Stability analysis and stabilization of LPV systems with jumps and

(piecewise) differentiable parameters using continuous and

sampled-data controllers

Corentin Briat∗

Abstract

Linear Parameter-Varying (LPV) systems with jumps and piecewise differentiable parameters is aclass of hybrid LPV systems for which no tailored stability analysis and stabilization conditions havebeen obtained so far. We fill this gap here by proposing an approach relying on the reformulationof the considered LPV system as an impulsive system that will incorporate, through a suitable stateaugmentation, information on both the dynamics of the state of the system and the considered class ofparameter trajectories. Conditions for the stability of the hybrid system, and hence that of the associatedLPV system, under both constant and minimum dwell-time are established. Those results are based onthe use of a clock- and parameter-dependent Lyapunov function that is enforced to be decreasing alongthe flow and the jumps of the system. An interesting adaptation of this result consists of a minimumdwell-time stability condition for LPV switched impulsive systems with time-varying dimensions. Theminimum dwell-time stability condition is notably shown to naturally generalize and unify the well-knownquadratic and robust stability criteria all together. Those conditions are then adapted to address thestabilization problem via timer-dependent and a timer- and/or parameter-independent (i.e. robust) state-feedback controllers, the latter being obtained from a relaxed minimum dwell-time stability conditioninvolving slack-variables. Finally, the last part addresses the stability of LPV systems with jumps undera range dwell-time condition which is then used to provide stabilization condition for LPV systemsusing a sampled-data state-feedback gain-scheduled controller. The obtained stability and stabilizationconditions are formulated as infinite-dimensional semidefinite programming problems which are solvedusing a relaxation approach based on sum of squares programming. Examples are given for illustrationof the results.Keywords: LPV systems; hybrid systems; sampled-data control; dwell-time; sum of squares

1 Introduction

Linear Parameter-Varying (LPV) systems [1, 2] are an important class of linear systems that can be usedto model linear systems that intrinsically depend on parameters [3] or to approximate nonlinear systems inthe objective of designing gain-scheduled controllers [4]; see [2] for a classification attempt of the differentfamilies of parameters. They have been, since then, successfully applied to a wide variety of real-worldsystems such as automotive suspensions systems [5], aircrafts [6, 7], etc. The field has also been enrichedwith very broad theoretical results and numerical tools [2, 8–17].

As discussed in [2, 16], LPV systems are often separated into two categories depending on whether theparameter trajectories are continuously differentiable or vary arbitrarily fast (possibly including disconti-nuities) - a very strong dichotomy as those classes of trajectories are quite far apart. As a consequence,this classification seems rather unadapted to parameter trajectories admitting sporadic discontinuities whilebeing differentiable in between. Indeed, such trajectories cannot be considered as differentiable but, at thesame time, considering them to be arbitrarily varying is also very restrictive since we would ignore the factthat, in between discontinuities, the parameter trajectories are differentiable. This remark motivated the

∗corentin@briat.info; http://www.briat.info

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consideration of LPV systems with piecewise constant parameters in [16] in order to illustrate the fact thatconsidering a more accurate description of the parameter trajectories can, in fact, predict the stability ofcertain systems for which other methods would have proven inconclusive. As the parameters were piece-wise constant, it was possible to readily adapt methods developed for switched/impulsive systems [18–20] toobtain sufficient stability conditions for such LPV systems through the introduction of dwell-time constraints.

The objective of the paper (and that of its conference version [21]) is to extend those results to the case ofimpulsive LPV systems with piecewise differentiable parameters. Such parameter trajectories may arise whenan (impulsive) LPV system is used to approximate a nonlinear impulsive system or in linear systems wherethe parameters naturally have such a behavior [22]. Finally, they can also be used to approximate parametertrajectories that exhibit intermittent very fast, yet differentiable, variations. The approach developed inthis paper consists of an extension of that in [16] for piecewise constant parameters. Using a clock- andparameter-dependent Lyapunov function, both the case of periodic and aperiodic jumps in the parametertrajectories are considered through the use of the concepts of constant and minimum dwell-time. A relaxedresult is also provided to address the case of uncertain systems subject to polytopic uncertainties and anadaptation of the result to the case of switched impulsive LPV systems are also given for completeness. Wethen prove that the obtained minimum dwell-time stability conditions generalize and unify the well-knownquadratic stability and robust stability conditions, which can be recovered as extremal/particular cases.The obtained stability conditions stated in terms of infinite-dimensional semidefinite programming problemssimilar to those obtained using the same approach [18–20]. To make these conditions computationallyverifiable, we propose a relaxation approach based on sum of squares programming [23–25]; i.e. semidefiniteprogramming [26,27]. The analysis part is validated on two relevant systems from the literature, one of thembeing known to be not quadratically stable.

Unlike in the conference version [21] of this paper, we also extend here the approach to the case ofLPV systems with jumps and to the design of continuous-time and sampled-data gain-scheduled state-feedback controllers. The problem of designing continuous-time controllers for LPV systems with piecewisedifferentiable parameters has never been addressed until now and a full solution is provided here. A possiblelimitation of the approach is that it leads to controllers which depend on the clock/timer variable, whichmay be difficult to implement in practice, in particular if the gain of the controller varies a lot over shorttime-scales. To remedy this issue, the relaxed stability result previously derived for the analysis of uncertainsystems is exploited to provide stabilization conditions using a timer-independent controller, a result that isalso novel. A minor modification of this result can also be considered for the design of parameter-independent(i.e. robust) controllers.

On the other hand, the problem of designing sampled-data gain-scheduled state-feedback controllers forstandard LPV systems has been solved in [28] using a discretization approach (assuming the parametersare piecewise constant), in [29] using the input-delay approach and in [30, 31] using looped-functionals.The advantage of the proposed approach is that it relies on an exact representation of the problem intoan impulsive system and the use of clock-dependent Lyapunov functions which have been shown to be asuitable approach for the design of controllers for linear hybrid systems or their analysis when subject touncertainties; see e.g. [18, 32–35] and references therein. Unlike functionals such as looped-functionals, thisclass of Lyapunov functions naturally leads to convex design conditions (see e.g. the discussion in [33]) dueto the presence of a lower number of decision variables. Several examples are given for illustration.

The main drawback of the approach is certainly its complexity since the approach relies on infinite-dimensional semidefinite programs whose SDP approximations are typically large in size (i.e. in both thenumber of variables and the number constraints). This is the price to pay to obtain accurate stability andstabilization conditions. Indeed, unfair comparative examples tend to suggest that the proposed approachleads to better results than previously obtained in the literature. However, it is expected in a near futureto have improvements at the solver-level that will make this kind of approaches applicable to larger sys-tems. Note, finally, that LMI methods are only restricted to small to medium size problems and they are ingeneral not applicable to large systems unless the resulting semidefinite programs satisfy certain convenientstructural properties, such as chordal sparsity [36, 37], that can be exploited by the solver to reduce thecomplexity and speed-up the solving time..

2

Outline. The structure of the paper is as follows: in Section 2 preliminary definitions and results aregiven. Section 3 develops the main stability results of the paper which are then extended to continuous andsampled-data state-feedback control in Section 4 and Section 5, respectively. The examples are treated inthe related sections.Notations. The set of nonnegative and positive integers are denoted by Z≥0 and Z>0, respectively. Theset of symmetric matrices of dimension n is denoted by Sn while the cone of positive (semi)definite matricesof dimension n is denoted by (Sn0) Sn0. For some A,B ∈ Sn, the notation that A ()B means thatA − B is positive (semi)definite. The maximum and the minimum eigenvalue of a symmetric matrix Aare denoted by λmax(A) and λmin(A), respectively. For a square matrix A, we define Sym[A] = A + AT .For any differentiable function f(x, y), the partial derivatives with respect to the first and second argumentevaluated at (x, y) = (x∗, y∗) are denoted by ∂xf(x∗, y∗) and ∂yf(x∗, y∗), respectively. For a function f , theright-handed limit at a point t in its domain is defined as f(t+) = lims↓t f(s).

