Vehicle System DynamicsVol. 43, No. 8, August 2005, 561–579

Pole location control design of an active suspensionsystem with uncertain parameters

VALTER J. S. LEITE† and PEDRO L. D. PERES*‡

†UnED Divinópolis – CEFET/MG, R. Monte Santo, 319 – 35502-036, Divinópolis – MG, Brazil‡School of Electrical and Computer Engineering, University of Campinas, CP 6101, 13081-970,

Campinas – SP, Brazil

This paper addresses the problem of robust control design for an active suspension quarter-car modelby means of state feedback gains. Specifically, the design of controllers that assure robust pole locationof the closed-loop system inside a circular region on the left-hand side of complex plane is investigated.Three sufficient conditions for the existence of a robust stabilizing state feedback gain are presented aslinear matrix inequalities: (i) the quadratic stability based gain; (ii) a recently published condition thatuses an augmented space and has been here modified to cope with the pole location specification; (iii) acondition that uses an extended number of equations and yields a parameter-dependent state feedbackgain. Unlike other parameter-dependent strategies, neither extensive gridding nor approximations areneeded. In the suspension model, the sprung mass, the damper coefficient and the spring constant areconsidered as uncertain parameters belonging to a known interval (polytope type uncertainty). It isshown that the parameter-dependent gain proposed allows one to impose the closed-loop system polelocations that in some situations cannot be obtained with constant feedback gains.

Keywords: Active suspension control; Linear matrix inequalities; Robust control; Parameter dependentcontrol; Robust pole location; Uncertain systems

1. Introduction

The control strategies for active suspension systems are major topics of interest in controlconferences as well as in the automobile industry. A lot of effort has been made to developrealistic models for suspension systems and to define the design specifications that reflect themain objectives to be taken into account. In this sense, ride comfort, associated with the verticalacceleration, and road-holding ability, associated with the tyre deflection, are important factorsto be addressed by any control scheme.

A suspension system has some parameters that are inherently uncertain, such as the sprungmass, whose value depends on the total load of the vehicle. Moreover, dampers and springsmay lose efficiency very slowly during their lifetime. A good way to incorporate all theseinfluences is to consider uncertain parameters in the system model, and one of the most usefulrepresentations for this is undoubtedly the polytopic uncertainty [1].

*Corresponding author. Email: peres@dt.fee.unicamp.br

Vehicle System DynamicsISSN 0042-3114 print/ISSN 1744-5159 online © 2005 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/0042311042000266702

562 V. J. S. Leite and P. L. D. Peres

In the literature it is possible to find many results that deal with a precisely known system andalso with uncertain models in the context of active suspension design. Among the approachesthat consider some kind of uncertainty in the model plant, it is worth mentioning the results ofoptimal control which has dominated this area of application in the last decades [2–8]. Optimalcontrol techniques have been widely investigated, providing good results in many situations.However, besides optimizing the suspension system with respect to some criteria such as thesprung mass acceleration or the suspension and tyre deflections, the optimal state feedbackgains cannot cope with more stringent specifications, since the gains are fixed independentlyof the operating conditions, and do not take into account uncertain parameters. In [5], a gooddiscussion about the necessity of improving the dynamic behaviour of suspension systemsin the presence of uncertainty can be found. Issues that must be taken into account, such asrobustness, disturbance rejection and practical implementation, and the perspective of usingnew techniques such as optimal adaptive control or gain-scheduling are pointed out. This laterstrategy is directly related to the parameter-dependent feedback gain which is the main resultof this paper.

Also, there are several results based on robust control synthesis and guaranteed cost statefeedback gains like the H2 and H∞ norms available for active suspension control design [9],with some extensions to static output feedback [10]. Recently, an approach based on genetic-algorithm and fuzzy proportional-integral/proportional-derivative controllers [11] appearedin the literature. Another design, based on quantitative feedback theory, has been recentlyaddressed in [12] in which a nonlinear suspension model is used. Nonlinear controllers havebeen investigated in [13, 14] to improve suspension performance when the uncertain param-eters are time-varying. An approach based on controllers with non-integer order is presentedin [15]. In [16] the interested reader can find control simulations of semi-active suspensionsystems that consider hardware implementation effects with road adaptive control laws. Themain objective of all of those works is to deal with the uncertainty inherent to the model andto synthesize a control law that assures a minimal performance level of specifications such asuser comfort and road-holding ability.

In the context of linear systems with uncertain parameters described by state space equa-tions, many robust control approaches are based on quadratic Lyapunov functions [17]. In fact,quadratic stabilizability conditions have been largely used for robust state feedback controlin several contexts including performance indices such as H2 and H∞ norms. These prob-lems, in most cases, are formulated as convex linear matrix inequality (LMI) optimizationprocedures [18–20]. Extensions of these conditions for robust output feedback control canalso be encountered in the literature [21]. Although being specially adequate for arbitrarilyfast time-varying parameters, methods based on quadratic stability can produce conservativeresults since the same parameter-independent Lyapunov function must be used to assure theclosed-loop stability for the entire uncertainty domain.

One way to overcome this conservativeness is to consider a parameter-dependent Lyapunovfunction, which can be used to derive a robust feedback control law independent of the uncer-tainty or a controller which depends on the vector of uncertain parameters (gain scheduling).However, the conditions for the existence of a gain-scheduled control are, in general, difficult tosolve, sometimes requiring an extensive gridding on the uncertain parameter space, matchingconditions or a linear fractional structure of the uncertainty [22–24]. For a gain-scheduledapproach to active suspension systems see [13, 14]. Note that, in general, the controllerobtained through a gain-scheduling strategy assures the desired closed-loop properties onlyat the gridding points since an approximation is used for all other points.

