Static Output-Feedback Control for Vehicle Suspensions: A Single-Step Linear Matrix Inequality Approach

Abstract

In this paper, a new strategy to design static output-feedback controllers for a class of vehicle suspension systems is presented. A theoretical background on recent advances in output-feedback control is first provided, which makes possible an effective synthesis of static output-feedback controllers by solving a single linear matrix inequality optimization problem. Next, a simplified model of a quarter-car suspension system is proposed, taking the ride comfort, suspension stroke, road holding ability, and control effort as the main performance criteria in the vehicle suspension design. The new approach is then used to design a static output-feedback 𝐻_∞ controller that only uses the suspension deflection and the sprung mass velocity as feedback information. Numerical simulations indicate that, despite the restricted feedback information, this static output-feedback 𝐻_∞ controller exhibits an excellent behavior in terms of both frequency and time responses, when compared with the corresponding state-feedback 𝐻_∞ controller.

1. Introduction

In recent decades, vehicle suspension systems have been attracting a growing interest. In particular, much research effort has been devoted to designing different kinds of passive, active, and semiactive vehicle suspensions using a wide variety of control strategies. Some relevant instances of control strategies used in this field are fuzzy control, optimal control, 𝐻_∞ control, gain scheduling, adaptive control, and model predictive control [1–5]. The development of these control strategies has been closely related to the emergence of computational tools and efficient numerical algorithms, which allow solving complex and sophisticated control problems in a reasonably short time.

When designing a feedback control system, the amount of information available for feedback purposes is an element of particular importance. In the ideal case that the entire state vector is available, many advanced state-feedback controller designs can be formulated as linear matrix inequality (LMI) optimization problems and efficiently computed using standard computational tools as those provided by the MATLAB Robust Control Toolbox [6]. In a more realistic situation, however, the complete state vector is rarely accessible and the available feedback information consists only in a reduced set of linear combinations of the states. In this context, static output-feedback control strategies are an excellent option to facilitate a simple implementation in practice.

From a computational perspective, static output-feedback controller designs lead to challenging problems. Typically, this kind of problems has been solved using multistep numerical algorithms such as those based on random search [7], or those consisting in iterative procedures [8–11]. In both cases, complex matrix equations or LMI optimization problems need to be solved at each step. To avoid the high computational cost associated with the multistep methods, some single-step strategies have also been proposed [12–15], which formulate the static output-feedback controller design in terms of a single LMI optimization problem. These single-step methods are based on a proper transformation of the state variables and present the drawback of being highly problem-dependent, in the sense that a complete derivation of the LMI optimization problem must be carried out for most controller designs.

Recently, a new single-step strategy has been presented in [16], which can be applied to any control problem that admits an LMI-based state-feedback controller design. In this case, an LMI formulation to compute the output-feedback control gain matrix can be easily derived by means of a simple change of variables. The new design methodology is computationally effective, conceptually simple, and easy to implement. Moreover, it can be an excellent tool to design static output-feedback controllers in a wide variety of problems by taking advantage of the rich literature on LMI formulations for state-feedback controller design. Some preliminary works, with successful applications to the field of vibration control of large structures, can be found in [17–21].

The main objective of this paper is to explore the potential applicability of the new design methodology in the field of vehicle suspensions. Additionally, we are also interested in providing a clear and practical presentation of the main theoretical elements of the new approach, which we believe can be of general interest for control engineers in different fields. To this end, a static output-feedback 𝐻_∞ controller is designed for a simplified quarter-car suspension system. A state-feedback 𝐻_∞ controller is also designed, and it is used as a reference in the performance assessment. Moreover, the LMI formulation of the state-feedback design serves as a natural starting point to derive the LMI formulation for the output-feedback design.

The paper is organized as follows. In Section 2, the fundamental elements of the new strategy for static output-feedback controller design are summarily discussed. In Section 3, a suitable mathematical model for a quarter-car suspension system is provided, and the general ideas of Section 2 are applied to the particular case of 𝐻_∞ controller design. In Section 4, a static output-feedback 𝐻_∞ controller is designed for a particular quarter-car suspension system, and a suitable set of frequency and time responses are computed to assess the effectiveness of the proposed controller. Finally, in Section 5, some conclusions and future lines of research are briefly presented.