2 Preliminaries

LPV systems with state-jumps are dynamical systems which can be described as

x(t) = A(t− tk, ρ(t))x(t), t ∈ (tk, tk+1], k ∈ Z≥0

x(t+k ) = J(ρ(tk))x(tk), k ∈ Z>0

x(0+) = x(0) = x0, t0 = 0(1)

where x, x0 ∈ Rn are the state of the system and the initial condition, respectively. The trajectories ofthe above system are left-continuous with right-handed limit (χ(t+), τ(t+)). The matrix-valued functionsA(·, ·), J(·) ∈ Rn×n are assumed to be piecewise continuous, hence bounded. The parameter vector trajectoryρ : R≥0 → P ⊂ RN , P compact, is assumed to be piecewise differentiable with derivative in D ⊂ RN , whereD is also compact and connected. When the parameters are independent of each other, then we can assumethat P is a box and that the following decompositions hold P =: P1× . . .×PN where Pi := [ρ

i, ρi], ρi ≤ ρi

and D =: D1 × . . . × DN where Di := [νi, νi], νi < 0 < νi. We also define the set of vertices of D as Dv;i.e. Dv := ν1, ν1 × . . . νN , νN. Note, however, that these sets are only considered here to simplifythe exposition of the results but, as it will be shown in the examples, any other set that can be describedby a set of polynomial inequalities (e.g. ellipses) can also be considered. It is also worth mentioning thatdiscontinuities are assumed to only occur at the times tk for any positive integer k and that no discontinuityoccurs at t0 = 0. This assumption allows one to simplify the exposition of the results by considering thedwell-time values Tk := tk+1 − tk where k ∈ Z≥0. If we were allowing such a discontinuity, we would simplyneed to change the initial condition to J(ρ(t0))x(t0) for some T−1 ∈ Z≥0 but, since the system is linear,stability properties of the solution would remain unchanged.

We can associate with the system (1) the following embedded discrete-time systems:

x(tk+1) = Φρ(tk+1, tk)J(ρ(tk))x(tk), k ≥ 1x(t1) = Φρ(t1, 0)x0

(2)

andx(t+k+1) = J(ρ(tk+1))Φρ(tk+1, tk)x(t+k ), k ≥ 0x(t+0 ) = x0

(3)

where Φρ(tk+1, tk) ∈Mk(ρ(t+k )) where

Mk($) :=

Ψ(tk+1)Ψ(tk)−1 : Ψ(s) = A(s, ρ(s))Ψ(s),Ψ(0) = I, ρ(tk) = $, ρ(s) ∈ P, ρ(s) ∈ Q(ρ(s))

(4)

and Q(ρ) = Q1(ρ)× . . .×QN (ρ) with

Qi(ρ) :=

Di if ρi ∈ (ρ

i, ρi),

Di ∩ R≥0 if ρi = ρi,

Di ∩ R≤0 if ρi = ρi.

(5)

3

We also define the setMk :=

⋃$∈P

Mk($). (6)

The setMk($) is the set of all the possible transition matrices mapping x(t+k ) 7→ x(tk+1) for every parametertrajectories obeying the boundedness and continuous differentiability assumptions and having $ as initialvalue; i.e. ρ(t+k ) = $. This set is very complex and cannot be explicitly computed even if the trajectory forthe parameter would be known in advance except in some very special and uninteresting cases, such as thescalar case with periodic parameter trajectory. In this regard, establishing the stability of the discrete-timesystems directly from their discrete-time formulation is formidable task. Some methods have been developedto address and, in fact, circumvent this problem. In particular, the looped-functional framework initiatedin [38] and adapted to the impulsive systems framework in [39] has been shown to overcome this difficulty byproviding discrete-time stability conditions which are affine in the data of the system and do not necessitatethe computation of the state-transition matrix. This made the overall framework directly applicable touncertain systems subject to both time-invariant and time-varying uncertainties. Another approach is basedon the use of so-called clock-dependent (or timer-dependent) Lyapunov functions which have been shownto achieve the same feat with the extra feature of being readily applicable to controller design using verysimple nonconservative algebraic manipulations; see e.g. [33] for more details. This latter approach will beconsidered in this paper.

3 Stability analysis of LPV systems with jumps and piecewisedifferentiable parameters

The objective of this section is to present the main stability analysis result of the paper. Section 3.1 presentsthe main result pertaining on the stability analysis under a constant dwell-time condition whereas Section3.2 addresses the minimum dwell-time case. This latter result is then connected to the usual quadratic androbust stability conditions in Section 3.5. Finally, some computational discussions are provided in Section3.6. Examples are treated in Section 3.7.

3.1 Stability under constant dwell-time

We will consider in this section the following family of periodically changing piecewise differentiable parametertrajectories

PT :=

ρ : R≥0 7→ P

∣∣∣∣ ρ(t) ∈ Q(ρ(t)), t ∈ (tk, tk+1], Tk = T ,t0 = 0, ρ(t0) = ρ(t+0 ), k ∈ Z≥0

(7)

where T > 0, and Tk := tk+1 − tkIn other words, the trajectories contained in this family can only exhibitjumps at the times tk = kT , k ∈ Z>0 and hence the distance between two potential successive discontinuities– the so-called dwell-time – is given by Tk := tk+1 − tk = T and is constant, whence the name constantdwell-time. Although not the most interesting case, the case constant dwell-time case is the simplest oneand allows us to easily set-up the main ideas. It can also be useful to treat the case of periodic parametersystems with periodic discontinuities as also considered in [33]. It is important to finally note that, for thissystem, we have thatMk defined in (6) verifiesM0 =Mk =Mk+1 for all k ≥ 1. This leads to the followingresult:

Theorem 1 (Constant dwell-time) Let T ∈ R>0 be given and assume that there exist a bounded contin-uously differentiable matrix-valued function P : [0, T ]× P 7→ Sn, P (0, θ) ∈ Sn0 and a scalar ε > 0 such thatthe conditions

∂τP (τ, θ) +

N∑i=1

∂ρiP (τ, θ)µi + Sym[P (τ, θ)A(τ, θ)] 0 (8)

andJ(η)ᵀP (0, θ)J(η)− P (T , η) + ε In 0 (9)

4

hold for all (τ, θ, η, µ) ∈ [0, T ]× P × P × Dv. Then, the LPV system (1) with parameter trajectories in PT

is uniformly exponentially stable. M

Proof : Let us consider the Lyapunov function V (x, τ, ρ) = xᵀP (τ, ρ)x. From the definition of P , thereexists positive parameters α1, α2 > 0 such that

α1||x||22 ≤ V (x, 0, ρ) ≤ α2||x||22 (10)

for all ρ ∈ P. From (13), we have that ε ||x||22 ≤ V (x, T , ρ) and because it is a quadratic form and P isbounded, there also exists a positive scalar α′2 such that V (x, T , ρ) ≤ α′2||x||22. Note also that, by virtue ofconvexity of the conditions, the condition (8) is feasible for all (τ, θ, η, µ) ∈ [0, T ]×P ×P ×Dv if and only ifit is feasible for all (τ, θ, η, µ) ∈ [0, T ]×P×P×D. Pre- and post-multiplying (8) by x(tk+τ)ᵀ and x(tk+τ),letting (θ, µ)← (ρ(tk + τ), ρ(tk + τ)), and integrating from 0 to Tk with respect to τ yields

V (x(tk+1), T , ρ(tk+1))− V (x(t+k , 0, ρ(t+k ))) ≤ 0. (11)

Using now the fact that x(tk+1) = Φρ(tk+1, tk)x(t+k ) where Φρ(tk+1, tk) ∈M(ρ(t+k )), we then obtain

x(t+k )ᵀ(Φρ(tk+1, tk)ᵀP (T , ρ(tk+1))Φρ(tk+1, tk)− P (0, ρ(t+k ))

)x(t+k ) ≤ 0 (12)

which holds for all x(t+k ) ∈ Rn, all Φρ(tk+1, tk) ∈ M(ρ(t+k )), and all ρ(t+k ) ∈ P. Considering now (13) withθ = ρ(t+k+1) and η = ρ(tk+1) yields

J(ρ(tk+1))ᵀP (0, ρ(t+k+1))J(ρ(tk+1)) + ε In P (T , ρ(tk+1)) (13)

which together with (12) imply that

x(t+k )ᵀ(Φ+(ρ(tk+1, tk))ᵀP (0, ρ(t+k+1))Φ+(ρ(tk+1, tk))− P (0, ρ(t+k ))

)x(t+k ) ≤ − ε ||Φρ(tk+1, tk)x(t+k )||22 (14)

where Φ+(ρ(tk+1, tk)) := J(ρ(tk+1))Φρ(tk+1, tk). This is equivalent to say that

V (x(t+k+1), 0, ρ(t+k+1))− V (x(t+k , 0, ρ(t+k ))) ≤ − ε β||x(t+k )||22 (15)

where β := min ||M0||22. Letting Vk := V (x(t+k ), 0, ρ(t+k )), the above expression can be reformulated asVk+1 − Vk ≤ − ε β||x(t+k )||22 which, together with (10), imply that

Vk+1 ≤ ψVk (16)

where ψ := 1− ε β/α1 ∈ (0, 1). Therefore, we have that Vk ≤ ψkV0 and that ||x(t+k )||22 ≤α2