The aim here is to analyse robust control techniques that can be applied to the activesuspension system with uncertain parameters assuring to the closed-loop system a desireddynamical behaviour. For this, three conditions that assure a robust pole location by means of

Pole location of a suspension system 563

a state feedback control law are presented. The state variables are supposed to be available forfeedback, and thus the state estimation is not addressed in this paper (see [25–27] for detailson different approaches for estimators and [28] for an active suspension system application).To cope with the design specifications, a circular region located inside the left-hand side ofthe complex plane is imposed. As discussed in [29], the system modes damp asymptoticallyat desired rates if the circle centre and radius are adequately chosen.

All three conditions presented here are formulated in terms of LMIs and can be applied tolinear systems with uncertain parameters, in both dynamic and control matrices, belongingto a convex bounded domain (polytope type uncertainty). The main purpose of this paperis to apply these conditions for the control design of an uncertain active suspension systemand to compare the closed-loop eigenvalue location that can be achieved by each method. Theuncertain parameters are supposed to vary slowly enough to allow for the definition of a lineartime-invariant system for each evaluation of the parameters.

The first condition uses the well-known concept of quadratic stability [30], where a staticstate feedback gain can be obtained from the fixed Lyapunov matrix that assures the closedloop pole location for the entire uncertain system. Besides being capable of handling arbitrarilyfast time varying parameters, the quadratic stability based results are in general conservative.

The second result presented is based on a recently developed approach that uses an aug-mented variable space to compute a fixed state feedback gain and a parameter-dependentLyapunov matrix [31], yielding less conservative results than the ones obtained throughquadratic stability.

Finally, a condition based on an augmented number of linear matrix inequality equationsis given. From a LMI feasibility test, a parameter-dependent Lyapunov function, as well as aparameter-dependent state feedback control assuring the desired pole location, are obtained.This condition exhibits less conservative results than the former two, allowing one to imposeon the closed-loop system more stringent design specifications, which in many cases cannotbe implemented by means of a constant feedback gain.

Differing from other gain-scheduling strategies, the parameter-dependent feedback gainproposed here does not require an exhaustive gridding on the uncertain parameter space. Onlythe vertices are used to compute off-line the controller matrices. The on-line controller isobtained by means of an analytical formula involving the uncertain parameter (which needs tobe measured or estimated) and the matrices already computed off-line. Therefore, no approx-imation is used and the specified closed-loop properties are assured for the entire uncertaintydomain. Note that standard linear parameter varying or gain-scheduling techniques usuallyinterpolate among controllers computed off-line for a fixed number of grid points, and theon-line controller cannot assure the desired behaviour outside these grid points.

The three conditions presented in this paper are applied to the control design of an activesuspension quarter-car model. The sprung mass, the damper coefficient and the spring constantare considered as uncertain parameters. The design options provided by these three differenttechniques are analysed and results showing the pole location limits of each technique arepresented.

2. Vehicle model and definitions

A simple two-degree-of-freedom model is used in this paper to represent a quarter-car suspen-sion system which is borrowed from [2]. Other works consider more sophisticated actuators inthe system, as in [32, 33], and more complex vehicle suspension models can be found in [5],but the aim here is to investigate the pole location capabilities of different control techniques

564 V. J. S. Leite and P. L. D. Peres

Figure 1. Schematic quarter-car model.

applied to a simpler model. Figure 1 shows a schematic representation of the suspension sys-tem which combines the classical passive system with the active one. This kind of structureassures a minimal level of performance and safety. In this configuration, the power demandfrom the active device is usually smaller than in a scheme with no passive elements. Thequarter-car model consists of a sprung mass m1, an unsprung mass m2, a damper c, a linearspring k and an active device that controls the force u. The tyre is modeled as a linear springkt and the model has as exogenous input the vertical velocity w imposed by the road.

The state-space model is given by [2]

x = Ax + Bu + Ew (1)

with A ∈ Rn×n, B ∈ R

n×m and E ∈ Rn×1. For the model depicted in figure 1, x ∈ R

4 (x1 isthe suspension deflection, x2 is the tyre deflection, x3 is the sprung mass velocity and x4 is theunsprung mass velocity), u ∈ R and w ∈ R. With the inclusion of an extra state, x5, which isthe integral of x1, matrices A, B and E are given by

A =

0 0 −1 1 0

0 0 0 −1 0k

m10 − c

m1

c

m10

− k

m2

kt

m2

c

m2− c

m20

1 0 0 0 0

, (2)

B =[

0 01

m1− 1

m20

]′; E = [

0 1 0 0 0]′

. (3)

The inclusion of an integral action is usually performed, in order to avoid steady-state error inthe suspension deflection. The nominal parameters (typical for passenger cars) of this model,obtained from [8], are shown in table 1.

Pole location of a suspension system 565

Table 1. Nominal parameter values forquarter-car model.

Parameter Nominal value Unit

m1 288.9 kgm2 28.58 kgk 10000 N m−1

kt 155900 N m−1

c 850 N s m−1

Consider that matrices A and B in (1) have uncertain parameters, in such a way that thepair (A, B)(α) belongs to a polytope type uncertain domain D given by

D =(A, B)(α):(A, B)(α) =

N∑j=1

αj (A, B)j ;N∑

j=1

αj = 1; αj ≥ 0

. (4)

Any matrix pair inside the domain D can be written as a convex combination of the vertices(A, B)j of the uncertainty polytope. It is assumed that the vector of uncertain parameters α doesnot depend explicitly on the time variable. The aim here is to determine a parameter-dependentstate feedback control law

u(t) = K(α)x(t) (5)

with K(α) ∈ R5×1 such that the closed-loop uncertain system

Acl(α) = A(α) + B(α)K(α)

has all eigenvalues inside the circle at (−σ, 0) with radius r , shown in figure 2, for all(A, B)(α) ∈ D. The case K(α) = K (constant state feedback gain) is also addressed withtwo approaches: one is based on quadratic stability (fixed Lyapunov function) and the otherbased on an augmented space and a parameter-dependent Lyapunov function.

