The third basic property of dynamical systems is stability. The most complete contribution to the stability analysis of nonlinear dynamical systems is due to Lyapunov [47].
Lyapunov’s indirect or linearization method is concerned with the local stability of nonlinear systems. It is a formalization of the intuition that a nonlinear system should behave similarly to
its linearized approximation for small ranges. Moreover, the global asymptotic stability property is implied by the local asymptotic stability in all the operation points as follows [43]:
• If the linearized system is strictly stable i.e, if all eigenvalues ofAare strictly in the left-half complex plane, then the equilibrium point is asymptotically stable for the actual nonlinear system.
• If the linearized system is unstable i.e, if at least one eigenvalue of A is strictly in the right-half complex plane, then the equilibrium point is unstable for the nonlinear system.
• If the linearized system is marginally stable i.e, all eigenvalues of A are in the left-half complex plane, but at least one of them is on the imaginary axis, then one cannot conclude anything from the linear approximation. The equilibrium point may be stable, asymptotically stable, or unstable for the nonlinear system.
Lyapunov’s direct method, along with the Barbashin-Krasovskii-LaSalle invariance principle [48,49,50], provide more powerful framework for analyzing the stability of nonlinear dynamical systems. In particular, Lyapunov’s direct method can provide local and global stability con-clusions of an equilibrium point of a nonlinear dynamical system if a smooth positive-definite function of the nonlinear system states (Lyapunov function) can be constructed for which its time rate of change due to perturbations in a neighborhood of the system’s equilibrium is always negative or zero, with strict negative definiteness ensuring asymptotic stability. Alternatively, using the Barbashin-Krasovskii-LaSalle invariance principle the positive-definite condition on the Lyapunov function as well as the strict negative- definiteness condition on the Lyapunov derivative can be relaxed while assuring asymptotic stability. In particular, if a smooth function defined on a compact invariant set with respect to the nonlinear dynamical system can be con-structed whose derivative along the system’s trajectories is negative semi-definite and no system trajectories can stay indefinitely at points where the function’s derivative identically vanishes, then the system’s equilibrium is asymptotically stable [51].
The definitions and theorems in terms of stability are given below [41,43,40,46].
The stability analysis is concerned with general differential equation in the form
˙
x=f(x, t), x(x0) =x0, (4.10) wherex∈Rn andt≥0. The system defined byEq. (4.10) is said to be autonomous. Moreover, it is assumed thatf(x, t) is piecewise continuous with respect tot.
Definition 4.3.1. Equilibrium point. x∗ is said to be an equilibrium point of Eq. (4.10) if f(x∗, t)≡0 for all t >0.
It should be noted thatx0 can be made an equilibrium point of Eq. (4.10)by translating the origin to the equilibrium point x∗.
Definition 4.3.2. Stability in the sense of Lyapunov. The equilibrium pointf(x∗, t) = 0 is called a stable equilibrium point of Eq. (4.10) if for all t0 ≥0 and ǫ > 0, there exist δ(t0, ǫ) such that
|x0|< δ(t0, ǫ)⇒ |x(t)|< ǫ∀t≥t0, (4.11) wherex(t) is the solution ofEq. (4.10) starting for x0 at t0.
Definition 4.3.3. Uniform stability. The equilibrium pointf(x∗, t) =0 is called a uniformly stable equilibrium point of Eq. (4.10)if in the preceding definition δ can be chosen independent oft0.
Definition 4.3.4. Asymptotic stability. The equilibrium pointf(x∗, t) =0 is an asymptoti-cally stable equilibrium point of Eq. (4.10)if
• f(x∗, t) =0 is a stable equilibrium point ofEq. (4.10),
• f(x∗, t) =0 is attractive, that is for allt0 ≥0there exists aδ(t0) such that
|x0|< δ(t0, ǫ)⇒ lim
t→∞|x(t)|=0. (4.12) Definition 4.3.5. Uniform asymptotic stability. The equilibrium point f(x∗, t) =0 is an uniformly asymptotically stable equilibrium point ofEq. (4.10) if
• f(x∗, t) =0 is a uniformly stable equilibrium point ofEq. (4.10),
• The trajectoryx(t) converges uniformly to0, that is, there exists δ >0and a function γ(τ,x0) :R+×Rn7→R+such that lim
τ→∞γ(τ,x0) = 0 for allx0∈Bδ, whereBδis a ball of radiusδ centered at the origin, and
|x0|< δ⇒ |x(t)| ≤γ(t−t0,x0) ∀t≥t0. (4.13) The previous definitions are local, since they concern neighborhoods of the equilibrium point.