α1ψk||x0||22. This

proves that Vk → 0 as k → ∞ and that ||x(t+k )|| → 0 as k → ∞. As a result, the system (3) is uniformlyexponentially stable whenever Tk = T . To prove that this also implies the stability of the system (1)-(7),let t ∈ (tk, tk+1], then from (8), we get that V (x(t), t − tk, ρ(t)) is nonincreasing on t ∈ (tk, tk+1] and sinceV (x(t), 0, ρ(t)) and V (x(t), T , ρ(t)) are both positive definite, then V (x(t), t− tk, ρ(t)) is positive definite aswell on t ∈ (tk, tk+1]. Since it also satisfies

V (x(t), t− tk, ρ(t)) ≤ V (x(t+k , 0, ρ(t+k ))), t ∈ (tk, tk+1], k ≥ 0 (17)

andV (x(t+k+1), 0, ρ(t+k+1)) ≤ ψV (x(t+k , 0, ρ(t+k ))). (18)

immediate calculations show that

V (x(t), t, ρ(t)) ≤ ψt/T−1V (x0, 0, ρ(0)) (19)

where V (x(t), t, ρ(t)) = V (x(t), t − tk, ρ(t)) for t ∈ (tk, tk+1] which proves the uniform exponential stabilityof the LPV system (1) with parameter trajectories in PT . ♦

It is interesting to note that even if we only impose the Lyapunov function to be positive definite at τ = 0and not for all timer values, we still have the Lyapunov function is still positive definite for all timer values.This is interesting as it reduces the computational complexity of the approach. The same remark is raisedin [18].

5

3.2 Stability under minimum dwell-time - Nominal case

Let us consider now the family of piecewise differentiable parameter trajectories given by

P>T :=

ρ : R≥0 7→ P

∣∣∣∣ ρ(t) ∈ Q(ρ(t)), t ∈ (tk, tk+1], Tk ≥ Tt0 = 0, ρ(t0) = ρ(t+0 ), k ∈ Z≥0

(20)

where Tk := tk+1 − tk and T > 0. We also make the assumption that the matrix of the system A(τ, θ) issuch that A(T + s, θ) = A(T ) for all s ≥ 0. This assumption is made since it will be of practical use forstabilizing an impulsive LPV system using a state-feedback controller.

This leads to the following result:

Theorem 2 (Minimum dwell-time) Assume that A(T + s, θ) = A(T ) for all s ≥ 0 and let T ∈ R>0 begiven. Assume further that there exist a matrix-valued function P : [0, T ] × P 7→ Sn, P (0, θ) ∈ Sn0, and ascalar ε > 0 such that the conditions

∂τP (τ, θ) +

N∑i=1

∂ρiP (τ, θ)µi + Sym[P (τ, θ)A(τ, θ)] 0, (21)

N∑i=1

∂ρiP (T , θ)µi + Sym[P (T , θ)A(T , θ)] + ε I 0 (22)

andJ(η)ᵀP (0, θ)J(η)− P (T , η) + ε I 0 (23)

hold for all (τ, θ, η, µ) ∈ [0, T ]×P ×P ×Dv. Then, the LPV system (1) with parameter trajectories in P>T

is uniformly exponentially stable.

Proof : As the proof is similar to that of Theorem 1, it will be only sketched and details will be onlygiven at places where the proofs differ. Let V (x, τ, ρ) := xᵀP (τρ)x as for Theorem 1, the Lyapunov functionis positive definite for all (τ, ρ) ∈ [0, T ] × P. Pre- and post-multiplying (42) by x(tk + τ)ᵀ and x(tk + τ),letting (θ, µ)← (ρ(tk + τ), ρ(tk + τ)) and integrating from 0 to T with respect to τ yields

V (x(tk + T ), T , ρ(tk + T ))− V (x(t+k ), 0, ρ(t+k )) ≤ 0 (24)

or, using the definition of V ,

x(t+k )ᵀ(Φρ(tk + T , tk)ᵀP (T , ρ(tk + T ))Φρ(tk + T , tk)− P (0, ρ(t+k ))

)x(t+k ) ≤ 0 (25)

which must both hold for all x(t+k ) ∈ Rn, all Φρ(tk+1, tk) ∈Mk(ρ(t+k ), and all ρ(t+k ) ∈ P.Pre- and post-multiply now (43) by x(t)ᵀ and x(t), substituting θ = ρ(t) for t ∈ [tk + T , tk+1] shows that

is equivalent to saying that

∂

∂tV (x(t), T , ρ(t)) ≤ − ε ||x(t)||22 ≤ −

ε

α2V (x(t), T , ρ(t)), t ∈ (tk + T , tk+1]. (26)

Therefore, we have that

V (x(t), T , ρ(t)) ≤ exp

(− ε

α2(t− tk − T )

)V (x(tk + T ), T , ρ(tk + T )), t ∈ [tk + T , tk+1], (27)

and this must hold for all x(tk + T ) ∈ Rn, all ρ(tk + T ) ∈ P and all ρ satisfying the boundedness andcontinuous differentiability conditions. Together with the previous result, we obtain that

V (x(t), T , ρ(t)) ≤ exp

(− ε

α2(t− tk − T )

)V (x(t+k ), 0, ρ(t+k )), t ∈ [tk + T , tk+1]. (28)

6

We now consider (44) with η ← ρ(tk+1), θ ← ρ(t+k+1), and pre-/post-multiply the expression by x(tk+1)ᵀ

and x(t+k+1), respectively, to get

V (x(t+k+1), 0, ρ(t+k+1))− V (x(tk+1), T , ρ(tk+1)) ≤ − ε ||x(t+k+1)||22. (29)

Combining this with (28) evaluated at t = tk+1, gives

V (x(t+k+1), 0, ρ(t+k+1)) ≤ exp

(− (Tk − T ) ε

α2

)V (x(t+k ), 0, ρ(t+k ))− ε ||x(t+k+1)||22 (30)

which finally implies thatV (x(t+k+1), 0, ρ(t+k+1)) ≤ ψV (x(t+k ), 0, ρ(t+k )) (31)

where

ψ := exp

(− (Tk − T ) ε

α2

)− ε

α1. (32)

Hence, we have that V (x(t+k ), 0, ρ(t+k ))→ 0 and, therefore, ||x(t+k )|| → 0 as k →∞. To prove that this alsothe case for all times, then from (42) and (28), we have that

V (x(t),minT , t− tk, ρ(t)) ≤ exp

(−max0, t− tk − T ε

α2

)V (x(t+k ), 0, ρ(t+k )) (33)

for all t ∈ (tk, tk+1] and all k ≥ 0. Therefore, if x(t+k )→ 0 as k →∞, then so is

supt∈(tk,tk+1]

V (x(t),minT , t− tk, ρ(t))

as k →∞. This proves that the LPV system (1) with parameter trajectories in P>T is uniformly exponen-tially stable. ♦

The result above is interesting for several reasons. One can see first that compared to the constantdwell-time case, one has only the extra condition (43) which characterizes the long run uniform exponentialstability of the flow part of the system. The condition (43), on the other hand, addresses the case where thedwell-time is larger than its minimum value where we only require the Lyapunov function to be nonincreasing.Another interesting point is that despite the fact that dwell-time can be arbitrarily large, one just needsto verify the conditions for all dwell-time values in the interval [0, T ], which is obviously bounded. Thisdramatically simplifies the problem as the boundedness of the Lyapunov function will trivially follow fromits continuity. If we were to check the conditions for dwell-time values in [0,∞), however, we would run intothe difficulty that no polynomial Lyapunov function xᵀP (τ, ρ)x can be bounded on τ ∈ [0,∞). This wouldalso restrict a lot the families of matrices A(τ, θ) that could be considered.