Note that the conditions stated in Lemma 2 and Theorem 1 presented in the section remainvalid if the parameter variation is slow enough to be accommodated by the dynamics of thesystem under control [34].

Figure 2. Circular region for pole location.

566 V. J. S. Leite and P. L. D. Peres

3. Conditions for robust pole location

For a precisely known autonomous system, a necessary and sufficient condition such that allthe eigenvalues of Acl lie inside the circular region depicted in figure 2 is given by the existenceof a positive definite symmetric matrix W such that (see [29])

(Acl + dI)W + W(Acl + dI)′ + 1

r(Acl + dI)W(Acl + dI)′ < 0 (6)

with

d = σ − r. (7)

If Acl belongs to an uncertain polytope D described by its vertices, a sufficient conditionassuring the desired pole location for all Acl ∈ D is given by the existence of a commonmatrix W = W ′ > 0 satisfying (6) at the vertices of D, which can be viewed as an extensionof quadratic stability.

LEMMA 1 (Quadratic Stabilizability) If there exists a symmetric positive definite matrixW ∈ R

n×n and a matrix Z ∈ Rm×n such that

[(AjW + WA′

j + BjZ + Z′B ′j + 2dW) AjW + BjZ + dW

WA′j + Z′B ′

j + dW −rW

]< 0; j = 1, . . . , N

(8)

then K = ZW−1 assures for the uncertain closed-loop system the pole location specified infigure 2 for all (A, B)(α) ∈ D.

Proof In fact, equation (8) comes from (6) with (Aj + BjK) replacing the system matrixAcl, Z = KW and applying the Schur complement [35]. �

In the literature, several extensions of the quadratic stability condition have been usedfor robust control and filter design including pole location specifications [36–38]. However,quadratic stabilizability can be sometimes conservative, in the sense that there are pole locationspecifications that cannot be attained by means of a quadratic stabilizing state feedback gain.

Recently, sufficient conditions for robust stabilizability by means of a state feedback gainand a parameter-dependent Lyapunov function have appeared in the context of discrete-timeuncertain systems [31]. These conditions are here modified in order to cope with the polelocation specification discussed in this paper. Note that the use of a parameter-dependentLyapunov function W(α) = W(α)′ > 0 such that

(Acl(α) + dI)W(α) + W(α)(Acl(α) + dI)′ + 1

r(Acl(α) + dI)W(α)(Acl(α) + dI)′ < 0

(9)

can provide less conservative results than the ones that impose a fixed matrix W .

Pole location of a suspension system 567

LEMMA 2 (Augmented space) If there exist N positive definite matrices Pj ∈ Rn×n and

matrices L ∈ Rm×n and G ∈ R

n×n such that

[ −rPj AjG + BjL + σG

G′A′j + L′B ′

j + σG′ r(Pj − G − G′)

]< 0; j = 1, . . . , N, (10)

then K = LG−1 is a robust state feedback gain and P(α), given by

P(α) =N∑

j=1

αjPj ; αj ≥ 0;N∑

j=1

αj = 1, (11)

is a parameter-dependent Lyapunov matrix assuring the uncertain closed-loop system polelocation specified in figure 2 for all (A, B)(α) ∈ D.

Proof Note that with L = KG, (10) is equivalent to

[ −Pj (Aj + BjK + σ I)G/r

G′(Aj + BjK + σ I)′/r Pj − G − G′

]< 0; j = 1, . . . , N, (12)

which multiplied on the left by T = [I (Aj + BjK + σ I)/r], on the right by T ′ and summingon j (see [31] for details) yields

1

r2(A(α) + B(α)K + σ I)P (α)(A(α) + B(α)K + σ I)′ − P(α) < 0. (13)

Now, multiplying (13) by r , with σ = r + d and W(α) = P(α)/r , it is possible to write

(A(α) + B(α)K + dI) W(α) + W(α) (A(α) + B(α)K + dI)′

+ 1

r(A(α) + B(α)K + dI) W(α) (A(α) + B(α)K + dI)′ < 0, (14)

which assures, with the parameter-dependent Lyapunov function W(α), the pole locationspecified in figure 2 (see equation (9)). �

Lemmas 1 and 2 presented two LMI based solutions for the problem of robust pole locationimposing a fixed (constant) state feedback gain. The results of Lemma 2 are less conserva-tive, since a parameter-dependent Lyapunov function is used, encompassing the conditions ofLemma 1 as a particular case. However, there are uncertain systems that cannot be stabilizedby means of a fixed robust gain or, sometimes, the desired closed-loop pole location requiresa parameter-dependent feedback control to be attained. The following theorem presents suf-ficient conditions for the existence of a parameter-dependent state feedback gain assuring therobust pole location for uncertain continuous-time systems in convex bounded domains. Theconditions are formulated in terms of a set of LMIs described at the vertices of the uncertaintypolytope D as in Lemmas 1 and 2. Despite the larger number of LMIs, the feasibility of theentire set can be solved in polynomial time by interior point based algorithms [39].

