzy supervisor, which is adapted to obtain the desired closed loop system performances by benefiting from the advantage of the 𝐻₂ in the approaching phase, minimizing the energy of impulse response and the ability of the 𝐻_∞ control to eliminate the chattering and to guarantee the system robustness. So, the 𝐻₂ mainly acts in the transient phase providing a fast dynamic response and enlarging the stability limits of the system, while the 𝐻_∞ control acts mainly in the steady state to reduce chattering and to maintain the tracking performances. Furthermore, the global stability of the system even if the system switches from one configuration to another (transient to steady state and vice versa) is guaranteed.

The structure of the paper is as follows. Section 2 presents the system definition and the controllers used. In Section 3, the fuzzy supervisor and the proposed control law are described. Stability analysis is demonstrated in Section 4. The design procedure is explained in Section 5 and an example is given to illustrate the efficiency of the proposed method, followed by conclusions in Section 6.

Notation 1. Throughout the paper, the superscript “𝑇” stands for matrix transposition, 𝑅^𝑛 denotes the 𝑛 dimensional Euclidean space, 𝑅^𝑛×𝑚 is the set of all 𝑛×𝑚 real matrices, 𝑁 is the set of natural numbers and the notation 𝑃>0, (resp., 𝑃≥0) for 𝑃∈𝑅^𝑛×𝑛 means that 𝑃 is symmetric and positive definite (resp., semipositive definite). The variables {𝜁_𝑘} and {𝜈_𝑘} are zero-mean real scalar white-noise sequences that satisfy 𝐸{𝜈_𝑘𝜈_𝑗}=𝛿_𝑘𝑗, 𝐸{𝜁_𝑘𝜁_𝑗}=𝛿_𝑘𝑗, 𝐸{𝜁_𝑘𝜈_𝑗}=𝛿_𝑘𝑗, for all 𝑘,𝑗≥0. By 𝐿²(Ω,𝑅^𝑛), the space of square-summable 𝑅^𝑛-valued functions on the probability space (Ω,𝔽,𝑃) is denoted, where Ω is the sample space, 𝔽 is a 𝜎 algebra of a subset of Ω called events, and 𝑃 is the probability measure on 𝔽. By (𝔽_𝑘)_𝑘∈𝑁, an increasing family of 𝜎-algebras 𝔽_𝑘⊂𝔽 is denoted which is generated by 𝜈_𝑗,𝜁_𝑗,𝑗≤𝑘−1. 𝑙²̃(𝑁;𝑅^𝑛) is the space of non-anticipative stochastic processes {𝑓_𝑘}={𝑓_𝑘}_{𝑘∈[0,∞]}∈𝑙²̃(𝑁;𝑅^𝑛) in 𝑅^𝑛 with respect to (𝔽_𝑘)_{𝑘∈[0,∞]} satisfying ‖‖𝑓_𝑘‖‖²𝑙²̃=𝐸^∞₀‖‖𝑓_𝑘‖‖²=^∞₀𝐸‖‖𝑓_𝑘‖‖²<∞,(1.1) where ‖⋅‖ is the standard Euclidean norm. By 𝛿_𝑖𝑗, the Kronecker delta function is denoted.

2. Problem Statement

The following linear system is considered:𝑥_𝑘+1=𝐴+𝐷𝜈_𝑘𝑦𝑥_𝑘+𝐵₁𝜔_𝑘+𝐵₂+𝐺𝜁_𝑘𝑢_𝑘,𝑥₀=0,_𝑘=𝐶₂𝑥_𝑘+𝐷₂₁𝑛_𝑘,(2.1) with the objective vector𝑧_𝑘=𝐶₁𝑥_𝑘+𝐷₁₂𝑢_𝑘,(2.2) where {𝑥_𝑘}∈𝑅^𝑛 is the system state vector, {𝜔_𝑘}∈𝑅^𝑞 is the exogenous disturbance signal, {𝑛_𝑘}∈𝑅^𝑝 is the measurement noise sequence, {𝑢_𝑘}∈𝑅^𝑙 is the control input, {𝑦_𝑘}∈𝑅^𝑚 is the measured output, and {𝑧_𝑘}∈𝑅^𝑟⊂𝑅^𝑛 is the state combination (objective function signal) to be regulated. The state-multiplicative white-noise sequences are defined in the notation subsection. The matrices in (2.1) and (2.2) are assumed to be constant matrices of appropriate dimensions.