3.3 Stability under minimum dwell-time - Uncertain case

It seems interesting to address the uncertain case where the matrices of the system are subject to constantpolytopic uncertainties, that is, with a slight abuse of language, the matrices of the system are now consideredto be given by

A(τ, θ, λ) =

M∑j=1

λjAj(τ, θ) and J(θ, λ) =

M∑j=1

λjJj(θ) (34)

where λ ∈ ΛM where ΛM is the M -unit simplex defined as

ΛM :=λ ∈ RM≥0 : ||λ||1 = 1

. (35)

Dealing with such a type of uncertainties can be done using a Lyapunov function that depends affinely onthe polytopic parameters. However, due to the product between the Lyapunov matrix P and the matrices

7

of the system A and J , the resulting LMI would be quadratic in the polytopic parameter and nonconvexin general. An elegant way for solving this issue is through the use of lifted conditions and the use of slackvariables: The, we have the following result:

Theorem 3 (Minimum dwell-time) Assume that Ai(T + s, θ) = Ai(T ), i = 1, . . . ,M , for all s ≥ 0 andlet T ∈ R>0 be given and assume that there exist matrix-valued functions Pj : [0, T ]×P 7→ Sn, Pj(0, θ) ∈ Sn0,j = 1, . . . ,M , X1, X2 : [0, T ] × P × Dv 7→ Rn×n, Z1, Z2 : P × P 7→ Rn×n, and a scalar ε > 0 such that theconditions [

−Sym[X1(τ, θ, µ)] X1(τ, θ, µ)ᵀAj(τ, θ)−X2(τ, θ, µ) + Pj(τ, θ)

? ∂τP (τ, θ) +∑Ni=1 ∂ρiPj(τ, θ)µi + Sym[X2(τ, θ, µ)ᵀAj(τ, θ)]

] 0 (36)

[−Sym[X1(T , θ, µ)] X1(T , θ, µ)ᵀAj(T , θ)−X2(T , θ, µ) + Pj(R, θ)

?∑Ni=1 ∂ρiPj(T , θ)µi + Sym[X2(T , θ, µ)ᵀAj(T , θ)] + ε I

] 0 (37)

and [Pj(0, θ)− Sym[Z1(θ, η)] Z1(θ, η)ᵀJj(η)− Z2(θ, η)

−Pj(T , η) + Sym[Z2(θ, η)ᵀJj(η)] + ε I

] 0 (38)

hold for all (τ, θ, η, µ) ∈ [0, T ]×P ×P ×Dv and all j = 1, . . . ,M . Then, the uncertain LPV system (1)-(34)with parameter trajectories in P>T is uniformly exponentially stable.

Proof : Multiplying the conditions by λj and summing over j = 1, . . . ,M yields the same conditions asin Theorem 2 with (Pj(τ, θ), Aj(τ, θ), Jj(θ)) replaced by (P (τ, θ, λ), A(τ, θ, λ), J(θ, λ)) where P (τ, θ, λ) =∑Mj=1 Pj(τ, θ), A(τ, θ, λ) =

∑Mj=1Aj(τ, θ) and J(θ, λ) =

∑Mj=1 λjJj(θ, λ). The rest of the proof is based on

a direct application of Finsler’s lemma [40]. Indeed, the condition (59) can be reformulated as[0 P (τ, θ, λ)

P (τ, θ, λ) ∂τP (τ, θ, λ) +∑Ni=1 ∂ρiP (τ, θ, λ)µi

]+ Sym

([X1(τ, θ, µ)ᵀ

X2(τ, θ, µ)ᵀ

] [−I A(τ, θ)

]) 0. (39)

By virtue of Finsler’s Lemma, this inequality is equivalent to[A(τ, θ, λ)

I

]ᵀ [0 P (τ, θ, λ)

P (τ, θ, λ) ∂τP (τ, θ, λ) +∑Ni=1 ∂ρiP (τ, θ, λ)µi

] [A(τ, θ, λ)

I

] 0 (40)

which is readily shown to be identical to (59) with A(τ, θ) and P (τ, θ) replaced by A(τ, θ, λ) and P (τ, θ, λ),respectively. The equivalence of the other conditions with (60) and (61) are proved in the exact same way.♦

Interestingly, the decoupling between the Lyapunov matrix P and the matrices of the system A andJ is not only useful for efficiently dealing with polytopic uncertainties but can also be used to enforcecertain constraints on the controller when stabilization is aimed. The explanation is that when structuralconstraints are enforced on the controller, keeping the problem convex can only be achieved by enforcinga similar structure on the Lyapunov function, thereby making the approach more conservative. However,when using the relaxed condition, the structural conditions are only imposed on the slack-variables X1 andX2, and the Lyapunov function can be left unchanged, thereby limiting the increase of the conservatism.This will be further details in Section 4.

3.4 Connection with switched impulsive LPV systems

If we extend the original system by adding a piecewise constant parameter σ which takes values in 1, . . . ,M,we obtain the following class of systems taking the form of a switched impulsive LPV system with piecewise

8

differentiable parameters: x(t) = A(t− tk, ρ(t))x(t)ρ(t) ∈ Q(ρ(t))σ(t) = 0

∣∣∣∣∣∣ if t ∈ (tk, tk+1], k ≥ 0x(t+k ) = J(ρ(tk), σ(tk), σ(t+k ))x(tk)ρ(t+k ) ∈ Pσ(t+k ) ∈ 1, . . . ,M

∣∣∣∣∣∣ if k ≥ 1 .

(41)

Note that the jump map depend on both the current value of the discrete-parameter and the next one.While this may look non-causal at first sight, this is actually not the case since one can decide first the nextvalue for the parameters ρ and σ and then make the state jump accordingly. Whenever, J = In we recovera standard LPV switched system. It is interesting to note that the above system can be used to representsystems with time-varying dimensions in both the state of the system and the parameter vector ρ by suitablychoosing the matrices J(ρ(tk), σ(tk), σ(t+k )) and the set P which can be now made mode-dependent.

This leads to the following result:

Corollary 4 (Minimum dwell-time - Switched LPV systems) Assume that Aj(T + s, θ) = Aj(T ),j = 1, . . . ,M , for all s ≥ 0 and let T ∈ R>0 be given. Assume further that there exist a matrix-valuedfunction P : [0, T ]× P × 1, . . . ,M 7→ Sn, P (0, θ, σ) ∈ Sn0, and a scalar ε > 0 such that the conditions

∂τP (τ, θ, σ) +

N∑i=1

∂ρiP (τ, θ, σ)µi + Sym[P (τ, θ, σ)A(τ, θ, σ)] 0, (42)

N∑i=1

∂ρiP (T , θ, σ)µi + Sym[P (T , θ, σ)A(T , θ, σ)] + ε I 0 (43)

andJ(η, σ, σ+)ᵀP (0, θ, σ+)J(η, σ, σ+)− P (T , η, σ) + ε I 0 (44)

hold for all σ, σ+ ∈ 1, . . . ,M, σ+ 6= σ, and all (τ, θ, η, µ) ∈ [0, T ] × P × P × Dv. Then, the switchedimpulsive LPV system (41) with parameter trajectories in P>T is uniformly exponentially stable.

3.5 Connection with quadratic and robust stability

Following the same lines as in [16], it can be shown that the minimum dwell-time result stated in Theorem2 naturally generalizes and unifies the quadratic and robust stability conditions for LPV systems withJ(ρ) = In through the concept of minimum dwell-time. We recall first the notions of quadratic and robuststability:

Definition 5 The LPV system (1) with J(ρ) = In and parameter trajectories defined in P>T in (20) is

(a) quadratically stable if there exists a matrix P ∈ Sn0 such that

A(θ)ᵀP + PA(θ) ≺ 0 (45)

holds for all θ ∈ P.

(b) robustly stable if there exists a continuously differentiable matrix-valued function P : P 7→ Sn0 suchthat

N∑i=1

∂ρiP (θ)µi +A(θ)ᵀP (θ) + P (θ)A(θ) ≺ 0 (46)

holds for all θ ∈ P and all µ ∈ Dv.

9

We then have the following result:

Theorem 6 Let us consider the LPV system (1) with J(ρ) = In and parameter trajectories defined in P>T

in (20). Then, the following statements hold:

(a) When T = 0 and tk → ∞ as k → ∞, then the conditions of Theorem 2 reduce the quadratic stabilitycondition in Definition 5.

(b) When T =∞, then the conditions of Theorem 2 reduce to robust stability condition in Definition 5.

Proof : We prove first statement (a). First note that since T = 0 and tk →∞ as k →∞, then there is noaccumulation point in the sequence of jumping instants and, therefore, the solution of the hybrid system iscomplete. Clearly, if T = 0, then we will have that P (T , θ) = P (0, θ) and P (τ, θ) = P (T , θ) for all τ ≥ T .Hence, we obtain that P (τ, θ) is constant for each θ and equal to P (0, θ). Substituting that expression inthe jump condition yields

P (0, θ)− P (τ, η) = P (0, θ)− P (0, η) 0,P (0, η)− P (τ, θ) = P (0, η)− P (0, θ) 0

(47)

where it is assumed that η 6= θ and where the second one is obtained by swapping the parameter values.Therefore, for those expressions above to be satisfied, we just need to have that P (0, η) − P (0, θ) = 0 forall η 6= θ. This can only be true if S is actually independent of the parameter and hence we need to haveP (τ, θ) = P for some P 0. Substituting that in (43)-(42) yield the quadratic stability condition (45).