2. Theoretical Background

Let us consider a linear matrix inequality that depends on a symmetric positive-definite matrix 𝑋∈ℝ^𝑛×𝑛, a matrix 𝑌∈ℝ^𝑚×𝑛, and possibly other scalar or matrix variables. Such an LMI can be written in the form: 𝑋>0,𝐹(𝑋,𝑌,𝜁)>0, (1) where the vector 𝜁∈ℝ^𝑝×1 collects the free entries of the matrices distinct from 𝑋 and 𝑌, together with the remaining LMI variables, and 𝐹 is a given affine map that makes the matrix inequality an LMI. This kind of LMI formulation is very common in practice and appears in a large number of state-feedback control problems [22]; some recent works can be found in [23–28]. More precisely, the LMI formulation (1) arises naturally in a wide variety of state-feedback controller designs, where the state control gain matrix 𝐺∈ℝ^𝑚×𝑛 is explicitly given by 𝐺=𝑌𝑋⁻¹, (2) where 𝑋 is usually the inverse of a Lyapunov matrix or a scaling of it, and 𝑌 comes from a previous change of variables defined by 𝑌=𝐺𝑋. (3)

Static output-feedback control problems can be seen as static state-feedback problems with the additional constraint that the state control gain matrix 𝐺 admits a factorization of the form: 𝐺=𝐾𝐶_𝑦, (4) where 𝐾∈ℝ^𝑚×𝑞 is the output control gain matrix and 𝐶_𝑦∈ℝ^𝑞×𝑛 is the observed-output matrix, which is assumed to be a given full row-rank matrix with 𝑞<𝑛. Consequently, if a static state-feedback control problem can be formulated in terms of an LMI of the form (1) with the state gain matrix given in (2), then the corresponding static output-feedback version of the same control problem can be reduced to a nonconvex problem, consisting in finding matrices 𝑋,𝑌, and 𝜁 satisfying 𝑋>0,𝐹(𝑋,𝑌,𝜁)>0,(𝑋,𝑌)∈ℳ, (5) where ℳ is the set of all pairs of matrices (𝑋,𝑌) for which there exists an 𝑚×𝑞 matrix 𝐾 satisfying 𝑌𝑋⁻¹=𝐾𝐶_𝑦. (6)

Recently, a systematic and easy-to-implement strategy to obtain feasible solutions of (5) has been proposed in [16]. This strategy considers an 𝑛×(𝑛−𝑞) matrix 𝑄, whose columns are a basis of ker(𝐶_𝑦), and a matrix 𝑅 defined by 𝑅=𝐶^†_𝑦+𝑄𝐿, (7) where 𝐿 is a given (𝑛−𝑞)×𝑞 matrix and 𝐶^†_𝑦=𝐶^𝑇_𝑦𝐶_𝑦𝐶^𝑇_𝑦⁻¹ (8) is the Moore-Penrose pseudoinverse of 𝐶_𝑦. Next, the following linear transformations are introduced: 𝑋=𝑄𝑋_𝑄𝑄^𝑇+𝑅𝑋_𝑅𝑅^𝑇,𝑌=𝑌_𝑅𝑅^𝑇, (9) where 𝑋_𝑄∈ℝ^{(𝑛−𝑞)×(𝑛−𝑞)} and 𝑋_𝑅∈ℝ^𝑞×𝑞 are symmetric matrices and 𝑌_𝑅∈ℝ^𝑚×𝑞 is an arbitrary matrix. The design of static output-feedback controllers is based on the following result.

Theorem 1 (see [16]). If the matrix 𝑋 in (9) is nonsingular, then 𝑋_𝑅 is also nonsingular and the matrix equation 𝑌𝑋⁻¹=𝐾𝐶_𝑦 holds with 𝐾=𝑌_𝑅𝑋_𝑅⁻¹. (10) Moreover, 𝑋 is positive-definite if and only if the matrices 𝑋_𝑄 and 𝑋_𝑅 are both positive-definite.

This theorem provides a systematic methodology to obtain solutions of (5). Indeed, we only need to choose a suitable (𝑛−𝑞)×𝑞 matrix 𝐿 in order to define 𝑅=𝐶^†_𝑦+𝑄𝐿, and solve the following LMI with variables 𝑋_𝑄, 𝑋_𝑅, 𝑌_𝑅, and 𝜁: 𝑋_𝑄>0,𝑋_𝑅𝐹>0,𝑄𝑋_𝑄𝑄^𝑇+𝑅𝑋_𝑅𝑅^𝑇,𝑌_𝑅𝑅^𝑇,𝜁>0, (11) which has been obtained from (1) by using the transformations (9), using also the fact that the condition 𝑋>0 can be replaced by 𝑋_𝑄>0 and 𝑋_𝑅>0. If a feasible solution to the LMI (11) is achieved by the matrices 𝑋_𝑄, 𝑋_𝑅, 𝑌_𝑅, and 𝜁̃ then, for the corresponding matrices 𝑋, 𝑌 defined in (9) and the vector 𝜁̃ we obtain a feasible solution of (5) that, at the same time, satisfies the constraint (6) with the output gain matrix: 𝐾=𝑌_𝑅𝑋_𝑅⁻¹. (12)