In each stage, a constant output-feedback controller𝑢_𝑘=𝐾𝑦_𝑘(2.3) is sought to achieve a certain performance requirement. The following performance criterion is treated.

The stochastic 𝐻₂/𝐻_∞ control problem: {𝜔_𝑘},{𝑛_𝑘} are realizations of a unit variance, stationary, white-noise sequences that are uncorrelated with {𝜈_𝑘},{𝜁_𝑘}. The exogenous disturbance signal is energy bounded; the following performance index should be minimized, which is useful to handle stochastic aspects such as measurement noises or random disturbances:𝐽₂=𝐸_𝜔,𝑛‖‖𝑧^𝑘‖‖²𝑙₂̃,(2.4) while for a prescribed scalar 𝛾>0 and for all nonzero {𝜔_𝑘}∈𝑅^𝑞, {𝑛_𝑘}∈𝑅^𝑝, guarantees that 𝐽_∞<0 where 𝐽_∞=‖‖𝑧_𝑘‖‖²𝑙₂̃−𝛾²‖‖𝜔^𝑘‖‖²𝑙₂̃+‖‖𝑛_𝑘+1‖‖²𝑙₂̃.(2.5)

That is useful for disturbance rejection, reference tracking, low-energy consumption, bandwidth limitation, low steady state control error, and robust stability. 𝐻_∞ norm measures the system input-output gain for finite energy or finite rms input signals.

Interconnection of (2.1) and (2.2) is denoted by the Redheffer star product to include 𝑦_𝑘. The augmented state vector 𝜉_𝑘=col{𝑥_𝑘,𝑦_𝑘} is defined, and the following representation is obtained to the closed loop system:𝜉_𝑘+1=𝐴𝜉_𝑘+𝐵𝜔_𝑘𝑧+𝐷𝜉_𝑘𝜈_𝑘+𝐺𝜉_𝑘𝜁_𝑘,𝜉₀=0,_𝑘=𝐶𝜉_𝑘,(2.6) where𝜔_𝑘=⎡⎢⎢⎣𝜔^𝑘𝑛_𝑘+1⎤⎥⎥⎦,𝐴=⎡⎢⎢⎣𝐶𝐴₂𝐵₂𝐾𝐺=⎡⎢⎢⎣00𝐶𝐺₂𝐾𝐴𝐶₂𝐵₂𝐾⎤⎥⎥⎦,𝐷=⎡⎢⎢⎣𝐶𝐷𝐷0₂0⎤⎥⎥⎦,𝐺𝐾⎤⎥⎥⎦,𝐵=⎡⎢⎢⎣𝐵¹𝐶₂0𝐵₁𝐷₂₁⎤⎥⎥⎦,𝐶=𝐶₁𝐷₁₂𝐾.(2.7) The Lyapunov functions 𝑉_𝑖=𝜉^𝑇𝑃_𝑖𝜉, 𝑖=2,∞ are considered with:𝑃_𝑖=⎡⎢⎢⎣𝑃^𝑖−𝛽⁻¹𝑃_𝑖𝐶^𝑇2𝛽⁻¹𝐶₂𝑃𝑃_𝑖⎤⎥⎥⎦,𝑃_𝑖∈𝑅^𝑚×𝑚𝑃_𝑖∈𝑅^𝑛×𝑛,𝑃_𝑖>0.(2.8) While the parameter 𝛽 is an important positive scalar tuning parameter and𝑄_𝑖=𝑃_𝑖−1=⎡⎢⎢⎣𝑄^𝑖𝐶^𝑇2𝑄_𝑖𝑄_𝑖𝐶₂𝛽𝑄_𝑖⎤⎥⎥⎦,𝑄_𝑖∈𝑅^𝑛×𝑛𝑄_𝑖∈𝑅^𝑚×𝑚.(2.9)