To prove statement (b), it is enough to remark that when T = ∞, the jump set D is never reachedand the system reduces to a continuous-time system. Dropping then the timer dependence i.e. (lettingP (τ, θ) = P (θ)) and ignoring the condition (44) yield the robust stability condition as the conditions (43)-(42) both reduce to (46) in this case. ♦

3.6 Computational considerations

Box 1: SOS program associated with Theorem 2

Find polynomial matrices S,Γ0,Γi,Ωi : [0, T ]×P 7→ Sn, Θi,Ξi : P ×P 7→ Sn, Υ0,Υi : [0, T ]×P ×Dv 7→ Sn,i = 1, . . . ,M , such that

• Γ0,Γi,Θi,Ξi,Υi,Υ0,Ωi, i = 1, . . . ,M , are SOS for all µ ∈ Dv,

• P (τ, θ)−∑Mi=1 Γi(τ, θ)gi(θ)− Γ0(τ, θ)τ(T − τ)− ε In is SOS

• −∑Ni=1 ∂ρiP (τ, θ)µi−∂τP (τ, θ)−Sym[P (τ, θ)A(θ)]−

∑Mi=1 Υi(τ, θ, µ)gi(θ)−Υ0(τ, θ, µ)τ(T −τ) is SOS

for all µ ∈ Dv

• P (T , η)− J(η)ᵀP (0, θ)J(η)−∑Ni=1 Θi(θ, η)gi(θ)−

∑Mi=1 Ξi(θ, η)gi(η)− ε In is SOS

• −∑Ni=1 ∂ρiP (T , θ)µi − Sym[P (T , θ)A(θ)]−

∑Mi=1 Ωi(θ, µ)gi(θ)− ε In is SOS for all µ ∈ Dv

The conditions formulated in Theorem 2 are infinite-dimensional semidefinite programs which can notbe solved directly. To make them tractable, we propose to consider an approach based on sum of squaresprogramming [24] that will result in an approximate finite-dimensional semidefinite program which can thenbe solved using standard solvers such as SeDuMi [27]. The conversion to a semidefinite program can beperformed using the package SOSTOOLS [25] to which we input the SOS program corresponding to theconsidered conditions. We illustrate below how an SOS program associated with some given conditions canbe obtained. The set P defined in Section 2 can be described as

P =:θ ∈ RN : gi(θ) ≥ 0, i = 1, . . . ,M

(48)

10

for some polynomials gi : RN 7→ R, i = 1, . . . ,M and further note that

[0, T ] =τ ∈ R : f(τ) := τ(T − τ) ≥ 0

. (49)

In what follows, we say that a symmetric polynomial matrix Θ(·) is a sum of squares matrix (SOS matrix)or is SOS, for simplicity, if there exists a polynomial matrix Ξ(·) such that Θ(·) = Ξ(·)TΞ(·). The followingresult provides the sum of squares formulation of Theorem 2:

Proposition 7 Let ε, T > 0 be given and assume that the sum of squares program in Box 1 is feasible.Then, the conditions of Theorem 2 hold with the computed polynomial matrix P (τ, θ) and the system (1) isasymptotically stable for all ρ ∈P>T .

Remark 8 When the parameter set P is also defined by equality constraints hi(θ) = 0, i = 1, . . . ,M ′, theseconstraints can be simply added in the sum of squares programs in the same way as the inequality constraints,but with the particularity that the corresponding multiplier matrices be simply symmetric instead of being SOSmatrices.

3.7 Examples

We consider now two examples. The first one is a 2-dimensional toy example considered in [41] whereasthe second one is a 4-dimensional system considered in [3] and inspired from an automatic flight controldesign problem. The numerical calculations have been performed using the package SOSTOOLS [25] andthe semidefinite solver SeDuMi [27] on a PC equipped with 12GB of RAM and a processor Intel i7-950 @3.07Ghz.

Example 9 Let us consider here the system (1) with the matrices J(ρ) = In and [16, 41]

A(ρ) =

[0 1

−2− ρ −1

](50)

where the time-varying parameter ρ(t) takes values in P = [0, ρ], ρ > 0. It is known [41] that this systemis quadratically stable if and only if ρ ≤ 3.828 but it is was later proven in the context of piecewise constantparameters [16] that this bound can be improved provided that discontinuities do not occur too often. Wenow apply the conditions of Theorem 2 in order to characterize the impact of parameter variations betweendiscontinuities. To this aim, we consider that |ρ(t)| ≤ ν with ν ≥ 0 and that ρ ∈ 0, 0.1, . . . , 10. For eachvalue for the parameter upper-bound ρ in that set, we solve for the conditions of Theorem 2 to get estimates(i.e. upper-bounds) for the minimum dwell-times. We use here ε = 0.01 and polynomials of degree 4 inthe sum of squares programs. Note that we have, in this case, M = 1, M ′ = 0 and g1(θ) = θ(ρ − θ). Thecomplexity of the approach can be evaluated here through the number of primal and dual variables of thesemidefinite program which are 2409 and 315, respectively. The average preprocessing and solving times aregiven by 6.04sec and 1.25sec, respectively. The results are Fig. 1 where we can see that the obtained minimumvalues for the dwell-times increase with the rate of variation ν of the parameter, which is an indicator of thefact that increasing the rate of variation of the parameter tends to destabilize the system and, consequently,the dwell-time needs to be increased in order to preserve the overall stability of the system.

Example 10 Let us consider now the system (1) with the matrices J(ρ) = In and [3, p. 55]:

A(ρ) =

3/4 2 ρ1 ρ2

0 1/2 −ρ2 ρ1

−3υρ1/4 υ (ρ2 − 2ρ1) −υ 0−3υρ2/4 υ (ρ1 − 2ρ2) 0 −υ

(51)

where υ = 15/4 and ρ ∈ P = z ∈ R2 : ||z||2 = 1. It has been shown in [3] that this system is not quadrati-cally stable but was proven to be stable under minimum dwell-time equal to 1.7605 when the parameter trajec-tories are piecewise constant [16]. We propose now to quantify the effects of smooth parameter variations be-tween discontinuities. Note, however, that the set P is not a box as considered along the paper but, as the next

11

7;0 2 4 6 8 10

7 T

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.68 = 08 = 18 = 38 = 58 = 10

Figure 1: Evolution of the computed minimum upper-bound on the minimum stability-preserving minimumdwell-time using Theorem 2 for the system (1)-(50) with |ρ| ≤ ν using an SOS approach with polynomialsof degree 4.

Table 1: Evolution of the computed minimum upper-bound on the minimum dwell-time using Theorem 2for the system (1)-(51) with |β| ≤ ν using an SOS approach with polynomials of degree d. The number ofprimal/dual variables of the semidefinite program and the preprocessing/solving time are also given.

ν = 0 ν = 0.1 ν = 0.3 ν = 0.5 ν = 0.8 ν = 0.9 primal/dual vars. time (sec)

d = 2 2.7282 2.9494 3.5578 4.6317 11.6859 26.1883 9820/1850 20/27d = 4 1.7605 1.8881 2.2561 2.9466 6.4539 num. err. 43300/4620 212/935

calculations show, this is not a problem since a proper set for the values of the derivative of the parameters canbe defined. To this aim, let us define the parametrization ρ1(t) = cos(β(t)) and ρ2(t) = sin(β(t)) where β(t)is piecewise differentiable. Differentiating these equalities yields ρ1(t) = −β(t)ρ2(t) and ρ2(t) = β(t)ρ1(t)where β(t) ∈ [−ν, ν], ν ≥ 0, at all times where β(t) is differentiable. In this regard, we can consider β as anadditional parameter that enters linearly in the stability conditions and hence the conditions can be checkedat the vertices of the interval, that is, for all β ∈ −ν, ν. Note that, in this case, we have M = 0, M ′ = 1and h1(ρ) = ρ2

1 + ρ22 − 1.

We now consider the conditions of Theorem 2 and we get the results gathered in Table 1 where we can seethat, as expected, when ν increases then the minimum dwell-time has to increase to preserve stability. Usingpolynomials of higher degree allows to improve the numerical results at the expense of an increase of thecomputational complexity. As a final comment, it seems important to point out the failure of the semidefinitesolver due to too important numerical errors when d = 4 and ν = 0.9.

12

4 Stabilization using hybrid state-feedback LPV controllers

We consider in this section the following extension for the system (1)

x(t) = A(t− tk, ρ(t))x(t) +Bc(t− tk, ρ(t))uc(t), t ∈ (tk, tk+1], k ∈ Z≥0

x(t+k ) = J(ρ(tk))x(tk) +Bd(ρ(tk))ud(k), k ∈ Z>0

x(t+0 ) = x(t0) = x0, t0 = 0

(52)

where x, x0 ∈ Rn, uc ∈ Rmc and ud ∈ Rmd are the state of the system, the initial condition, the continuous-time control input and the discrete-time control input, respectively. The same assumptions as for the system(1) are made for the above system.