The main features of this strategy are its generality, conceptual simplicity, and ease of implementation. Moreover, it can also be applied to optimization problems of the form: ()()()minimizeℎ𝑋,𝑌,𝜁subjectto𝑋>0,𝐹𝑋,𝑌,𝜁>0,𝑋,𝑌∈ℳ, (13) where the objective function ℎ is assumed to be linear. This kind of optimization problem arises when some performance criterion needs to be optimized. In this case, the optimization problem (13) can be transformed into the following LMI optimization problem with variables 𝑋_𝑄, 𝑋_𝑅, 𝑌_𝑅, and 𝜁: minimizeℎ𝑋_𝑄,𝑋_𝑅,𝑌_𝑅,𝜁subjectto𝑋_𝑄>0,𝑋_𝑅𝐹>0,𝑄𝑋_𝑄𝑄^𝑇+𝑅𝑋_𝑅𝑅^𝑇,𝑌_𝑅𝑅^𝑇,𝜁>0, (14) where the new objective function ℎ is defined as ℎ𝑋_𝑄,𝑋_𝑅,𝑌_𝑅,𝜁=ℎ𝑄𝑋_𝑄𝑄^𝑇+𝑅𝑋_𝑅𝑅^𝑇,𝑌_𝑅𝑅^𝑇,𝜁. (15) Clearly, given an optimal solution to the optimization problem (14), a corresponding triplet (𝑋,𝑌,𝜁̃) of matrices can be computed, which minimizes the objective function ℎ(𝑋,𝑌,𝜁) on a set that satisfies all the constraints in (13). In particular, we will have 𝑌𝑋⁻¹=𝐾𝐶_𝑦 with 𝐾=𝑌_𝑅𝑋_𝑅⁻¹.

Remark 2. In general, the objective function ℎ in (13) depends on the variables 𝑋, 𝑌, and 𝜁. However, an objective function ℎ(𝜁) that only depends on 𝜁 will be encountered in the following sections. In this case, we have ℎ(𝑋_𝑄,𝑋_𝑅,𝑌_𝑅,𝜁)=ℎ(𝜁) and the corresponding LMI optimization problem (14) takes the simplified form: minimizeℎ(𝜁)subjectto𝑋_𝑄>0,𝑋_𝑅𝐹>0,𝑄𝑋_𝑄𝑄^𝑇+𝑅𝑋_𝑅𝑅^𝑇,𝑌_𝑅𝑅^𝑇,𝜁>0. (16)

Remark 3. For simplicity, positive semidefinite terms are not considered in the LMI (1). If necessary, this kind of terms can be easily included by adding a new matrix inequality of the form 𝐹₀(𝑋,𝑌,𝜁)≥0.

3. Static Output-Feedback Control for Vehicle Suspensions

In this section, the general methodology introduced in Section 2 is applied to design an output-feedback 𝐻_∞ control system for a quarter-car suspension model. More precisely, in Section 3.1, a first-order state-space model for a quarter-car together with a suitable vector of controlled outputs are presented. An LMI formulation to compute static output-feedback 𝐻_∞ controllers is provided in Section 3.2.

3.1. Mathematical Model

Let us consider the quarter-car suspension model schematically depicted in Figure 1. The motion equations can be written as 𝑚_𝑠𝑧̈_𝑠(𝑡)=−𝑐_𝑠𝑧̇_𝑠(𝑡)−𝑧̇_𝑢(𝑡)−𝑘_𝑠𝑧_𝑠(𝑡)−𝑧_𝑢(𝑡(𝑡),𝑚)+𝑢_𝑢𝑧̈_𝑢(𝑡)=𝑐_𝑠𝑧̇_𝑠(𝑡)−𝑧̇_𝑢(𝑡)+𝑘_𝑠𝑧_𝑠(𝑡)−𝑧_𝑢(𝑡)−𝑘_𝑢𝑧_𝑢(𝑡)−𝑧_𝑟(𝑡)−𝑢(𝑡), (17) where 𝑚_𝑠 and 𝑚_𝑢 are the sprung and unsprung masses representing the chassis mass and wheel mass, respectively; 𝑘_𝑠 and 𝑐_𝑠 are, respectively, the stiffness and damping of the suspension system; 𝑘_𝑢 stands for the tire stiffness; 𝑧_𝑟(𝑡) represents the vertical road displacement; 𝑧_𝑠(𝑡) and 𝑧_𝑢(𝑡) are the vertical displacements of the sprung and unsprung masses, respectively; and 𝑢(𝑡) is the active input of the suspension system. By defining the state variables 𝑥₁(𝑡)=𝑧_𝑠(𝑡),𝑥₂(𝑡)=𝑧_𝑢(𝑡),𝑥₃(𝑡)=𝑧̇_𝑠(𝑡),𝑥₄(𝑡)=𝑧̇_𝑢(𝑡), (18) a first-order state-space model in the form: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝐵_𝑤𝑤(𝑡) (19) can be derived, where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡),𝑥₄(𝑡)]^𝑇 is the vector of states, 𝑢(𝑡) is the control input, 𝑤(𝑡)=𝑧_𝑟(𝑡) is the road disturbance input, and the matrices 𝐴, 𝐵, and 𝐵_𝑤 are given by −𝐴=𝑘00100001_𝑠𝑚_𝑠𝑘_𝑠𝑚_𝑠−𝑐_𝑠𝑚_𝑠𝑐_𝑠𝑚_𝑠𝑘_𝑠𝑚_𝑢−𝑘_𝑠+𝑘_𝑢𝑚_𝑢𝑐_𝑠𝑚_𝑢−𝑐_𝑠𝑚_𝑢,00𝐵=1𝑚_𝑠−1𝑚_𝑢,𝐵_𝑤=000𝑘_𝑢𝑚_𝑢. (20) To define the vector of controlled outputs, we consider the ride comfort, suspension stroke, road holding ability, and the required control effort as main performance criteria in the vehicle suspension design. These criteria can be quantified using the sprung mass acceleration 𝑧̈_𝑠(𝑡), the suspension deflection 𝑧_𝑠(𝑡)−𝑧_𝑢(𝑡), the tire deflection 𝑧_𝑢(𝑡)−𝑧_𝑟(𝑡), and the control force 𝑢(𝑡), respectively, and should be made as small as possible in order to have good vehicle suspension characteristics [29, 30]. Therefore, we consider the following vector of controlled outputs: 𝑧(𝑡)=𝑧̈_𝑠(𝑡)𝛼𝑧_𝑠(𝑡)−𝑧_𝑢(𝑡)𝛽𝑧_𝑢(𝑡)−𝑧_𝑟(𝑡)𝜂𝑢(𝑡), (21) where 𝛼, 𝛽, and 𝜂 are adjustable weights that should manage the tradeoff between the above performance requirements. Using (17) to isolate the sprung mass acceleration, the controlled output 𝑧(𝑡) can be written as 𝑧(𝑡)=𝐶𝑥(𝑡)+𝐷𝑢(𝑡)+𝐷_𝑤𝑤(𝑡) (22) with −𝑘𝐶=^𝑠𝑚_𝑠𝑘_𝑠𝑚_𝑠−𝑐_𝑠𝑚_𝑠𝑐_𝑠𝑚_𝑠,1𝛼−𝛼000𝛽000000𝐷=𝑚_𝑠00𝜂,𝐷_𝑤=000.−𝛽 (23)