Lemma 2.1 (the stochastic 𝐻₂ control). Consider systems (2.1), (2.2). The output-feedback control law (2.3) achieves a prescribed 𝐻₂-norm bound 𝛿>0, if there exist 𝑄∈𝑅^𝑛×𝑛,𝑄∈𝑅^𝑚×𝑚,𝑌∈𝑅^𝑙×𝑚, and 𝐻∈𝑅^{(𝑞+𝑝)×(𝑞+𝑝)} that, for some tuning scalar 𝛽>0, the following LMIs are satisfied: [2] Γ=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑄−𝐶^𝑇2∗−𝛽𝑄𝑄ΓΓ₁₃₂₃Γ₁₄00000∗∗−𝑄−𝐶Γ_24𝑇200000𝑄Γ₃₅𝑄𝐷^𝑇Γ₃₇𝐶^𝑇2𝑌^𝑇𝐺^𝑇Γ₃₉∗∗∗−𝛽𝑄Γ₄₅𝑄𝐶₂𝐷^𝑇Γ₄₇𝛽𝑌^𝑇𝐺^𝑇Γ₄₉∗∗∗∗∗∗∗∗−∗𝐼_𝑟−0𝑄−𝐶0_𝑇200∗∗∗∗∗∗∗∗∗∗∗∗−𝛽∗𝑄𝑄−00𝑄−𝐶00_𝑇2∗Γ<0∗,⎡⎢⎢⎢⎢⎢⎢⎣∗𝐻₁₁∗∗∗∗∗−𝛽𝑄𝑄⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(2.10)𝐻₁₂𝐵^𝑇1𝐵^𝑇1𝐶^𝑇2∗𝐻₂₂0𝐷^𝑇21∗∗∗∗𝑄∗𝐶𝛽^𝑇2𝑄𝑄⎤⎥⎥⎥⎥⎥⎥⎦>0,trace(𝐻)<𝛿²,(2.11) where 𝐻=⎡⎢⎢⎣𝐻¹¹𝐻₁₂𝐻₂₁𝐻₂₂⎤⎥⎥⎦, Γ₃₇=𝑄𝐷^𝑇𝐶^𝑇2,Γ₂₃=𝐶₂𝐴𝑄+𝐶₂𝐵₂𝑌𝐶₂−𝑄𝐶₂Γ₁₃+𝑄𝐶²,=𝐴𝑄+𝐵Γ₂𝑌₂₄𝐶₂,Γ₃₉=𝐶^𝑇2𝑌^𝑇𝐺^𝑇𝐶^𝑇2,Γ₄₇=𝑄𝐶₂𝐷^𝑇𝐶^𝑇2,Γ=𝐶₂𝐴𝐶^𝑇2+𝛽𝐶₂𝐵₂𝑌−𝛽𝑄+𝛽𝑄,₄₅Γ₁₄=𝛽𝑌^𝑇𝐷^𝑇12+𝑄𝐶₂𝐶^𝑇1,Γ₄₉=𝛽𝑌^𝑇𝐺^𝑇𝐶^𝑇2,=𝛽𝐵₂𝑌+𝐴𝐶^𝑇2𝑄,Γ₃₅=𝑄𝐶^𝑇1+𝐶^𝑇2𝑌^𝑇𝐷^𝑇12.(2.12) If a solution to the latter LMIs exists, the gain matrix 𝐾 that stabilizes the system and achieves the required performance is given by 𝐾₂=𝑌𝑄⁻¹.

Lemma 2.2 (the stochastic 𝐻_∞ problem). Consider the system of (2.1), (2.2). The control law (2.3) achieves a prescribed 𝐻_∞-norm bound 𝛾>0, if there exist, 𝑄∈𝑅^𝑛×𝑛,𝑄∈𝑅^𝑚×𝑚,𝑌∈𝑅^𝑙×𝑚 that, for some scalar 𝛽>0, the following LMI is satisfied: [2] ⎡⎢⎢⎢⎢⎢⎢⎣Γ⎡⎢⎢⎢⎢⎣𝐶𝐵₁₂0𝐵₁𝐷₂₁∗−0𝛾₂0⎤⎥⎥⎥⎥⎦𝐼_𝑞+𝑝⎤⎥⎥⎥⎥⎥⎥⎦<0,(2.13) where Γ is defined in (2.11) and 𝐾_∞=𝑌𝑄⁻¹.