Global asymptotic stability and global uniform asymptotic stability are defined as follows:
Definition 4.3.6. Global asymptotic stability. The equilibrium point f(x∗, t) = 0 is a globally asymptotically stable equilibrium point of Eq. (4.10) if it is asymptotically stable and
tlim→∞x(t) =0 for allx0 ∈Rn.
Definition 4.3.7. Global uniform asymptotic stability. The equilibrium point f(x∗, t) =0 is a globally, uniformly, asymptotically stable equilibrium point of Eq. (4.10) if it is globally asymptotically stable and if in addition, the convergence to the origin of trajectories is uniform in time, that is to say that there is a function γ(τ,x0) :R+×Rn7→R+ such that
|x(t)| ≤γ(t−t0,x0)∀t≥t0. (4.14) Definition 4.3.8. Positive definite function. A scalar continuous functionV(x) is said to be locally positive definite if V(0) =0 and, in a ballBδ x6= 0 ⇒ V(x) >0. If V(0) =0 and the above property holds over the whole state space, then V(x) is said to be globally positive definite.
A few related concepts can be defined similarly, in a local or global sense, i.e., a function V(x) is negative definite if−V(x) is positive definite;V(x) is positive semi-definite ifV(0) =0 and V(x) ≥0 for x 6=0; V(x) is negative semi-definite if −V(x) is positive semi-definite. The prefix ”semi” is used to reflect the possibility ofV being equal to zero for (x)6=0.
Definition 4.3.9. Lyapunov function. If, in a ball Bδ, the function V(x)is positive definite and has continuous partial derivatives, and if its time derivative along any state trajectory of system Eq. (4.10) is negative semi-definite, i.e., V˙(x) ≤ 0 then V(x) is said to be a Lyapunov function for the system Eq. (4.10).
The relations between Lyapunov functions and the stability of systems are made precise in a number of theorems in Lyapunov’s direct method. Such theorems usually have local and global versions. The local versions are concerned with stability properties in the neighborhood of equilibrium point and usually involve a locally positive definite function.
Theorem 4.3.1. Local stability. If, in a ball Bδ, there exists a scalar function V(x) with continuous first partial derivatives such that
• V(x) is positive definite (locally in Bδ)
• V˙(x) is negative semi-definite (locally in Bδ)
then the equilibrium point 0 is stable. If, actually, the derivativeV˙(x) is locally negative definite in Bδ, then the stability is asymptotic.
The above theorem applies to the local analysis of stability. In order to assert global asymp-totic stability of a system, one might naturally expect that the ballBδ in the above local theorem has to be expanded to be the whole state-space. This is indeed necessary, but it is not enough. An additional condition on the functionV(x) has to be satisfied: V(x) must be radially unbounded, i.e. V(x)→ ∞ as|x| → ∞.
Theorem 4.3.2. Global stability. Assume that there exists a scalar function V(x) with con-tinuous first partial derivatives such that
• V(x) is positive definite
• V˙(x) is negative semi-definite
• V(x)→ ∞ as |x| → ∞
then the equilibrium at the origin is globally stable. If, actually, the derivative V˙(x) is negative definite, then the stability is globally asymptotic.
Asymptotic stability of a system is usually a very important property to be determined.
However, the theorems just described are often difficult to apply in order to assert this property.
The reason is that it often happens thatV˙(x), the derivative of the Lyapunov function candidate, is only negative semi-definite. In this kind of situation, it is still possible to draw conclusions on asymptotic stability, with the help of the powerful invariant set theorems, attributed to La’Salle.
The central concept in these theorems is that of invariant set, a generalization of the concept of equilibrium point.