4.1 Stabilization by timer-dependent controllers

We consider in this section the following class of state-feedback controllers

uc(tk + τ) =

Kc(τ, ρ(tk + τ))x(tk + τ), τ ∈ [0, T ],

Kc(T , ρ(tk + τ))x(tk + τ), τ ∈ (T , Tk],

ud(k) = Kd(ρ(tk))x(tk)

(53)

where Kc(·, ·) ∈ Rmc×n and Kd(·) ∈ Rmd×n are the gains of the controller we would like to determine. Themotivation for considering this structure stems from the fact that it mimics the structure of the conditions inTheorem 2 where the condition (42) defined over [0, T ] is timer-dependent and the condition (43) for dwell-times τ ≥ T is timer-independent for which the value is locked to T . Such a structure has been consideredbefore, for instance, in [18–20,32]in the context of switched and (stochastic) impulsive systems, respectively.

As the constant dwell-time case can be easily recovered by dropping the timer-independent condition inthe minimum dwell-time conditions, only the latter one is addressed here:

Theorem 11 (Minimum dwell-time) Assume that A(T+s, θ) = A(T ) and let T ∈ R>0 be given. Assumefurther that there exist a continuously differentiable matrix-valued function P : [0, T ]×P 7→ Sn, P (0, θ) ∈ Sn0,a matrix-valued function Uc : [0, T ]× P 7→ Rmc×n, a matrix-valued function Ud : P 7→ Rmd×n, and a scalarε > 0 such that the conditions

−N∑i=1

∂ρi P (T , θ)µi + Sym[A(T , θ)P (T , θ) +Bc(T , θ)Uc(T , θ)] + ε In 0, (54)

− ∂τ P (τ, θ)−N∑i=1

∂ρi P (τ, θ)µi + Sym[A(τ, θ)P (τ, θ) +Bc(τ, θ)Uc(τ, θ)] + ε In 0 (55)

and [−P (0, η) [J(θ)P (T , θ) +Bd(θ)Ud(θ)]

ᵀ

? −P (T , θ)

] 0 (56)

hold for all (τ, θ, η, µ) ∈ [0, T ] × P × P × Dv. Then, the LPV system (52)-(53) with parameter trajectoriesin P>T is uniformly exponentially stable with the controller gains Kc(τ, θ) = Uc(τ, θ)P (τ, θ)−1 and Kd(θ) =Ud(θ)P (T , θ)−1. M

Proof : The state matrix of the continuous-time part of the closed-loop system (52)-(53) is given by

Acl(τ, θ) =

A(θ) +Bc(θ)Kc(τ, θ) τ ∈ [0, T ]A(θ) +Bc(θ)Kc(T , θ), τ ∈ (T , Tk],

(57)

while the state-matrix of the discrete-time part is given by Jcl(θ) := J(θ) + Bd(θ)Kd(θ). Substitutingthis matrix in the conditions of Theorem 2 and performing a congruence transformation with respect to

13

the matrix P (τ, θ) = P (τ, θ)−1 yields the conditions (54)-(55) where we have used the change of vari-ables Uc(τ, θ) = Kc(τ, θ)P (τ, θ) and the facts that −∂τ P (τ, θ) = P (τ, θ)∂τP (τ, θ)P (τ, θ) and −∂ρP (τ, θ) =

P (τ, θ)∂ρP (τ, θ)P (τ, θ). Finally, the condition (56) is obtained from (44) using successively a Schur com-

plement, a congruence transformation with respect to diag(P (0, θ), I) and the change of variables Ud(θ) =Kd(θ)P (T , θ). This proves the desired result. ♦

As for the previously obtained results, the conditions in Theorem 11 can be checked using sum of squaresand convex programming since the conditions are convex in the decision variables P , Uc and Ud. We can alsosee that the chosen structure for the continuous-time controller matches perfectly the stability conditions andthis matching allows us to use elementary and well-known linearizing change of variables. Should we havechosen a different structure, finding a linearizing change of variables would have been way more challenging.A difficulty that may arise when implementing timer-dependent controllers is that the initial timer valuemay be unknown and there my be a mismatch between the actual timer value and the implemented one.This may result in an unstable initial transient phase of finite duration. However, this will not affect thelong-term properties of the system. Another difficulty with the use of timer-dependent controllers is thatthey cannot be easily discretized, especially when the sensitivity of the gain with respect to the timer ishigh. As the timer has unit-rate, we can solve this problem by choosing a sufficiently small sampling periodand/or a small minimum dwell-time value for the design. Another solution, addressed in the next section, isthe consideration of timer-independent controller.

4.2 Stabilization by timer-independent controllers

Due to the possible difficulty of implementing timer-dependent controllers, we suggest to design timer-independent ones taking the form

uc(t) = Kc(ρ(t))x(t), t 6= tk, k ≥ 0ud(k) = Kd(ρ(tk))x(tk), k ≥ 1

(58)

where Kc(·) ∈ Rmc×n and Kd(·) ∈ Rmd×n are the gains of the controllers. Even though the structure ofthis controller is simpler than that of the timer-dependent controller, its design is more cumbersome as itsstructure is not adapted to that of the clock-dependent stability conditions. Indeed, the difficulty in thedesign of such a controller lies in the fact that, in Theorem (11), the gain Kc of the controller is obtained usingthe expression Kc(τ, θ) = Uc(τ, θ)P (τ, θ)−1. It is immediate to see that a timer-independent controller can beobtained by setting both Uc(τ, θ) and P (τ, θ) to be timer-independent. However, making this simplificationwould not result in a stability condition under minimum dwell-time but in a stability result under arbitrarydwell-time, which is unfeasible in most scenarios. In this regard, it is not reasonable to set the Lyapunovfunction to be timer-independent. Adding the constraint that ∂τKc(τ, θ) = 0 for all θ ∈ P is definitelyanother interesting option but this constraint is clearly nonconvex and, therefore, difficult to consider. Tothe best of the author’s knowledge, this has never been considered. A rather well-known solution to thisproblem is the transfer of the timer-independent constraint to another decision variable than the Lyapunovmatrix. This can be possibly achieved through the use of so-called slack-variables [2,42], which leads to thefollowing result:

Theorem 12 (Minimum dwell-time - Timer-independent conditions) Assume that A(T + s, θ) =A(T ) and let T ∈ R>0 be given. Assume further that there exist a continuously differentiable matrix-valuedfunction P : [0, T ] × P 7→ Sn0, a matrix-valued function Uc : P 7→ Rmc×n, a matrix-valued functionUd : P 7→ Rmd×n, and a scalar ε > 0 such that the conditions[

−(X + Xᵀ) A(τ, θ)X +Bc(τ, θ)Uc(θ)− Xᵀ + P (τ, θ)

? ∂τ P (τ, θ) +∑Ni=1 ∂ρi P (τ, θ)µi + Sym[A(τ, θ)X +Bc(τ, θ)Uc(θ)]

] 0 (59)

[−(X + Xᵀ) A(T , θ)X +Bc(T , θ)Uc(θ)− Xᵀ + P (τ, θ)

?∑Ni=1 ∂ρi P (T , θ)µi + Sym[A(T , θ)X +Bc(T , θ)K(θ)] + ε I

] 0 (60)

14

and [P (0, θ)− (X + Xᵀ) J(η)X +Bd(η)Ud(η)− Xᵀ

? −P (T , η) + Sym[J(η)X +Bd(η)Ud(η)] + ε I

] 0 (61)

hold for all (τ, θ, η, µ) ∈ [0, T ]×P ×P ×Dv. Then, the LPV system (52)-(77) with parameter trajectories inP>T is uniformly exponentially stable with the controller gains Kc(θ) = Uc(θ)X

−1 and Kd(θ) = Ud(θ)X−1.

M

Proof : The proof is based on the substitution of the matrices of the closed-loop system in the conditionsof Theorem 3 with the simplification X1 = X2 = Z1 = Z2 = X for some constant matrix X ∈ Rn×n.Performing a congruence transformation on the conditions with respect to diag(X, X) where X = X−1 andletting Uc(θ) = Kc(θ)X, Ud(η) = Kd(θ)X P (·, ·) = XᵀP (·, ·)X yield the conditions. ♦

It is worth mentioning that the choice that X1 = X2 = Z1 = Z2 = X allows us to obtain convexdesign conditions at the expense of some conservatism. Other possibilities exist but would introduce extrascalar parameters to be tuned manually, which would have resulted in a more complicated design procedure.However, we will see in the example of the next section that the above theorem may still yield interestingresults. It is also worth mentioning that other constraints can be enforced such as a block-structuredcontroller can be obtained by enforcing the matrix X to have the same block-structure (which should bestable by inversion). A simple example is the case of block diagonal controller where Uc and X are set to beblock diagonal. Robust (i.e. parameter independent) controllers can also be designed using this method byrestricting the matrices Uc and Ud to be constant matrices.