568 V. J. S. Leite and P. L. D. Peres

THEOREM 1 If there exist positive definite matrices Wj = W ′j ∈ R

n×n, j = 1, . . . , N andmatrices Zj ∈ R

n×m, j = 1, . . . , N such that

[AdjWj + WjA

′d j

+ BjZj + Z′jB

′j �

WjA′d j

+ Z′jB

′j −rWj

]<

[−I 00 0

]; j = 1, . . . , N, (15)

AdjWj + WjA′d j

+ BjZj + Z′jB

′j

+ AdjWk + AdkWj + WjA′d k

+ WkA′d j

+ BjZk + BkZj + Z′jB

′k + Z′

kB′j

�

WjA′d j

+ Z′jB

′j + WjA

′d k

+ WkA′d j

+ Z′jB

′k + Z′

kB′j

−r(2Wj + Wk)

<

1

(N − 1)2

[I 00 0

],

j = 1, . . . , N; k = 1, . . . , N; k �= j, (16)

AdjWk + AdkWj + WjA′d k

+ WkA′d j

+ BjZk + ZjBk

+ Z′jB

′k + Z′

kB′j + AdjW� + Ad�Wj + WjA

′d �

+ W�A′d j

+ BjZ� + B�Zj + Z′jB

′� + Z′

�B′j + Ad�Wk + AdkW�

+ W�A′d k

+ WkA′d �

+ B�Zk + BkZ� + Z′�B

′k + Z′

kB′�

�

WjA′d k

+ WkA′d j

+ WjA′d �

+ W�A′d j

+ WkA′d �

+ W�A′d k

+ Z′jB

′k + Z′

kB′j + Z′

jB′� + Z′

�B′j + Z′

kB′� + Z′

�B′k

−2r(Wj + Wk + W�)

<6

(N − 1)2

[I 00 0

], j = 1, . . . , N − 2; k = j + 1, . . . , N − 1;

� = k + 1, . . . , N (17)

with � replacing symmetric terms in the LMIs and Adj= Aj + dI, then, the parameter-

dependent state feedback control gain given by

K(α) = Z(α)W(α)−1 (18)

with

Z(α) =N∑

j=1

αjZj ; W(α) =N∑

j=1

αjWj ;N∑

j=1

αj = 1; αj ≥ 0 (19)

assures for the uncertain closed-loop system the pole location specified in figure 2; that is, theparameter-dependent Lyapunov function W(α) satisfies (9) for all (A, B)(α) ∈ D.

Proof It is clear that W(α) given by (19) is a positive definite parameter-dependentLyapunov matrix. From equation (9), using the Schur complement and the change of variables

Pole location of a suspension system 569

Z(α) = K(α)W(α), it is possible to write (by multiplying the right-hand side by∑N

j=1 αj = 1) A(α)W(α) + W(α)A(α)′

+B(α)Z(α) + Z(α)′B(α)′ + 2dW(α)�

A(α)W(α) + B(α)Z(α) + dW(α) −rW(α)

+N∑

j=1

α3j

[AdjWj + WjA

′d j

+ BjZj + Z′jB

′j �

WjA′d j

+ Z′jB

′j −rWj

]

+N∑

j=1

N∑k �=j,k=1

α2j αk

AdjWj + WjA′d j

+ BjZj + Z′jB

′j

+ AdjWk + AdkWj + WjA′d k

+ WkA′d j

+ BjZk + BkZj + Z′jB

′k + Z′

kB′j

�

WjA′d j

+ Z′jB

′j + WjA

′d k

+ WkA′d j

+ Z′jB

′k + Z′

kB′j

−r(2Wj + Wk)

+N−2∑j=1

N−1∑k=j+1

N∑�=k+1

αjαkα�

AdjWk + AdkWj + WjA′d k

+ WkA′d j

+BjZk + ZjBk + Z′jB

′k + Z′

kB′j

+AdjW� + Ad�Wj + WjA′d �

+ W�A′d j

+BjZ� + B�Zj + Z′jB

′� + Z′

�B′j

+Ad�Wk + AdkW� + W�A′d k

+ WkA′d �+B�Zk + BkZ� + Z′

�B′k + Z′

kB′�

�

WjA′d k

+ WkA′d j

+ WjA′d �

+ W�A′d j

+WkA′d �

+ W�A′d k

+ Z′jB

′k + Z′

kB′j

+Z′jB

′� + Z′

�B′j + Z′

kB′� + Z′

�B′k

−2r(Wj+Wk + W�)

.

(20)

Using conditions (15)–(17) and taking into account that αj ≥ 0 one has A(α)W(α) + W(α)A(α)′

+ B(α)Z(α) + Z(α)′B(α)′ + 2dW(α)�

A(α)W(α) + B(α)Z(α) + dW(α) −rW(α)

−( N∑

j=1

α3j − 1

(N − 1)2

N∑j=1

N∑k �=j ;k=1

α2j αk − 6

(N − 1)2

N−2∑j=1

N−1∑k=j+1

N∑�=k+1

αjαkα�

)

×[

I 00 0

]≤ 0. (21)

This last inequality can be verified by noting that (N − 1)� + � ≥ 0 with � and � definedas

� �N∑

j=1

N∑k=1

αj (αj − αk)2 = (N − 1)

N∑j=1

α3j −

N∑j=1

N∑k �=j ;k=1

α2j αk ≥ 0,

� �N∑

j=1

N−1∑k �=j ;k=1

N∑��=j,k;�=2

αj (αk − α�)2

= (N − 2)

N∑j=1

N∑k �=j ;k=1

α2j αk − 6

N−2∑j=1

N−1∑k=j+1

N∑�=k+1

αjαkα� ≥ 0.

This concludes the proof of the theorem. �

570 V. J. S. Leite and P. L. D. Peres

Theorem 1 provides an easy-to-test sufficient LMI condition for the desired pole locationby means of a parameter-dependent state feedback gain. It can be viewed as an extension ofthe recently developed robust stability conditions based on parameter-dependent Lyapunovequations [40, 41].

4. Active suspension control design

The aim now is to apply the preceding results in the design of a state feedback control law forsystem (1). The parameters associated with the sprung mass, m1, the damper coefficient, c, andthe spring constant, k, of the quarter-car model (1) are assumed to be uncertain, i.e. the valuesof these parameters can vary 20% around their nominal values (shown in table 1). Furthermore,it is supposed that the parameters c and k vary as a function of a common mechanical devicewhich is affected by age (represented by the parameter β). Note that these parameters affectboth the dynamical and input matrices. The numerical intervals are given by

231.12 kg ≤ m1 ≤ 346.68 kg,

850 − 17β Ns m−1 ≤ c ≤ 850 + 17β Ns m−1,

1000 − 20β N m−1 ≤ k ≤ 1000 + 20β N m−1,

0 ≤ β ≤ 10.