3. Fuzzy Supervisor

𝐻₂ control provides a fast dynamic response, a stable control system, and a simple implementation. Conversely, this control strategy has some drawbacks that appear in the steady state. The 𝐻_∞ techniques are alternatives that can guarantee the robustness and the global stability. In order to take advantage of both controllers, 𝐻₂ during the transient time and 𝐻_∞ control during the steady state, their control actions are combined by means of a weighting factor, 𝛼∈01, representing the output of a fuzzy logic supervisor that takes the tracking error e and its time derivatives 𝑒,̇𝑒,̈…,𝑒^𝑛−1 as inputs. The global control scheme of the proposed approach is illustrated in Figure 1.

Figure 1 

The control scheme of the proposed method.

The fuzzy system is constructed from a collection of fuzzy rules whose 𝑗th component can be given in the formIf𝑒is𝐻^𝑗1And…And𝑒^𝑛−1is𝐻^𝑗𝑛Then𝛼=𝛼_𝑗,(3.1) where 𝐻^𝑗𝑖 is a fuzzy set and 𝛼_𝑗 is a singleton.

It is easy to see that it can be considered as a fuzzy rule of a Takagi-Sugeno fuzzy system. The fuzzy implication uses the product operation rule. The connective AND is implemented by means of the minimum operation, whereas fuzzy rules are combined by algebraic addition. Defuzzification is performed using the centroid method, which generates the gravity centre of the membership function of the output set. Since the membership functions that define the linguistic terms of the output variable are singletons, the output of the fuzzy system is given by𝛼=∑^𝑚𝑖=1𝛼_𝑖∏^𝑛𝑗=1𝜇^𝑗𝑖∑^𝑚𝑖=1∏^𝑛𝑗=1𝜇^𝑗𝑖,(3.2) where 𝜇^𝑗𝑖 is the degree of membership of 𝐻^𝑗𝑖 and 𝑚 is the number of fuzzy rules used.

The objective of this fuzzy supervisor is to determine the weighting factor, 𝛼, which gives the participation rate of each control signal. Indeed, when the norm of the tracking error 𝑒 and its time derivatives 𝑒,̇𝑒,̈…,𝑒^𝑛−1 are small, the plant is governed by the 𝐻_∞ controller (𝛼=1). Conversely, if the error and its derivatives are large, the plant is governed by the 𝐻₂ (𝛼=0). The control action, 𝑢, is determined by 𝑢=(1−𝛼)𝑢_𝐻2+𝛼𝑢_𝐻∞.(3.3)

Remark 3.1. In the case of a large rule base, some techniques can be employed to significantly reduce the number of rules activated at each sampled time by using the system position in the state space. Indeed, it is demonstrated that using a strict triangular partitioning allows guaranteeing that, at each sampling time, each input variable is described with two linguistic terms at the most [6]. Thus, the output generated by the fuzzy system with 𝑛 inputs is then reduced to that produced by the subsystem composed of the 2^𝑛 fired rules.

4. Stability Analysis

The theorem of Essounbouli et al. [7] is used to prove the global stability of the system governed by the control law (3.3). Similar to [7], using 𝐻₂ and 𝐻_∞ controls, this theorem is rewritten as follows.

Theorem 4.1. Consider a combined fuzzy logic control system as described in this work. If (1)there exists a positive definite, continuously differentiable, and radially unbounded scalar function 𝑉 for each subsystem;(2)every fuzzy subsystem gives a negative definite 𝑉̇ in its active region;(3)the weighted sum defuzzification method is used, such that for any output 𝑢min𝑢_𝐻2,𝑢_𝐻∞≤𝑢≤max𝑢_𝐻2,𝑢_𝐻∞.(4.1)
Then the resulting control 𝑢, given by (3.3), guarantees the global stability of the closed loop system.