4.3 Example

Let us consider back the example from [16]

x =

[3− ρ 11− ρ 2 + ρ

]x+

[1

1 + ρ

]uc, J = In (62)

where P = [0, ρmax], ρmax = 1, and D = [−ν, ν]. It was proved in [16] that this system cannot be stabilizedquadratically. This latter property makes it a perfect example to illustrate the proposed approach sinceneither quadratic nor robust stabilization results can be used here. Applying then Theorem 11 with T = 0.05,we find that the conditions are feasible for ν ∈ 0, 0.1, 0.3 for d = 2 and ν ∈ 0.5, 0.7, 0.9, 1, 2 for d = 3.When d = 2 the number of primal/dual variables is 834/180 whereas, when d = 3, this number is 2414/315.Finally, when d = 2, it takes roughly 2.62sec to solve the problem whereas, in the case d = 3, it takes around6.31sec. For simulation purposes, we consider the parameter trajectory

ρ(tk + τ) =ρmax

2

[1 + sin

(2ν(tk + τ)

ρmax+ ϕk

)], k ∈ Z≥0 (63)

where ϕk is a uniform random variable taking values in [0, 2π] and we generate a random sequence ofinstants satisfying the minimum dwell-time condition. At each of these time instants, we draw a new valuefor the random variable ϕk, which introduces a discontinuity in the trajectory. Note, however, that betweendiscontinuities we have that |ρ(tk + τ)| ≤ ν for all τ ∈ (0, Tk], k ∈ Z≥0. We then obtain the results depictedin Fig. 2 for the case d = 3, ρmax = 1, and ν = 1 where we can see that stabilization is indeed achieved forthis system.

We now use Theorem 12 for the design of a timer-independent controller. Due to the conservatism ofthe relaxed conditions, it is not possible to stabilize the system for such a broad range of constraints as inthe timer-dependent case but it is possible to find a stabilizing controller in the case where T = 0.05, d = 2,ν = 1, and ρmax = 0.6. Note that the LMI constraint (61) is changed here to S(0, θ)− S(T , η) + ε I 0 heresince the state does not jump. The obtained matrices are given

X =

[10.028 30.29215.114 52.193

],K(θ) =

[−19.455− 8.4526θ − 19.131θ2

7.7740 + 3.9694θ + 9.1796θ2

]ᵀ, P (τ, θ) = [Pij(τ, θ)]

2i,j=1 (64)

15

0 0.5 1 1.5 2 2.5-2

0

2

4

6

8

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

Figure 2: Trajectories of the closed-loop system (62)-(53)-(63) with ν = 1, ρmax = 1 and T = 0.05.

whereP11(τ, θ) = 29.962 + 13.055τ + 0.43986θ + 0.34533τ2 − 19.903τθ + 0.044874θ2

P12(τ, θ) = 44.005 + 15.763τ + 0.62755θ − 0.66049τ2 − 24.719τθ + 0.10091θ2

P22(τ, θ) = 81.148 + 19.004τ + 0.59441θ − 0.072018τ2 − 22.198τθ + 0.31172θ2.

(65)

The simulation results are depicted in Figure 3 where we can observe that the controller stabilizes the systembut that it takes longer to reach equilibrium. Note also that the fact that we can find a timer-independentcontroller may be due to the fact that the minimum dwell-time is small and, hence, the range of values for τis small and, as a result, we can approximate fairly well a timer-dependent controller by a timer-independentone. The following case shows that the problem remains feasible for larger dwell-times. Indeed, the problemis still solvable with T = 1, e d = 2, ν = 1, ρmax = 0.6, and the matrices

X =

[7.6849 23.04511.582 39.644

],K(θ) =

[−19.233− 10.112θ − 15.726θ2

7.6689 + 4.8734θ + 7.3227θ2

]ᵀ, P (τ, θ) = [Pij(τ, θ)]ij (66)

whereP11(τ, θ) = 22.041 + 1.5769τ + 0.70604θ + 0.20096τ2 − 0.97129τθ − 0.30546θ2

P12(τ, θ) = 32.532 + 2.0088τ + 1.1276θ + 0.2084τ2 − 1.1802τθ + 0.57069θ2

P22(τ, θ) = 60.359 + 2.8099τ + 1.6351θ + 0.1849τ2 − 1.3145τθ + 1.9863θ2.

(67)

The simulation results are depicted in Figure 4 where we can observe that the controller stabilizes the system.

5 Stabilization using sampled-data LPV controllers

Let us consider now the case of sampled-data controllers. In this case, it seems reasonable to consider thefollowing class of continuously differentiable parameters:

P∞ :=ρ : R≥0 7→ P

∣∣ ρ(t) ∈ Q(ρ(t)), t ≥ 0. (68)

16

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2

0

2

4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

Figure 3: Trajectories of the closed-loop system (62)-(53)-(63)-(64) with ν = 5/3 and ρmax = 3/5 andT = 0.05.

0 1 2 3 4 5 6 7 8 9 10-2

0

2

4

6

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

Figure 4: Trajectories of the closed-loop system (62)-(53)-(63)-(66) with ν = 5/3 and ρmax = 3/5 and T = 1.

17

Discontinuities in the parameter trajectories have been removed in this context as having discontinuitiesoccurring at the same time as control updates does not motivate the use of any gain-scheduled controllerbecause the controller would be scheduled with an incorrect parameter value right at the beginning of thesampling interval. Hence, the use of gain-scheduled controllers is, in this case, not beneficial over that ofrobust controllers from a stabilization perspective. Note, however, that parameter discontinuities that wouldoccur at different times than control updates can be incorporated via the introduction of an extra clock/timervariable, at the expense of an increase of the model complexity and computational burden. This case is notbe treated here for brevity.

5.1 A preliminary stability result

We are interested here in deriving a stability result under a range dwell-time constraint for the sequence ofjumping instants, that is, for all sequences of jumping instants in the set

T :=tkk∈Z≥0

∣∣ Tk := tk+1 − tk ∈ [Tmin, Tmax], t0 = 0, k ∈ Z≥0

(69)

for some 0 ≤ Tmin ≤ Tmax <∞. The corresponding impulsive system is given byx(t) = A(ρ(t))x(t)ρ(t) ∈ Q(ρ(t))

∣∣∣∣ if t ∈ (tk, tk+1], k ≥ 0x(t+k ) = J(ρ(tk))x(tk)ρ(t+k ) = ρ(tk)

∣∣∣∣ if k ≥ 1 .(70)

With these definitions in mind, the we can now state the main stability result of the section:

Theorem 13 (Range dwell-time) Let the scalars 0 < Tmin ≤ Tmax < ∞ be given and assume that thereexist a continuously differentiable matrix-valued function P : [0, Tmax] × P 7→ Sn0 and a scalar ε > 0 suchthat the conditions

− ∂τP (τ , θ) +

N∑i=1

∂ρiP (τ , θ)µi + Sym[P (τ , θ)A(θ)] 0 (71)

andJ(θ)P (σ, θ)J(θ)− P (0, θ) + ε In 0 (72)

hold for all θ ∈ P, all µ ∈ Dv, all τ ∈ [0, Tmax] and all σ ∈ [Tmin, Tmax].Then, the LPV system (1) with parameter trajectories in P∞ (i.e. the system (70)) is uniformly expo-

nentially stable under the range dwell-time constraint [Tmin, Tmax]; i.e. for all sequences of jumping instantsin T . M

Proof : The proof of this result follows the same lines as the proof of Theorem 2 with the difference thatwe consider here the Lyapunov function

V (x, τ, ρ, T ) = xᵀP (T − τ, ρ)x (73)

where P (·, ·) is defined as in Theorem 13. Pre- and post-multiplying (71) by x(tk + τ)ᵀ and x(tk + τ)ᵀ andletting τ = Tk − τ , θ = ρ(tk + τ), µ = ρ(tk + τ), and integrating from 0 to Tk yields

V (x(tk+1), Tk, ρ(tk+1), Tk)− V (x(t+k ), 0, ρ(tk), Tk) ≤ 0 (74)

or, equivalently,

x(tk)ᵀJ(ρ(tk))ᵀ [Φρ(tk+1, tk)ᵀP (0, ρ(tk+1))Φρ(tk+1, tk)− P (Tk, ρ(tk))] J(ρ(tk))x(tk) ≤ 0 (75)

which together with (72) implies that with

x(tk)ᵀ [J(ρ(tk))ᵀΦρ(tk+1, tk)ᵀP (0, ρ(tk+1))Φρ(tk+1, tk)J(ρ(tk))− P (0, ρ(tk))]x(tk) ≤ − ε ||x(tk)||22 (76)