Figure 1 

Quarter-car suspension model with active suspension.

3.2. Static Output-Feedback 𝐻_∞ Controller Design

Now, let us consider the system: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝐵_𝑤𝑤(𝑡),𝑧(𝑡)=𝐶𝑥(𝑡)+𝐷𝑢(𝑡)+𝐷_𝑤𝑤(𝑡), (24) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector, 𝑢(𝑡)∈ℝ^𝑚 is the control input, 𝑤(𝑡)∈ℝ^𝑟 is the disturbance input, 𝑧(𝑡)∈ℝ^𝑑 is the controlled output, and 𝐴, 𝐵, 𝐵_𝑤, 𝐶, 𝐷, and 𝐷_𝑤 are constant matrices with appropriate dimensions. Given a state-feedback controller: 𝑢(𝑡)=𝐺𝑥(𝑡) (25) with state gain matrix 𝐺∈ℝ^𝑚×𝑛, the following closed-loop system results: 𝑥̇(𝑡)=𝐴_𝐺𝑥(𝑡)+𝐵_𝑤𝑤(𝑡),𝑧(𝑡)=𝐶_𝐺𝑥(𝑡)+𝐷_𝑤𝑤(𝑡), (26) where 𝐴_𝐺=𝐴+𝐵𝐺,𝐶_𝐺=𝐶+𝐷𝐺. (27) For a given control gain matrix 𝐺, the 𝐻_∞-norm of the closed-loop system (26) is given by 𝛾_𝐺=𝑇_𝐺∞=sup_𝜔∈ℝ𝜎_max𝑇_𝐺𝑗𝜔, (28) where 𝜎_max[⋅] denotes the maximum singular value and 𝑇_𝐺 is the transfer function from the disturbance input to the controlled output: 𝑇_𝐺(𝑠)=𝐶_𝐺𝑠𝐼−𝐴_𝐺⁻¹𝐵_𝑤+𝐷_𝑤. (29)