(22)

Initially, the limits of the closed-loop performance associated with the circle specificationfor pole location of the uncertain active suspension system (1) are investigated. Using Lemmas1, 2 and Theorem 1, the aim is to determine, for each centre value −σ as depicted in figure 2,what is the minimum radius such that a feasible state feedback exists.

The maximum percentage ratio d/σ attained by each method is shown in figure 3, whereonly m1 is considered as an uncertain parameter, and in figure 4, with m1, c and k belonging tothe intervals given by (22). Note that d/σ = 100% means that r = 0 (all eigenvalues assignedto the same point (−σ, 0)) and d/σ = 0 means d = 0 and r = σ (i.e. the circle touches theimaginary axis). Greater values of d/σ indicate that a more stringent pole location can bespecified (smaller circles located far from the imaginary axis), while negative values implythat the circle has passed into the unstable right-hand side of the complex plane. In both figures,it can be noted that the behaviour of the results of Lemmas 1 and 2 is quite similar, while theresults of Theorem 1 provide the more stringent pole locations.

Figure 3. Percentage ratios d/σ as a function of the centre −σ , for Lemmas 1 (dotted line) and 2 (dashed line) andTheorem 1 (solid line). Only m1 has been considered as an uncertain parameter.

Pole location of a suspension system 571

Figure 4. Percentage ratios d/σ as a function of the centre −σ , for Lemmas 1 (dotted line) and 2 (dashed line) andTheorem 1 (solid line). The uncertain parameters m1, c and k are limited as in (22).

Figures 3 and 4 illustrate the capabilities of each method regarding closed-loop pole locationspecifications. In the sequel, the methods are applied for the control design of systems (1)–(4)with the uncertain parameters m1, c and k given by (22), which yields a four vertices uncertainsystem given by the matrices

A1 =

0 0 −1 1 00 0 0 −1 0

1.00 0 −1.00 1.00 0−8.09 5454.86 8.09 −8.09 0

1 0 0 0 0

; B1 =

00

0.0043−0.0350

0

, (23)

A2 =

0 0 −1 1 00 0 0 −1 0

1.50 0 −1.50 1.50 0−12.13 5454.86 12.13 −12.13 0

1 0 0 0 0

; B2 =

00

0.0043−0.0350

0

, (24)

A3 =

0 0 −1 1 00 0 0 −1 0

0.67 0 −0.67 0.67 0−8.09 5454.86 8.09 −8.09 0

1 0 0 0 0

; B3 =

00

0.0029−0.0350

0

, (25)

A4 =

0 0 −1 1 00 0 0 −1 0

1.00 0 −1 1 0−12.13 5454.86 12.13 −12.13 0

1 0 0 0 0

; B4 =

00

0.0029−0.0350

0

. (26)

First of all, a traditional pole placement technique has been used to compute a state feedbackgain for the nominal system. A set of real poles has been chosen as: −36, −38, −40, −42 and−44 assuring no oscillatory modes and fast enough dynamics. The control gain is given by

Kn = [0.6628 0.4386 −0.0220 0.0026 5.3556] × 106. (27)

Besides providing a precise pole location specification for the nominal system, the controlgain (27) is not able to cope with the parameter variations. Figure 5 shows the closed-looppoles for the uncertain system A(α) + B(α)Kn, far away from their nominal specifications. A

572 V. J. S. Leite and P. L. D. Peres

Figure 5. Eigenvalues of A(α) + B(α)Kn with Kn given in (27) and the vertices of the uncertain system given by(23)–(26). The eigenvalues of the closed-loop system vertices are represented by �.

circle centred at (−40, 0) with radius r = 31 has been included, since this is the smallest circleattained by the parameter-dependent control through the results of Theorem 1 (see figure 4).

The robust pole location inside a circle centred at (−40, 0) with radius r = 31 (d = 9) canbe attained by means of a parameter-dependent state feedback control gain K(α) obtainedfrom the results of Theorem 1. This circular region has been chosen in order to provide goodrates of damping and oscillatory behaviour to the controlled system. Note that other designspecifications can be imposed and thus different trade-offs between road-holding ability andpassenger comfort could be achieved (see figure 4). The feasibility test of the conditions ofTheorem 1 yield matrices Wi , Zi , i = 1, . . . , 4 given by

W1 =

0.0123 −0.0021 0.1184 −0.0166 −0.0010−0.0021 0.0011 −0.0272 0.0219 0.00010.1184 −0.0272 1.2893 −0.1850 −0.0084

−0.0166 0.0219 −0.1850 1.0229 −0.0003−0.0010 0.0001 −0.0084 −0.0003 0.0001

× 105, (28)

W2 =

0.0122 −0.0021 0.1176 −0.0163 −0.0010−0.0021 0.0011 −0.0271 0.0221 0.00010.1176 −0.0271 1.2795 −0.1788 −0.0084

−0.0163 0.0221 −0.1788 1.0463 −0.0003−0.0010 0.0001 −0.0084 −0.0003 0.0001

× 105, (29)

Pole location of a suspension system 573

W3 =

0.0936 −0.0183 0.8735 −0.1426 −0.0078−0.0183 0.0091 −0.2249 0.1704 0.00070.8735 −0.2249 9.0984 −1.8987 −0.0606

−0.1426 0.1704 −1.8987 7.5338 −0.0028−0.0078 0.0007 −0.0606 −0.0028 0.0008

× 104, (30)