18

holds for all x(tk) ∈ Rn, all Φρ(tk+1, tk) ∈ Mk(ρ(tk)), ρ(tk), ρ(tk+1) ∈ P, and all tk+1 − tk ∈ [Tmin, Tmax].Using the same arguments as in the proof of the other results, one can prove that the discrete-time system(3) is uniformly exponentially stable under the range dwell-time [Tmin, Tmax], which also implies that theLPV system (1) with parameter trajectories in P∞ (i.e. the system (70)) is uniformly exponentially stableunder the range dwell-time constraint [Tmin, Tmax]. The proof is completed. ♦

It seems important to mention the presence of the negative term in front of the timer-derivative. Thiscomes from the fact that we consider here the Lyapunov function V (x, τ, ρ, T ) = xᵀP (T − τ, ρ)x, whichintroduces a negative sign when deriving it by τ . It is interesting to note that the timer still measures thetime elapsed since the last jump but that the Lyapunov function depends on the remaining time until thenext jump. This is different from the constant and minimum dwell-time stability and stabilization resultswhere the Lyapunov functions directly depended on the timer variable. Using such a ”reverse condition” willallow us to obtain stabilization conditions using a dwell-time-independent sampled-data controller withoutthe need for using relaxed conditions and slack variables.

5.2 Main stabilization result

We consider here the following class of sampled-data gain-scheduled state-feedback controllers

u(tk + τ) = K1(ρ(tk))x(tk) +K2(ρ(tk))u(tk) (77)

where τ ∈ (0, Tk], Tk ∈ [Tmin, Tmax] and where K1(·) ∈ Rm×n and K2(·) ∈ Rm×m are the gains of thecontroller to be determined. The sequence of update times for the control input is assumed to satisfy arange dwell-time condition; i.e. tkk∈Z>0

∈ T . Before stating the stabilization result, it is necessary toreformulate the closed-loop system (52)-(77) as a hybrid system, which is given below: x(t) = A(ρ(t))x(t) +B(ρ(t))u(t)

u(t) = 0ρ(t) ∈ Q(ρ(t))

∣∣∣∣∣∣ if t 6= tk

x(t+) = J(ρ(t))x(t)u(t+) = K1(ρ(t))x(t) +K2(ρ(t))u(t)ρ(t+) = ρ(t)

∣∣∣∣∣∣ otherwise

.

(78)

For conciseness, we also define the following matrices:

A(ρ) :=

[A(ρ) B(ρ)

0 0

], J(ρ) :=

[J(ρ) 0

0 0

], B :=

[0Im

], and K(ρ) :=

[K1(ρ) K2(ρ)

]. (79)

We can now state the stabilization result of the section:

Theorem 14 Let the scalars 0 < Tmin ≤ Tmax < ∞ be given and assume that there exist a continuouslydifferentiable matrix-valued function P : [0, Tmax]×P 7→ Sn+m

0 , a matrix-valued function U : P 7→ Rm×(n+m)

and a scalar ε > 0 such that the conditions

∂τ P (τ , θ)−N∑i=1

∂ρi P (τ , θ)µi + Sym[A(θ)P (τ , θ)] + ε In 0 (80)

and [−P (0, θ) [J(θ)P (0, θ) + BU(θ)]ᵀ

? −P (σ, η)

] 0 (81)

hold for all θ ∈ P, all µ ∈ Dv, all τ ∈ [0, Tmax] and all σ ∈ [Tmin, Tmax].Then, the sampled-data LPV system (52)-(77) with parameter trajectories in P∞ is uniformly exponen-

tially stable under the range dwell-time condition [Tmin, Tmax] (i.e. for all sequences of jumping instants inT ) with the controller gain K(θ) = U(θ)R(T , θ)−1. M

19

0 1 2 3 4 5-20

0

20x1 x2 u

0 2 4 6 8 10-1

0

1x1 x2 u

Figure 5: Trajectories of the closed-loop system (62)-(53) (top) and (83)-(53) (bottom).

Proof : The proof follows from similar lines as the proof of Theorem 11 with the difference that we use herethe range dwell-time stability conditions of Theorem 13. The details are omitted for brevity. ♦

5.3 Examples

To illustrate the interest of the approach, we consider here three examples from the literature.

Example 15 Let us consider back the system (62) with the difference that we now aim at stabilizing itwith a gain-scheduled sampled-data state-feedback controller of the form (53). Solving for the sum of squaresconditions associated with the conditions stated in Theorem 14 with d = 2, ν = 1, Tmin = 0.01 and Tmax = 0.1yields the controller gain

K(θ) =1

den(θ)

3.01− 2.00θ + 5.52θ2 − 2.43θ3 − 0.59θ4 + 0.69θ5 + 0.04θ6

−0.74− 0.29θ + 0.77θ2 − 1.13θ3 + 0.10θ4 + 0.24θ5 + 0.08θ6

−0.002 + 0.014θ + 0.029θ2 − 0.46θ3 + 1.10θ4 − 0.95θ5 + 0.28θ6

ᵀ

,

den(θ) = −0.32 + 0.56θ − 1.20θ2 + 0.45θ3 + 1.15θ4 − 1.18θ5 + 0.23θ6.

(82)

Computational-wise, the underlying semidefinite program has 3078/525 primal/dual variables and is solvedin 7.88sec. The trajectory of the closed-loop system is depicted in the top panel of Figure 5 for the parametertrajectory ρ(t) = (1 + sin(2νt))/2 and initial condition x0 = (−1, 1), u0 = 0. The dwell-time values havebeen randomly selected in the interval [0.01, , 0.1].

Example 16 Let us consider the system [29]

x =

[2ρ 1.1 + ρ

−2.2 + ρ −3.3 + 0.1ρ

]x+

[2ρ

0.1 + ρ

]u (83)

20

0 2 4 6 8 10 12-2

-1

0

18 = 0:2

x1 x2 u

0 2 4 6 8 10 12-2

-1

0

18 = 1

x1 x2 u

Figure 6: Trajectories of the closed-loop system (84)-(53).

where ρ(t) = sin(0.2t), hence P = [−1, 1] and D = [−0.2, 0.2]. It was shown in [29] that this system couldbe stabilized at least up to Tmax = 0.6 using an input-delay model for the zero-order hold and a parameter-dependent Lyapunov-Krasovskii functional. Using polynomials of order 4 in the SOS conditions, we can solvethe SOS program associated with Theorem 14 and find a controller that makes the closed-loop system stablefor all Tk ∈ [0.001, 0.6]. The program has 9618/966 primal/dual variables and is solved in 36.26sec. Thesimulation results are depicted in Figure 5 where the dwell-time values were randomly selected in the interval[0.001, 0.6]. The initial conditions are chosen to be x0 = (−1, 1), u0 = 0.

Example 17 Let us consider now the system [30, 31]

x =

[0 1

0.1 0.4 + 0.6ρ

]x+

[01

]u (84)

where ρ(t) = sin(νt). Hence, P = [−1, 1] and D = [−ν, ν]. Using a looped-functional approach, it was shownin [30] that, for Tmin = 0.001, this system could be stabilized up to Tmax = 1.264 when ν = 0.2 and up toTmax = 0.8 when ν = 1. A refined approach from the same authors

Using Theorem 14 with d = 4, we can show that, for both ν = 0.2 and ν = 1, we can find a controllerthat stabilizes the system for all Tk ∈ [0.001, 1.3]. In fact, the system remains stabilizable up to at leastTmax = 2 using the proposed approach. The number of primal/dual variables is 9618/966 and the problemsolves in approximately 25sec. For simulation purposes, we set Tmax = 0.4 for both ν = 0.2 and ν = 1,and we design controllers using Theorem 14 with d = 2 (in this case, the number of primal/dual variablesis given by 3078/525 and the problem is solved in 7sec). Using the initial conditions x0 = (−1, 1), u0 = 0and random sequences of dwell-times in [0, 0.4], we get the trajectories depicted in Figure 6. Note that, asin [30], using a controller designed for Tmax = 1.3 would result in a very slow response for the closed-loopsystem which is not desirable.

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6 Future Works

Potential extensions include the consideration of different types of Lyapunov functions of polyhedral orhomogeneous types, the consideration of additional clocks in order to consider multiple types of discreteevents (such as control update and parameter discontinuities events) and the consideration of switchedimpulsive LPV systems. Converse results along the lines of [43, 44] for this class of systems could also bevery interesting to obtain. Finally, the derivation of convex stabilization conditions for the design of dynamicoutput-feedback controller is a problem which is worth investigating.

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