In the state-feedback 𝐻_∞ controller design approach, the objective is to obtain a controller of the form (25), which defines an asymptotically stable matrix 𝐴_𝐺 and, simultaneously, minimizes the corresponding 𝐻_∞-norm 𝛾_𝐺. According to the Bounded Real Lemma [22], for a given scalar 𝛾>0, the closed-loop state matrix 𝐴_𝐺 is asymptotically stable and 𝛾_𝐺<𝛾, if and only if there exists a symmetric positive-definite matrix 𝑃∈ℝ^𝑛×𝑛 such that the matrix inequality 𝑃𝐴_𝐺+𝐴^𝑇_𝐺𝐵𝑃∗∗_𝑇𝑤𝐶𝑃−𝛾𝐼∗_𝐺𝐷_𝑤−𝛾𝐼<0 (30) holds, where (∗) denotes the transposed entry. Pre- and postmultiplying both sides of (30) by the symmetric matrix diag[𝑋,𝐼,𝐼] with 𝑋=𝑃⁻¹, using the values of the matrices 𝐴_𝐺, and 𝐶_𝐺 in (27), and introducing the new variable 𝑌=𝐺𝑋, we arrive at the linear matrix inequality: 𝑆𝑋,𝑌,𝛾=𝐴𝑋+𝑋𝐴^𝑇+𝐵𝑌+𝑌^𝑇𝐵^𝑇𝐵∗∗_𝑇𝑤−𝛾𝐼∗𝐶𝑋+𝐷𝑌𝐷_𝑤−𝛾𝐼<0. (31) The state-feedback 𝐻_∞ controller can then be effectively computed by solving the following LMI optimization problem: minimize𝛾subjectto𝑋>0,𝛾>0,𝑆𝑋,𝑌,𝛾<0. (32) If the optimization problem (32) attains an optimal value 𝛾 for the matrices 𝑋 and 𝑌, then the state gain matrix 𝐺=𝑌𝑋⁻¹ defines a state-feedback controller 𝑢(𝑡)=𝐺𝑥(𝑡) with an asymptotically stable closed-loop state matrix 𝐴_𝐺 and an optimal 𝐻_∞-norm 𝛾_𝐺=𝛾.

Let us now consider the case of a static output-feedback controller: 𝑢(𝑡)=𝐾𝑦(𝑡), (33) where 𝐾∈ℝ^𝑚×𝑞 is the observed-output gain matrix and 𝑦(𝑡)∈ℝ^𝑞 is the vector of observed outputs, which can be written as 𝑦(𝑡)=𝐶_𝑦𝑥(𝑡), (34) for a given full row-rank matrix 𝐶_𝑦∈ℝ^𝑞×𝑛 with 𝑞<𝑛. From (33) and (34), we obtain 𝑢(𝑡)=𝐾𝐶_𝑦𝑥(𝑡), (35) and consequently, the output-feedback controller (33) can be considered as a state-feedback controller with an associated state gain matrix 𝐺_of=𝐾𝐶_𝑦. Therefore, the design of a static output-feedback 𝐻_∞ controller leads to the optimization problem (32) with the additional constraint 𝑌𝑋⁻¹=𝐾𝐶_𝑦, which is a particular case of the optimization problem (13) with 𝐹.𝜁=𝛾,ℎ𝑋,𝑌,𝛾=𝛾,𝑋,𝑌,𝛾=diag𝛾,−𝑆𝑋,𝑌,𝛾 (36)

According to the discussion presented in Section 2, the following LMI optimization problem results: minimize𝛾subjectto𝑋_𝑄>0,𝑋_𝑅𝑆>0,𝛾>0,𝑄𝑋_𝑄𝑄^𝑇+𝑅𝑋_𝑅𝑅^𝑇,𝑌_𝑅𝑅^𝑇,𝛾<0, (37) where the matrix inequality 𝑆(𝑄𝑋_𝑄𝑄^𝑇+𝑅𝑋_𝑅𝑅^𝑇,𝑌_𝑅𝑅^𝑇,𝛾)<0 takes the explicit form𝐴𝑄𝑋_𝑄𝑄^𝑇+𝑄𝑋_𝑄𝑄^𝑇𝐴^𝑇+𝐴𝑅𝑋_𝑅𝑅^𝑇+𝑅𝑋_𝑅𝑅^𝑇𝐴^𝑇+𝐵𝑌_𝑅𝑅^𝑇+𝑅𝑌^𝑇_𝑅𝐵^𝑇𝐵∗∗_𝑇𝑤−𝛾𝐼∗𝐶𝑄𝑋_𝑄𝑄^𝑇+𝐶𝑅𝑋_𝑅𝑅^𝑇+𝐷𝑌_𝑅𝑅^𝑇𝐷_𝑤−𝛾𝐼<0. (38)If the optimization problem defined in (37) attains an optimal solution 𝛾 for the matrices 𝑋_𝑄, 𝑋_𝑅, and 𝑌_𝑅, then the output gain matrix 𝐾=𝑌_𝑅𝑋_𝑅⁻¹ defines a static output-feedback controller 𝑢(𝑡)=𝐾𝑦(𝑡) with asymptotically stable matrix 𝐴_𝐺of and optimal 𝐻_∞-norm 𝛾_𝐺of≤𝛾, where 𝐺_of=𝐾𝐶_𝑦.