Proof. Satisfying the two first conditions guarantees the existence of a Lyapunov function in the active region which is a sufficient condition for ensuring the asymptotic stability of the system during the transition from the 𝐻₂ control to the 𝐻_∞ one. Consider the Lyapunov function 𝑉₂=𝜉^𝑇𝑃₂𝜉 where 𝑃₂=𝑃 is a positive definite matrix and the solution of (2.11) and we have 𝜆_min(𝑃₂)𝜉^𝑇𝜉≤𝜉^𝑇𝑃₂𝜉, where 𝜆_min(𝑃₂) is the minimal eigenvalue of 𝑃₂. In Lemma 2.1, it was shown that the synthesized 𝐻₂ control ensures the decrease of the Lyapunov function 𝑉₂. Consider the Lyapunov function 𝑉_∞=𝜉^𝑇𝑃_∞𝜉 where 𝑃_∞=𝑃 is a positive definite matrix and the solution of (2.13) and we have 𝜉^𝑇𝑃_∞𝜉≤𝜆_max(𝑃_∞)𝜉^𝑇𝜉, where 𝜆_max(𝑃_∞) is the maximal eigenvalue of 𝑃_∞. In Lemma 2.2, it was shown that the synthesized 𝐻_∞ control ensures the decrease of the Lyapunov function 𝑉_∞.
To satisfy the second condition of the theorem, it is enough to choose 𝑃₂,𝑃_∞ such that 𝜆_max𝑃_∞≤𝜆_min𝑃₂.(4.2) This condition guarantees that in the neighborhood of the steady state (𝐻_∞ control), the value of the Lyapunov function 𝑉₂ is greater than that of 𝑉_∞. To guarantee the third condition, the balancing term 𝛼 takes its values in the interval [01]. Consequently, the three conditions of the above theorem are satisfied and the global stability of the system is guaranteed. So, The Problem formulation (switching 𝐻₂/𝐻_∞ control) will be as: minimize𝐽₂(2.11)⟶𝐾₂subjectto∶⎧⎪⎨⎪⎩𝐽^∞<0(2.13)⟶𝐾_∞𝜆_max𝑃_∞≤𝜆_min𝑃₂(4.2)(4.3)𝑃₂and𝑃_∞, which influence the stability and relation (4.2) are dependent on 𝛽. Genetic algorithm is used to find optimal 𝛽 that solve the above problem.

Remark 4.2. It should be noted that the proof of stability in this case is similar to those used for switching system theory [7, 11]. Indeed, the energy of the system corresponding to the 𝐻_∞ controller is less than that for 𝐻₂, guaranteeing the stability of the closed loop system during the transition from 𝐻₂ to 𝐻_∞. In the event of large external disturbance, which forces the system back to a transient phase, the proposed controller adjusts the weighting factor in a way that the system remains stable in the new configuration until returning to the steady state, which implies a new variation of the control signal.

5. Design Procedure

In order to minimize the online computing time of the proposed method and to simplify its real time implementation, the design procedure implies an offline processing step and an online step during control execution. In the offline step, the gains and 𝛽 are defined in order to satisfy (4.3). The supervisor design is essentially based on the available information of the process under study. Indeed, when a sufficient amount of information is available, it becomes possible to reduce the number of inputs and the fuzzy rules.

To construct the fuzzy supervisor, firstly, the fuzzy sets are defined for each input (the error and its derivatives) and output; then, the rule base is elaborated. For the online step, the error vector is computed and then is injected in the supervisor to determine the value of 𝛼 to apply the global control signal.

Example 5.1. To demonstrate the solvability of the various LMIs, simplicity and low conservatives of the proposed method, a third-order, two-output, one-input example is considered and a switching output feedback controllers is sought 𝐵𝐴=⎡⎢⎢⎢⎢⎣00..90080153200..3904824−010..53.1496885866⎤⎥⎥⎥⎥⎦,𝐷=⎡⎢⎢⎢⎢⎣000000000.4₁⎤⎥⎥⎥⎥⎦,𝐵=⎡⎢⎢⎢⎢⎣00..00010908100..000013948−000..0.0011055168⎤⎥⎥⎥⎥⎦,𝐷₁₂=⎡⎢⎢⎢⎢⎣001⎤⎥⎥⎥⎥⎦,𝐷₂₁=0,₂=⎡⎢⎢⎢⎢⎣−0−0.41.05.4617024⎤⎥⎥⎥⎥⎦,𝐶₂=⎡⎢⎢⎣100100⎤⎥⎥⎦,𝐶₁=⎡⎢⎢⎢⎢⎣100010001⎤⎥⎥⎥⎥⎦,𝐺=0.(5.1) Based on the proposed control scheme, the following results are obtained:
applying the genetic algorithm and solving (4.3) a minimum 𝐻₂-norm bound of 𝛿=0.0449 and a minimum value of 𝛾=0.8916 is obtained for 𝛽=2.4. The corresponding controllers are 𝐾₂=0.34690.6216 and 𝐾_∞=0.35671.2622.