Definition 4.3.10. Invariant set. A set G is an invariant set for a dynamic system if every system trajectory which starts from a point inGremains in Gfor all future time.
For instance, any equilibrium point is an invariant set. The domain of attraction of an equilibrium point is also an invariant set. A trivial invariant set is the whole state-space. For an autonomous system, any of the trajectories in state-space is an invariant set.
The invariant set theorems make it possible to extend the concept of Lyapunov function so as to describe convergence to dynamic behaviors when V˙(x), the derivative of the Lyapunov function candidate, is only negative semi-definite.
Theorem 4.3.3. Local invariant set theorem. Consider an autonomous system of the form Eq. (4.10), with f(x) continuous, and let V(x) be a scalar function with continuous first partial derivatives. Assume that
• for somel >0, the region Ωl defined by V(x)< l is bounded
• V˙(x)≤0 for all xin Ωl
Let R be the set of all points within Ωl where V˙(x) = 0, and M be the largest invariant set in R. Then, every solution x(t) originating inΩl tends to Mas t→ ∞.
Corollary 4.3.1. Consider the autonomous system Eq. (4.10), with f(x) continuous, and let V(x) be a scalar function with continuous partial derivatives. Assume that in a certain neigh-borhood Ω of the origin
• V(x) is locally positive definite
• V˙(x) is negative semi-definite
• the set R defined by V˙(x) = 0 contains no trajectories of Eq. (4.10) other than the trivial trajectory x≡0
then, the equilibrium point 0 is asymptotically stable. Furthermore, the largest connected region of the formΩl (defined by V(x)< l) withinΩis a domain of attraction of the equilibrium point.
Indeed, the largest invariant set MinR then contains only the equilibrium point0.
The above invariant set theorem and its corollary can be simply extended to a global result, by requiring the radial unboundedness of the scalar function V(x) rather than the existence of a boundedΩl.
Theorem 4.3.4. Global invariant set theorem. Consider an autonomous system of the form Eq. (4.10), with f(x) continuous, and let V(x) be a scalar function with continuous first partial derivatives. Assume that
• V(x) is positive definite
• V(x)→ ∞ as |x| → ∞
• V˙(x)≤0 over the whole state-space
Let R be the set of all points where V˙(x) = 0, and M be the largest invariant set in R. Then, all solutions globally asymptotically converge toM as t→ ∞.
4.3.1 Stability analysis applying Lyapunov’s indirect method
A linear time-invariant (LTI) system (see the state space realization in Eq. (B.1)) is internally stable if and only if all the eigenvalues of the state matrixA have strictly negative real parts:
Re{λi(A)}<0, ∀i. (4.15)
For this purpose the control oriented model has been linearized around a typical equilibrium state. The linearization can be seen inSection B.1 ofAppendix B.
In order to prove that all the eigenvalues of A have strictly negative real parts the Routh-Hurwitz stability criterion is used [52]. This method helps to determine whether all the roots of the characteristic polynomial of the linear system P(λ) = det(λI−A), i.e. the eigenvalues of A, have negative real parts.
The general realization of the Routh-Hurwitz stability criterion is as follows [53]:
∆1>0, ∆2 >0, . . . , ∆n>0, (4.16) where
∆k=det
a1 1 0 0 0 0 · · · 0 a3 a2 a1 1 0 0 · · · 0 a5 a4 a3 a2 a1 0 · · · 0 ... ... ... ... ... ... . .. ...
a2k−1 a2k−2 a2k−3 a2k−4 a2k−5 a2k−6 · · · ak
. (4.17)
In case of the linearized model of the EPC system the characteristic polynomial is written as follows:
P(λ) =λ3+a1λ2+a2λ+a3, (4.18) where
a1 = kpst
mpst, (4.19)
a2 =
d
dxpstFl(x∗pst)
mpst + p∗chA2pst Vchd +Apstx∗pst
mpst and (4.20)
a3 = 0. (4.21)
From Eq. (4.16)andEq. (4.17)the Routh-Hurwitz stability criterion for third order systems is obtained as:
a1 > 0, (4.22)
a1a2−a3 > 0 and (4.23)
(a1a2−a3)a3 > 0. (4.24)
Since thekpstand mpst parameters for physical systems is always greater than zero, the first stability criterion defined byEq. (4.22) for the EPC system is inherently satisfied.