Remark 4. Note that the LMI optimization problem (37) allows us to design a static output-feedback 𝐻_∞ controller 𝑢(𝑡)=𝐾𝑦(𝑡) by solving a single LMI optimization problem. Moreover, the optimal value 𝛾 provides an upper bound of the 𝐻_∞-norm corresponding to 𝐺_of=𝐾𝐶_𝑦. The actual value of 𝛾_𝐺of can be obtained by maximizing the maximum singular value of the matrix 𝑇_𝐺of(𝑗𝜔), as indicated in (28). Alternatively, the value 𝛾_𝐺of can also be computed by solving the LMI optimization problem: ().minimize𝛾subjectto𝑃>0,𝛾>0,LMI30 (39)

4. Numerical Results

In this section, the ideas presented in Section 3.2 are applied to design a static output-feedback 𝐻_∞ controller for the quarter-car suspension model described in Section 3.1. This output-feedback controller only uses the suspension deflection and the sprung mass velocity as feedback information. A state-feedback 𝐻_∞ controller is also computed to be taken as a reference in the performance assessment. Next, three different configurations are considered: (i) uncontrolled system, (ii) controlled system using the state-feedback 𝐻_∞ controller, and (iii) controlled system using the static output-feedback 𝐻_∞ controller. For these configurations, a brief discussion on the corresponding frequency responses is presented in Section 4.2. The time responses to an isolated bump as road disturbance are presented and discussed in Section 4.3. Finally, some closing remarks are provided in Section 4.4. All computations have been carried out with the MATLAB Robust Control Toolbox [6], and the following particular values [29, 31] 𝑚_𝑠=504.5kg,𝑚_𝑢=62kg,𝑘_𝑠𝑘=13100N/m,_𝑢=252000N/m,𝑐_𝑠=400Ns/m, (40) have been taken as parameters of the quarter-car suspension model in the controllers design and numerical simulations.

4.1. Controllers Design

Let us consider the quarter-car state-space model defined by (19), (20), and the parameter values given in (40). To design a state-feedback 𝐻_∞ controller 𝑢(𝑡)=𝐺_sf𝑥(𝑡), (41) which uses the full state vector given in (18) as feedback information, we also consider the controlled output defined in (22), (23) and the following particular values of the weighting coefficients: 𝛼=8,𝛽=10,𝜂=1.5×10⁻³. (42) By solving the LMI optimization problem given in (32), we obtain the state gain matrix: 𝐺_sf=10³×1.4733−5.0315−2.88180.1018, (43) with an associated 𝐻_∞-norm: 𝛾_𝐺sf=528.32. (44)

Now, as proposed in [29], let us assume that the available feedback information only includes the suspension deflection and the sprung mass velocity. In this case, the vector of observed outputs is 𝑦(𝑡)=𝑧_𝑠(𝑡)−𝑧_𝑢(𝑡)𝑧̇_𝑠(𝑡)^𝑇, (45) which can be written in the form: 𝑦(𝑡)=𝐶_𝑦𝑥(𝑡), (46) where 𝑥(𝑡) is the state vector defined in (18), and 𝐶_𝑦=1−1000010 (47) is the observed-output matrix. To design a static output-feedback 𝐻_∞ controller 𝑢(𝑡)=𝐾𝑦(𝑡), (48) we compute the matrix 𝑄=10100001, (49) whose columns are a basis of ker(𝐶_𝑦) and the matrix 1𝑅=20−1200100, (50) which can be obtained from (7) for the particular choice 𝐿=0. By solving the LMI optimization problem (37) with the matrices 𝑄 and 𝑅 given in (49), (50) and the same matrices 𝐴, 𝐵, 𝐵_𝑤, 𝐶, 𝐷, and 𝐷_𝑤 used in the previous state-feedback controller design, we get the observed-output gain matrix: 𝐾=10³×−0.3585−8.9459, (51) and an optimal 𝛾-value: 𝛾=551.03. (52) According to Remark 4 and the 𝛾-value in (44), the 𝐻_∞-norm of the state gain matrix 𝐺_of=𝐾𝐶_𝑦 (53) associated with the observed-output gain matrix 𝐾, satisfies 528.32≤𝛾_𝐺of≤551.03. (54) By setting 𝐺=𝐺_of in (30) and solving the optimization problem (39), the following value of 𝛾_𝐺of results: 𝛾_𝐺of=532.74, (55) which is only a 0.84% greater than the optimal value 𝛾_𝐺sf achieved by the state-feedback 𝐻_∞ controller.

4.2. Frequency Response

In this subsection, we consider the frequency response for three different control configurations of the quarter-car suspension model: (i) uncontrolled system, (ii) controlled system using the state-feedback 𝐻_∞ controller defined in (41) and (43), and (iii) controlled system using the static output-feedback 𝐻_∞ controller defined in (45), (48), and (51). For these configurations, the frequency transfer functions from the road displacement 𝑤(𝑡) to the magnitudes used as performance criteria are displayed in Figure 2. Specifically, the frequency transfer functions from the road displacement 𝑤(𝑡) to the sprung mass acceleration 𝑧̈_𝑠(𝑡) are presented in Figure 2(a), where the black dotted line corresponds to the uncontrolled system (denoted as Passive in the legend), the blue dashed line pertains to the state-feedback controller (denoted as State-feedback in the legend), and the red solid line corresponds to the static output-feedback controller (denoted as Output-feedback in the legend). The frequency transfer functions from the road displacement 𝑤(𝑡) to the suspension deflection 𝑧_𝑠(𝑡)−𝑧_𝑢(𝑡), tire deflection 𝑧_𝑢(𝑡)−𝑧_𝑟(𝑡), and control force 𝑢(𝑡) are displayed in Figures 2(b), 2(c), and 2(d), respectively, using the same line styles and colors.