W4 =

0.0930 −0.0184 0.8716 −0.1424 −0.0077−0.0184 0.0092 −0.2266 0.1718 0.00070.8716 −0.2266 9.1022 −1.9003 −0.0602

−0.1424 0.1718 −1.9003 7.4344 −0.0029−0.0077 0.0007 −0.0602 −0.0029 0.0008

× 104, (31)

Z1 = [−0.2860 0.1555 −3.6196 3.8670 0.0110] × 108, (32)

Z2 = [−0.2706 0.1515 −3.4467 3.7909 0.0100] × 108, (33)

Z3 = [−0.2703 0.1448 −3.3264 3.4226 0.0099] × 108, (34)

Z4 = [−0.2611 0.1424 −3.2319 3.3102 0.0093] × 108. (35)

Figure 6 shows the eigenvalues of the uncertain closed-loop system for K(α) given by

K(α) = (α1Z1 + α2Z2 + α3Z3 + α4Z4)(α1W1 + α2W2 + α3W3 + α4W4)−1 (36)

Figure 6. Eigenvalues of the uncertain closed-loop system A(α) + B(α)K(α), with vertices given by (23)–(26)and K(α) given by (36). The eigenvalues of the closed-loop system vertices are represented by �.

574 V. J. S. Leite and P. L. D. Peres

with αi ≥ 0, i = 1, . . . , 4 and∑4

i=1 αi = 1. To have an idea of the magnitudes of theparameter-dependent control gain K(α) elements (which are similar to the ones of Kn givenby (27)), the following gains have been computed at the vertices:

K1 = [0.6559 0.7284 −0.0509 0.0244 2.8367] × 105, (37)

K2 = [0.7042 0.7048 −0.0533 0.0240 3.0563] × 105, (38)

K3 = [1.1116 0.7485 −0.0863 0.0296 4.8480] × 105, (39)

K4 = [1.0841 0.7652 −0.0830 0.0283 4.7389] × 105. (40)

As can be inferred from figure 4, both Lemmas 1 and 2 fail to provide a solution for thecircle centred at (−40, 0) with radius r = 31. If the pole location specification is somewhatrelaxed, for instance, imposing the radius r = 33.6, Lemma 2 yields a robust static feedbackgain given by

K33.6 = [0.5293 0.5968 −0.0515 0.0256 1.8121

] × 105.

A feasible solution is provided by Lemma 1 for a slightly larger radius, r = 33.7, yielding

K33.7 = [0.5159 0.5960 −0.0506 0.0255 1.7530

] × 105.

The results of Theorem 1 allow a more stringent pole location specification, at the priceof knowing the uncertain parameter α. As in other similar scheduled control strategies, theuncertain parameter α must be measured or estimated on-line; differently from other schemes,the vertices matrices Wj , Zj , j = 1, . . . , N can be off-line computed and easily stored in adigital memory. The actual gain K(α) is computed as an explicit function of α through equation(36), by means, for example, of a cheap standard microprocessor.

Besides providing more stringent design specifications than fixed gains, the strategy pro-posed in this paper is cheaper than, for instance, computing on-line a standard LQR (linearquadratic regulator) or carrying out with parameter-dependent matrix equations the require-ment to evaluate for every incremental change on α (in a gain-scheduling technique withoutapproximations).

Note that Theorem 1 guarantees that the specified pole location is preserved for anyoccurrence of the uncertain parameter α. If a standard pole placement were used, even aninfinitesimal change in the parameter value could cause the closed-loop poles to move out ofthe desired region. In many gain-scheduling strategies, a few controllers are computed off-lineand an approximation is done to construct the on-line controller as a function (usually linear)of the uncertain parameter and, therefore, the design specifications cannot be assured outsidethe grid points.

Of course, the closed-loop stability with the desired pole location can be assured only if theparameter variation is slow enough to be accommodated by the dynamics of the system or ifthe changes on α are not time dependent (changes in operating conditions, load, etc.). Thisis exactly the case for the active suspension system analysed here, since the sprung mass m1

varies due to load conditions (where changes occur from time to time) and the parameters c

and k are supposed to vary as a function of β (which can be interpreted as an aging factor).Finally, to obtain the parameter α different techniques can be used. For instance, the sprung

mass m1 can be directly measured by means of a sensor. The values of k and c are both relatedto the parameter β (aging factor) (22), and can be easily inferred from the displacementsx2 and x1. Other methods could be used, or combined with direct measures, such as, forinstance, ARMAX estimation [42]. The value of α is obtained from a simple linear equation

Pole location of a suspension system 575

involving the actual values of m1 and β (i.e. c and k) and the vertices of the uncertaintypolytope.

Two important factors for evaluating suspension systems are the vehicle stability (whichcan be associated with the tyre deflection x2) and the passenger ride comfort (related to thesprung mass velocity and acceleration). The time response of the states xi , i = 1, . . . , 4 andacceleration of the sprung mass (x3) for a unitary step of vertical velocity is shown in figure 7for several values of the uncertain parameters inside the intervals given in (22). As can beseen, the settling time (always smaller than 0.5 s) as well as the magnitudes are quite good foran active suspension system with uncertain parameters (for a comparison, see [2, 13, 14]).

Figure 8 shows the impulse response of x2 (tyre deflection), x3 (sprung mass velocity) and x3

(sprung mass acceleration) for several values of the uncertain parameters inside the intervalsgiven in (22). Again, the closed-loop time behaviour provided by the parameter-dependentfeedback gain is very good despite the presence of uncertainty.

A frequency diagram for each of the state variables with respect to the exogenous inputis shown in figure 9. The sprung mass acceleration in weighted rms values, according toISO2631 [43], is shown in figure 10 for two frequency ranges: from 1 to 80 Hz (top) andfrom 0.1 to 0.63 Hz (bottom). In the latter range of frequencies, associated with symptomssuch as motion sickness, it can be seen that the rms values obtained are far below the severediscomfort boundaries for an exposure time of 30 min [43]. Once again, good behaviour canbe seen, due to the pole location that was chosen and, in this case, can be achieved only bythe parameter-dependent feedback gain proposed in this paper. Of course, other pole locationcould be implemented with fixed gains, at the price of greater values of magnitude in thefrequency diagram and also greater settling times and overshoots.