The second criterion defined by Eq. (4.23) is written as follows:
d
dxpstFl(x∗pst) + p∗chA2pst
Vchd +Apstx∗pst >0. (4.25) The third criterion is obviously not satisfied since, from Eq. (4.18) and Eq. (4.21) one pole must be equal to zero. Hence, the analysis shows that the linearized model has a pole equal to zero and has two stable or unstable poles depending on its parameters and the equilibrium point used for the linearization. Thus the linearized model can be marginally stable or unstable.
In case of a given EPC system the area of the piston (Apst) is fixed, but the dead volume of the chamber (Vchd) and the stiffness of the clutch spring (dFl(xpst)/dxpst) can vary during the lifetime of the system, hence these parameter changes should be considered in the design of the system. In Fig. 4.1 different equilibrium points are shown over the pressure-position plane with increasing dead volume on the third axis to demonstrate the parameter dependance of the stability. The unstable and marginally stable equilibrium points are depicted with red and black points, respectively.
Consequently, the nonlinear system have locally unstable equilibrium points depending on its parameters and have equilibrium points, which may be stable, asymptotically stable, or unstable.
In order to conclude the stability of the EPC system further investigations are needed applying Lyapunov’s direct method.
4.3.2 Stability analysis applying Lyapunov’s direct method
The basic philosophy of Lyapunov’s direct method is the mathematical extension of a funda-mental physical observation: if the total energy of a system is continuously dissipated, then the system, whether linear or nonlinear, must eventually settle down to an equilibrium point. Thus, it may conclude the stability of a system by examining the variation of a single scalar function.
The Lyapunov function candidate is the total energy of the EPC system, namely, V(x,d) =
Z
mpstv˙pstdxpst+ Z
(pch−pamb)dVch+ Z
Fl(xpst)dxpst. (4.26)
0
0.005
0.01
0.015
0.02
0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 105 0 1 2 3 4
x 10−3
Vd ch[m3]
vpst∗ = 0
xpst [m]
pch[Pa]
Unstable equilibrium
points (x∗ = [pch∗ , vpst∗ , xpst∗ ])
Figure 4.1: Equilibrium points over the pressure-position plane with increasing dead volume The first term is the kinetic energy of the piston the next is the internal energy of the gas in the chamber and the third one is the potential energy stored in the springs of the clutch mechanism. The functionV(x,d) is a globally positive definite function.
Applying the substitution rule for integrals with dxpst = vpstdt and dVch = Apstvpstdt Eq. (4.26) can be rewritten as follows:
V(x,d) = Z
mpstv˙pstvpstdt+ Z
(pch−pamb)Apstvpstdt+ Z
Fl(xpst)vpstdt. (4.27) The rate of energy variation is obtained easily by differentiatingEq. (4.27)and usingEq. (3.20) as follows:
V˙(x,d) =mpstvpstv˙pst−(pch−pamb)vpstApst+Fl(xpst)vpst=−kpstvpst2 . (4.28) The function V˙(x,d) is a globally negative semi-definite function. This establishes the sta-bility (in Lyapunov sense) but not asymptotic stasta-bility. In order to conclude the asymptotic stability of the system the invariant set theorem is used.
According to the invariant set theorem an equilibrium point, which can be easily transformed to the origin with an appropriate coordinate transformation, is asymptotically stable if it is stable and contains no invariant sets other than the origin.
If Eq. (4.25) is satisfied for the EPC system over the whole state-space then there are no unstable equilibrium points. Only a single equilibrium point exists along a trajectory, which is globally asymptotically stable according to the invariant set theorem.
Since the convergence to the origin of trajectories of the EPC is uniform in time, i.e. the trajectories are independent oft0, the system has the uniform property, too.
Thus the EPC system is globally uniformly asymptotically stable if theEq. (4.25)is satisfied otherwise, the equilibrium points whereEq. (4.25)is not satisfied are unstable and the remained ones are locally uniformly asymptotically stable.