(a)

(b)

(c)

(d)

Figure 2 

Frequency transfer functions from road displacement to (a) sprung mass acceleration, (b) suspension deflection, (c) tire deflection, and (d) control effort, corresponding to the passive (black dotted line), state-feedback (blue dashed line), and static output-feedback (red solid line) configurations.

Looking at the graphics in Figure 2(a), it can be appreciated that a significant improvement on ride comfort is provided by the state-feedback and output-feedback controllers when compared with the passive system, especially in the sensitive frequency range of 0–65 rad/s [32]. A closer look at the graphics corresponding to the active control configurations in Figures 2(a) and 2(d) also indicates that, in this case, the static output-feedback controller behaves slightly better than the state-feedback controller requiring, moreover, similar levels of control effort. Regarding the suspension deflection and tire deflection, the graphics in Figures 2(b) and 2(c) show that both active controllers provide a significant improvement near the natural frequency of the sprung mass mode: 𝑓_𝑠=𝑘_𝑠𝑚_𝑠, (56) which for our particular model takes the value 𝑓_𝑠=5.1rad/s (0.81 Hz). It should be highlighted that, in the case of suspension deflection (Figure 2(b)), the passive control configuration exhibits the best behavior for frequencies inferior to 0.6 Hz. However, it should also be noted that in this case both active control configurations present negative dB-gains for frequencies below 0.6 Hz.

Remark 5. As indicated in [33], independently of the suspension type (passive or active), the frequency transfer function from road displacement to sprung mass acceleration has an invariant point at the wheel-hop frequency: 𝑓_𝑢=𝑘_𝑢𝑚_𝑢, (57) with magnitude 𝑘_𝑢/𝑚_𝑠. This fact reveals the difficulty of improving the ride comfort for frequencies around 𝑓_𝑢. For the parameter values in (40), we have a wheel-hop frequency of 𝑓_𝑢=63.75rad/s (10.15 Hz) with a magnitude of 𝑘_𝑢/𝑚_𝑠=499.50 (53.97 dB). The existence of an invariant point also applies to the frequency transfer function from road displacement to suspension deflection at the rattle-space frequency: 𝑓_𝑟=𝑘_𝑢𝑚_𝑠+𝑚_𝑢, (58) which for the parameters in (40) takes the value 𝑓_𝑟=21.10rad/s (3.36 Hz).

Remark 6. The graphics of frequency response in Figure 2 use the road displacement as disturbance input. For the sake of completeness, the graphics of frequency response using the road displacement velocity as disturbance input are presented in Figure 3. Note that the comments made to the graphics in Figure 2 also apply to this second set of graphics.

(a)

(b)

(c)

(d)

Figure 3 

Frequency transfer functions from road displacement velocity to (a) sprung mass acceleration, (b) suspension deflection, (c) tire deflection, and (d) control effort, corresponding to the passive (black dotted line), state-feedback (blue dashed line), and static output-feedback (red solid line) control configurations.

4.3. Time Response to a Bump Disturbance

To provide a more complete picture of the performance achieved by the proposed static output-feedback controller, in this subsection we present the time response of the quarter-car suspension system to a road disturbance. More precisely, we consider an isolated bump of the form: 𝑧_𝑟(𝑡)=𝐻21−cos2𝜋𝑉𝐿𝑡𝐿if0≤𝑡≤𝑉,0otherwise, (59) where 𝐻 and 𝐿 are the bump height and length, respectively; and 𝑉 is the vehicle velocity. The following particular values of the parameters [30]: 𝐻=0.1m,𝐿=5m,𝑉=12.5m/s, (60) have been used to conduct the numerical simulations. For this road disturbance, the magnitudes taken as performance criteria have been computed for the control configurations (i)–(iii) defined in Section 4.2, and the corresponding graphics are presented in Figure 4. In particular, the graphics in Figure 4(a) display the sprung mass acceleration 𝑧̈_𝑠(𝑡) for the uncontrolled system (black dotted line), the controlled system with state-feedback controller (blue dashed line), and the controlled system with static output-feedback controller (red solid line). The graphics of suspension deflection 𝑧_𝑠(𝑡)−𝑧_𝑢(𝑡), tire deflection 𝑧_𝑢(𝑡)−𝑧_𝑟(𝑡), and control force 𝑢(𝑡) for these three control configurations are displayed in Figures 4(b), 4(c), and 4(d), respectively, using the same line styles and colors.

(a)

(b)

(c)

(d)

Figure 4 

Time response to an isolated bump disturbance: (a) sprung mass acceleration, (b) suspension deflection, (c) tire deflection, and (d) control effort, corresponding to the passive (black dotted line), static state-feedback (blue dashed line), and static output-feedback (red solid line) control configurations.