Figure 7. Time-response of states xi , i = 1, . . . , 4 and acceleration of the sprung mass x3 for a unitary step appliedin w for several values of the uncertain parameters inside the intervals given by (22).

576 V. J. S. Leite and P. L. D. Peres

Figure 8. Impulse response of states x2, tyre deflection, x3, sprung mass velocity and acceleration of the sprungmass x3 for several values of m1, c and k inside the uncertainty intervals given by (22).

Figure 9. Frequency magnitude diagram of states xi , i = 1, . . . , 4 with respect to the exogenous input for severalvalues of m1, c and k inside the uncertainty intervals given by (22).

Pole location of a suspension system 577

Figure 10. Acceleration of the sprung mass x3 with respect to the exogenous input in weighted RMS valuesaccording to the ISO2631 [43] for frequency ranges from 1 to 80 Hz (top) and from 0.1 to 0.63 Hz (bottom).

Finally, note that different centres and radii could be chosen in order to cope with otherdesign specifications concerning road-holding ability and passenger comfort. However, onlythe parameter-dependent control strategy can be applied for circles with centre σ < 13 (seefigure 4).

5. Conclusion

Three conditions for pole location of uncertain continuous-time systems in polytope domainshave been applied for the control design of an active suspension system. A circular regioninside the left-hand half of the complex plane has been used, imposing on the closed-loopsystem the desired damping and oscillatory behaviour. In all three cases, the state feedbackgain is obtained from a LMI feasibility test which also provides the Lyapunov conditionassuring the desired pole location. The first condition is based on fixed gain and the Lyapunovfunction; the second condition provides a fixed gain and a parameter-dependent Lyapunovmatrix; the third condition furnishes a parameter-dependent gain and a parameter-dependentLyapunov function. It has been shown that a parameter-dependent control law can cope withmore stringent specifications which cannot be imposed by fixed gains, being an effectivealternative technique for the design of active suspension systems. Differing from other gain-scheduling strategies, the parameter-dependent control proposed in this paper assures thedesired dynamic behaviour for all values of parameters inside the uncertainty domain, as noapproximation is used since the actual control gain is computed on-line by means of a simpleanalytical expression.

578 V. J. S. Leite and P. L. D. Peres

Acknowledgements

This work is partially supported by the Brazilian agencies CAPES, CNPq, FAPEMIG (TEC1233/98) and FAPESP. The authors wish to thank the anonymous reviewers for their help-ful suggestions on this paper. Many thanks also to Professor Alberto Luiz Serpa, fromDMC/FEM/UNICAMP, for his help on mechanical vibration standardizations.

References

[1] Barmish, B.R., 1994, New Tools for Robustness of Linear Systems (NewYork: Macmillan Publishing Company).[2] ElMadany, M.M. and Al-Majed, M.I., 2001, Quadratic synthesis of active controls for a quarter-car model.

Journal of Vibration and Control, 7, 1237–1252.[3] Hrovat, D., 1990, Optimal active suspension structures for quarter-car vehicle models. Automatica, 26, 845–860.[4] Hrovat, D., 1993, Applications of optimal control to advanced automotive suspension design. ASME J. Dynamic

Systems, Measurement and Control, 115, 328–342.[5] Hrovat, D., 1997, Survey of advanced suspension developments and related optimal control applications.

Automatica, 33, 1781–1817.[6] Thompson, A.G., 1976, An active suspension with optimal linear state feedback. Vehicle System Dynamics, 5,

187–203.[7] Thompson, A.G., 1984, Optimal and suboptimal linear active suspensions for road vehicles. Vehicle System

Dynamics, 13, 61–72.[8] Thompson, A.G. and Davis, B.R., 1988, Optimal linear active suspensions with derivative constraints and output

feedback control. Vehicle System Dynamics, 17, 179–192.[9] Camino, J.F., Zampieri, D.E., Takahashi, R.H.C. and Peres, P.L.D., 1997, H2 and LQR active suspension

control schemes with uncertain parameters: a comparison, presented at the Proceedings of the 7th InternationalConference on Dynamic Problems in Mechanics, Vol. 1, Angra dos Reis, RJ, March, 1997, pp. 226–228.

[10] Hayakawa, K., Matsumoto, K., Yamashita, M., Suzuki, Y., Fujimori, K. and Kimura, H., 1999, Robust H∞-output feedback control of decoupled automobile active suspension systems. IEEE Transactions on AutomaticControl, 44, 392–396.

[11] Kuo, Y. and Li, T.S., 1999, GA-based fuzzy PI/PD controller for automotive active suspension system. IEEETransactions on Industrial Electronics, 46, 1051–1056.

[12] Liberzon, A., Rubinstein, D., and Gutman, P.O., 2001, Active suspension for single wheel station of off-roadtrack vehicle. International Journal of Robust and Nonlinear Control, 11, 977–999.

[13] Fialho, I. and Balas, G.J., 2002, Road adaptive active suspension design using linear parameter-varying gain-scheduling. IEEE Transactions on Control Systems Technology, 10, 43–54.

[14] Fialho, I.J. and Balas, G.J., 2000, Design of nonlinear controllers for active vehicle suspensions using parameter-varying control synthesis. Vehicle System Dynamics, 33, 351–370.

[15] Oustaloup, A., Moreau, X. and Nouillant, M., 1997, From fractal robustness to non integer approach in vibrationinsulation: the CRONE suspension, presented at the Proceedings of the 37th IEEE Conference on Decision andControl, Vol. 5, San Diego, CA, December, pp. 4979–4984.