Consider multiobjectiveminimize𝐽₂subjectto∶𝐽_∞<0for𝛾=0.8916(5.2) Solving (5.2) based on the Lyapunov-shaping paradigm that is given in [2] yields 𝛿=0.0528 as the best constrained 𝐻₂ performance, which is 17.69% higher than the optimal value 0.0449 due to the use of one controller for each objective and different Lyapunov functions. We can summarize the results in Table 1.

Table 1 

The simulation result.

According to Table 1, we can see that by a similar constraint on 𝐻_∞ norm of closed loop system, by using our proposed method 𝐻₂ norm of closed loop system will be 17.69% lower than the conventional Multiobjective method [2], that it is satisfactory.

The output response of Example 5.1 by using both conventional and our proposed switching method has been depicted in Figure 2. The solid line graph is our proposed method response, and the dash line graph is output response of conventional optimal method. As we see, it is clear that output regulation of our proposed method is much better than the conventional method, that it is satisfactory.

(a)

(b)

Figure 2 

The output response of  Example 5.1.

The fuzzy supervisor is constructed by using three fuzzy sets zero, medium, and large for the tracking error and its time derivative. The corresponding membership functions are triangular, as shown in Figure 3. For the output, five singletons are selected: very large (VL), large (L), medium (M), small (S), and zero (Z), corresponding to 1, 0.75, 0.5, 0.25, and 0, respectively. The fuzzy rule base is depicted in Figure 2. Rules are defined by a table; for example, a rule in the table can be stated as follows: “if the norm of the error is medium and the norm of the error derivative is large, then 𝛼 is zero.”

(a)

(b)

(c)

Figure 3 

The structure of the proposed fuzzy supervisor.

Results show that 𝐻₂ and the combined controller provide a fast dynamic response compared to 𝐻_∞ and that 𝐻_∞ and the combined controller provide a smooth variation of the control signal. Hence, the proposed control set-up benefits from the advantages of both 𝐻_∞ and 𝐻₂, in terms of tracking performance and the robustness to external perturbations, which is ensured by 𝐻_∞ control in the steady state (The fuzzy supervisor favors 𝐻_∞ to reach the steady state with a fast dynamic). As it is shown, the conservatism introduced by means of the proposed methodology is significantly decreased in comparison with the Lyapunov-shaping methods which oblige the designer to employ a common Lyapunov matrix for all the performance criteria and design one controller that satisfy all the objectives. The applied control signal forces the system to remain stable and attain the desired trajectory. Thus, an intermediate dynamics whose advantage is to have a compromise between the settling time and the actuator solicitations is obtained. Comparing the results shows that the proposed controller ensures a good convergence towards the desired trajectory. The conditions of Theorem 4.1. are satisfied, and the system global stability is guaranteed despite the configuration changing.

6. Conclusions

A convex programming method is presented which provides an efficient design of switching robust static output-feedback controllers for linear systems with state multiplicative noise. Sufficient conditions are derived for the existence of switching controller that stabilizes the system and achieves a prescribed bound on its performance. The stochastic 𝐻₂/𝐻_∞ performance criterion is considered.

In this work, a hybrid robust controller is developed. The main idea is the use of a fuzzy supervisor to manage efficiently the action of two controllers based on 𝐻₂ and 𝐻_∞, in a way that the system remains stable with good performance and low conservatives despite the plant switching from one mode to a new one. Furthermore, this structure allows us to take the advantage of both controllers and to efficiently eliminate their drawbacks. Simulation results showed the efficiency and the design simplicity of the proposed approach. Indeed, the 𝐻₂ provides good performances in the transient state (a fast dynamic response, enlarged stability limits of the system), while the 𝐻_∞ control acts mainly in the steady state (reduces chattering and the effect of the external disturbances). This work can be generalized to multiple controllers, more than two, managed by the same fuzzy supervisor. Indeed, the structure of the fuzzy supervisor allows partitioning the state into different substates. An adequate controller can be defined for each substate to ensure the desired performances. The rule base of the fuzzy supervisor will be reconstructed so that the premise part defines the subspace and the conclusion part defines the corresponding control law and the applied control signal will be a weighted sum of all the controllers used.