The graphics in Figures 4(a)–4(c) clearly show that a significant improvement in ride comfort, suspension deflection, and road holding ability is provided by the active controllers. It can also be appreciated that the static output-feedback controller achieves practically the same levels of vibrational response mitigation as the state-feedback controller. Moreover, the graphics in Figure 4(d) point out that there are no relevant differences between the levels of control effort required to operate the active controllers.

4.4. Closing Remarks

The numerical results obtained in this section indicate that the proposed static output-feedback 𝐻_∞ controller exhibits a remarkable performance in terms of both frequency and time responses when compared with the corresponding state-feedback 𝐻_∞ controller. In fact, from the point of view of 𝐻_∞ controller design, the values of the 𝐻_∞-norms in (44) and (55) show that the static output-feedback controller is practically optimal.

These outstanding numerical results are even more meaningful when considering some additional features of the new design methodology: (i) Generality: the proposed methodology can be applied to a wide variety of control problems, with the only requirement that the state-feedback version of the problem admits a standard LMI formulation. (ii) Conceptual simplicity: the ideas involved in the proposed change of variables are simple and transparent. As shown in Section 2, new LMI formulations for static output-feedback controller designs can be easily derived from existing state-feedback LMI formulations through a simple change of the LMI variables. (iii) Ease of implementation: the static output-feedback controller design is formulated in terms of LMI optimization problems, which can be directly solved using standard computational tools, as those provided by the MATLAB Robust Control Toolbox [6]. (iv) Computational efficiency: traditionally, static output-feedback gain matrices have been computed by means of multistep optimization algorithms that require solving complex matrix equations or LMI problems at each step. In contrast, in the new design methodology, the output-feedback gain matrix is computed by solving a single LMI optimization problem.

The property of computational efficiency deserves some additional considerations. As it is well known, the nonconvex nature of static output-feedback control problems makes them NP-hard [34, 35]. Consequently, heuristic strategies are commonly used to solve this kind of problems in a computationally effective way. These heuristic strategies can be based on a certain set of sufficient conditions for controller design or on sophisticated randomized algorithms. In any case, the design procedure depends critically on a suitable choice of certain parameters that, in principle, can take a wide range of possible values.

Obviously, the previous remark also applies to the new design strategy proposed in this paper, which requires a proper choice of the matrix 𝐿 in (7) to define the change of variables (9). The output-feedback controller design presented in Section 4.1 has been carried out by taking 𝐿 as a zero matrix, which leads to the simplified form 𝑅=𝐶^†_𝑦 in (7). The choice 𝐿=0 has also been used recently in the field of vibration control of large structures with positive results [17–21]. However, it is worth pointing out that certain feasibility problems typically appear when applying the proposed methodology to the design of static output-feedback controllers for structural vibration control, and that ongoing investigations indicate that a suitable choice of the 𝐿 matrix can play an important role in solving these feasibility issues.

5. Conclusions and Future Directions

In this work, a new strategy to design static output-feedback controllers for vehicle suspension systems has been presented. The proposed strategy is conceptually simple, easy to implement, and computationally efficient, and it can be applied to a wide variety of control problems, with the only requirement that the state-feedback version of the problem admits a standard LMI formulation. To illustrate the main elements of the new approach, a static output-feedback 𝐻_∞ controller has been designed for a simplified quarter-car suspension system. Numerical simulations show that the proposed static output-feedback 𝐻_∞ controller exhibits a remarkable behavior in terms of both frequency and time responses, when compared with the corresponding state-feedback 𝐻_∞ controller.

The main contribution of the paper is twofold: (i) to provide a clear and practical presentation of the main theoretical elements of the new strategy for static output-feedback controller design and (ii) to show the practical relevance of the proposed design strategy in the field of automotive suspensions. Indeed, the positive results obtained for the quarter-car suspension system clearly indicate that more research effort should be invested in applying the new design methodology to more complex scenarios. Specifically, more complete physical models can be considered, as half or full car models, seat suspension systems, and human-body models [4, 36, 37]. More sophisticated control strategies should be also explored, as those including multiobjective designs with 𝐻_∞ and generalized 𝐻₂ control strategies, fuzzy control, advanced 𝐻_∞ control, or model predictive control [2, 3, 28, 36, 38–43], combined with different mathematical complexities as, for example, uncertainties, input and output constraints, actuation saturations, delays and actuator faults [23, 24, 26, 30, 36, 44–46], and also combined with actuator dynamics and road excitation models [39, 47].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by the Spanish Ministry of Economy and Competitiveness under Grant DPI2012-32375/FEDER and by the Norwegian Center of Offshore Wind Energy (NORCOWE) under Grant 193821/S60 from the Research Council of Norway (RCN). NORCOWE is a consortium with partners from industry and science, hosted by Christian Michelsen Research.