[16] Sohn, H.-C., Hong, K.-S. and Hedrick, J.K., 2000, Semi-active control of the Macpherson suspension system:hardware-in-the-loop simulations, presented at the Proceedings of the 2000 IEEE International Conference onControl Applications, Anchorage, Alaska, September, pp. 982–987.

[17] Bernussou, J., Peres, P.L.D. and Geromel, J.C., 1989, A linear programming oriented procedure for quadraticstabilization of uncertain systems. Systems & Control Letters, 13, 65–72.

[18] Boyd, S., El Ghaoui, L., Feron, E. and Balakrishnan, V., 1994, Linear Matrix Inequalities in System and ControlTheory, SIAM Studies in Applied Mathematics (Philadelphia: SIAM).

[19] Geromel, J.C., Peres, P.L.D. and Souza, S.R., 1993, H2 guaranteed cost control for uncertain discrete-timelinear systems. International Journal of Control, 57, 853–864.

[20] Khargonekar, P.P. and Rotea, M.A., 1991, Mixed H2/H∞ control, a convex optimization approach. IEEETransactions on Automatic Control, 36 824–837.

[21] Scherer, C., Gahinet, P. and Chilali, M., 1997, Multiobjective output-feedback control via LMI optimization.IEEE Transactions on Automatic Control, 42, 896–911.

[22] Feron, E., Apkarian, P. and Gahinet, P., 1996, Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions. IEEE Transactions on Automatic Control, 41, 1041–1046.

[23] Gahinet, P., Apkarian, P. and Chilali, M., 1996, Affine parameter-dependent Lyapunov functions and realparametric uncertainty. IEEE Transactions on Automatic Control, 41, 436–442.

[24] Trofino, A., 1999, Parameter dependent Lyapunov functions for a class of uncertain linear systems: an LMIapproach, presented at the Proceedings of the 38th IEEE Conference on Decision and Control, Vol. 1, Phoenix,AZ, December, 1999, pp. 2341–2346.

[25] Daafouz, J., Bara, G.I. Kratz, F. and Ragot, J., 2000, State observers for discrete-time LPV systems: an inter-polation based approach, presented at the Proceedings of the 39th IEEE Conference on Decision and Control,Sydney, Australia, December, 2000, pp. 4571–4572.

Pole location of a suspension system 579

[26] Lyshevski, S.E., 1998, State-space model identification of deterministic nonlinear systems: nonlinear mappingtechnology and application of the Lyapunov theory. Automatica, 34, 659–664.

[27] Xiong, Y. and Saif, M., 2000, Sliding-mode observer for uncertain systems part I: linear system case, presentedat the Proceedings of the 39th IEEE Conference on Decision and Control, Vol. 1, Sydney, Australia, December,2000, pp. 316–321.

[28] Alleyne, A. and Hedrick. K., 1995, Nonlinear adaptive control of active suspensions. IEEE Transactions onControl Systems Technology, 3, 94–101.

[29] Haddad, W.M. and Bernstein, D.S., 1992, Controller design with regional pole constraints. IEEE Transactionson Automatic Control, 37, 54–69.

[30] Barmish, B.R., 1985, Necessary and sufficient conditions for quadratic stabilizability of an uncertain system.Journal of Optimization Theory and Applications, 46, 399–408.

[31] de Oliveira, M.C., Bernussou, J. and Geromel, J.C., 1999, A new discrete-time robust stability condition.Systems & Control Letters, 37, 261–265.

[32] Rajamani, R. and Hedrick, J.K., 1995, Adaptive observers for active automotive suspensions: theory andexperiment. IEEE Transactions on Control Systems Technology, 3, 86–93.

[33] Yokoyama, M., Hedrick, J.K. and Toyama, S., 2001, A model following sliding mode controller for semi-activesuspension systems with MR dampers, presented at the Proceedings of the 2001 American Control Conference,Vol. 5, Arlington, VA, June, 2001, pp. 2652–2657.

[34] Dahleh, M. and Dahleh, M.A., 1991, On slowly time-varying systems. Automatica, 27, 201–205.[35] Albert A., 1969, Conditions for positive and nonnegative definiteness in terms of pseudoinverses. SIAM Journal

on Applied Mathematics, 17, 434–440.[36] Chilali, M., Gahinet, P., and Apkarian, P., 1999, Robust pole placement in LMI regions. IEEE Transactions on

Automatic Control, 44, 2257–2270.[37] Garcia, G. and Bernussou, J., 1995, Pole assignment for uncertain systems in a specified disk by state-feedback.

IEEE Transactions on Automatic Control, 40, 184–190.[38] Palhares, R.M. and Peres, P.L.D., 1999, Robust H∞ filtering design with pole placement constraint via LMIs.

Journal of Optimization Theory and Applications, 102, 239–261.[39] Gahinet, P., Nemirovski, A., Laub, A.J. and Chilali, M., 1995, LMI Control Toolbox for Use with Matlab. User’s

Guide (The Math Works Inc: Natick, MA, USA).[40] Ramos, D.C.W. and Peres, P.L.D., 2001,A less conservative LMI condition for the robust stability of discrete-time

uncertain systems. Systems & Control Letters, 43, 371–378.[41] Ramos, D.C.W. and Peres, P.L.D., 2002, An LMI condition for the robust stability of uncertain continuous-time

linear systems. IEEE Transactions on Automatic Control, 47, 675–678.[42] Ljung, L., 1999, System Identification: Theory for the User, 2nd Edn (Englewood Cliffs, NJ: Prentice-Hall).[43] International Organization of Standardization, 1997, Mechanical Vibration and Shock-Evaluation of

Human Exposure to Whole-Body Vibration, ISO2631 (Geneva, Switzerland: International Organization ofStandardization).