7. Extensions
7.2 Hybrid Systems
7.2 Hybrid Systems
Chapter 7. Extensions
−5 0 5
−5 0 5
−11 −14 10
14 7 0
0 0 0
−1 −3 0 4 2 −4
0 0 0
0.1 −2 0 4 −0.5 0.1
0 0 0
−10 −11 0 10 9 −2
0 0 0
−1 2 0 1 −2 0
0 0 0
−10 −10 −2
10 5 0
0 0 0
−10 −11 2
1 1 2
0 0 0
−1 −2 0
0 1 2
0 0 0
−4.5 −10 1
10 5 2
0 0 0
Figure 7.3 Partition of the fuzzy system defined by the rules in Table 7.1 into operating regimes and interpolation regimes. The matrices ¯Aidefining the dynamics in each operating regime are also shown.
0 2 4 6 8 10
−4
−3
−2
−1 0 1 2 3 4
Figure 7.4 System response from a typical initial condition.
7.2 Hybrid Systems
−6 −2 −1 0 1 2 6
−6
−2
−1 0 1 2 6
Figure 7.5 Level surfaces of the computed Lyapunov function. The guaranteed region of attraction is given by the outermost level set.
be a particular case. Exponential stability will be established using Lya-punov functions that are discontinuous in x(t), but where the value of the Lyapunov function decreases every time there is a change in the discrete state. Similar to before, the analysis computations can be cast as convex optimization problems in terms of linear matrix inequalities.
A Class of Hybrid System
We consider piecewise affine systems on the form
˙x(t) Ai(t)x(t) +ai(t), i(t) ν{x(t),i(t−)} (7.8) with x∈Rnand i∈I⊂Z. This is a particular instance of the differential automaton defined in[136]. Here, the differential equation describes the continuous dynamics while the algebraic equation models the state of the decision-making logic. The discrete state i(t) ∈ I is piecewise constant.
The notation t− indicates that the discrete state is piecewise continuous from the right. The model associates one continuous affine dynamics to each value of the discrete state.
We will assume that changes in the discrete state are triggered by the evolution of the continuous dynamics. More precisely, a transition from the discrete state j to the discrete state k occurs when the continuous state hits the transition hyperplane
f¯jkT¯x(t) 0 provided that the enabling condition
E¯k¯x0
Chapter 7. Extensions
is satisfied. These systems can be conveniently represented as a finite automaton with a continuous dynamics associated to each discrete state, see Figure 7.6. The leftmost arrow indicates the initial discrete state. The
˙¯xA¯1¯x E¯1¯x0
˙¯xA¯2¯x E¯2¯x0 {f¯12T¯x0} ∩ {E¯2¯x0}
{f¯21T¯x0} ∩ {E¯1¯x0}
Figure 7.6 Piecewise linear hybrid system illustrated as an hybrid automaton.
following example illustrates the model class, and the need for extensions of the piecewise quadratic analysis from Chapter 4.
EXAMPLE7.1
Figure 7.7(left)shows a simulation of the system ˙x(t) Ai(t)x(t)
i(t)
(2, if i(t−) 1 and f12Tx(t) 0
1, if i(t−) 2 and f21Tx(t) 0 (7.9) with i(0) 1, switching boundaries
f12 [ −10 −1]T, f21 [2 −1]T and system matrices
A1 −1 −100 10 −1
, A2
1 10
−100 1
.
The simulations shown in Figure 7.7 indicate that the system is asymp-totically stable. From the simulated trajectory of the system, it is also clear that it is not possible to find a Lyapunov function that disregards the influence of the discrete state.
The state space of the model (7.8), RZ, can be thought of as a set of enumerated copies of Rn. From this perspective, a transition in the discrete state can be seen as the transfer from one copy of Rn to the other, see Figure 7.8. The simulation in Figure 7.7 is the projection of this trajectory onto the continuous state space.
7.2 Hybrid Systems
−3 −2 −1 0
−3
−2
−1 0
x1 x2
0 0.05 0.1 0.15 0.2 0.25
−3
−2
−1 0
0 0.05 0.1 0.15 0.2 0.25
1 2
t x(t)
i(t)
Figure 7.7 Sample trajectory of the hybrid system projected onto the continuous state space(left)and corresponding time plots(right).
−3
0 −3
0 1
2
x1
x2 i
Figure 7.8 State space of hybrid system illustrated as a number of enumerated copies ofRn. Changes in the discrete state transfers the state from one copy to the other.
With this basic understanding of the system class, we can now pro-ceed to state a more technical definition. We assume that the continuous dynamics is piecewise affine,
˙x(t) Ai(t)x(t) +ai(t) a.e. (7.10) We let I0 ⊆ I be the set of indices for which x(t) 0 is admissible, and let I1 I\I0. It is assumed that ai 0 for i ∈ I0. From the transition conditions for the discrete dynamics, we construct vectors fij and ¯fij for i,j ∈I such that fi,i0, ¯fi,i0 for i∈I and
fiT(t−)i(t)x(t) 0 ∀t, (7.11) f¯i(tT−)i(t)
x(t) 1
0 ∀t. (7.12)
To account for the enabling conditions, i.e., that certain discrete states may only be admissible for a subset of the continuous states, we construct
Chapter 7. Extensions
matrices Ei(t) and ¯Ei(t) such that
Ei(t)x(t) 0, i(t) ∈I0, E¯i(t)
x(t) 1
0, i(t) ∈I1.
Discontinuous Lyapunov Functions
In the analysis of hybrid systems it is sometimes desirable to relax the requirement that the Lyapunov function should be continuous. An in-teresting option is to use Lyapunov functions that have a discontinuous dependence on the discrete state, but where the value of the Lyapunov function decreases at the switching instants. This is the idea behind so-called ‘multiple Lyapunov functions’, see [22]. This possibility has been incoroporated in Lemma 4.1 by the requirement that V(t)be decreasing and piecewise
C
1.When the transition conditions are affine inequalities in the state, it is possible to formulate the search for this type of Lyapunov functions as a LMI problem. This can be seen by the following simple argument.
Let the Lyapunov function be V(x) ¯xTP¯i(t)¯x and let the discrete state initially have the value j. Assume that the condition for the discrete state to change value from j to k is given by ¯fjkT¯x0. Then, the requirement that the Lyapunov function should be decreasing at the switching instant,
¯xTPj¯x≥ ¯xTPk¯x for {xtf¯jkT¯x0}
can be expressed as the linear matrix inequality in ¯Pj,P¯k and ¯tjk
P¯j−P¯k+f¯jktTjk+tjkf¯jkT ≥0.
We let ¯Pi [I 0]TPi[I 0]for i∈I0, and state the following result.
THEOREM 7.1
Consider vectors tij and ¯tij, symmetric matrices Ui and Wi with
non-7.2 Hybrid Systems negative entries and symmetric matrices Pi and ¯Pi such that
(0>ATi Pi+PiAi+EiTUiEi
0<Pi−EiTWiEi i∈I0 (7.13) (0>A¯Ti P¯i+P¯iA¯i+E¯iTUiE¯i
0<P¯i−E¯iTWiE¯i
i∈I1 (7.14)
0<P¯i−P¯j+f¯ij¯tTij+¯tijf¯ijT i∈I1 or j ∈I1 (7.15) 0<Pi−Pj+fijtTij+tijfijT i,j∈I0 (7.16) where i6j, then every continuous, piecewise
C
1trajectory x(t)satisfying (7.10)tends to zero exponentially.Clearly, when applying Theorem 7.1 one only needs to consider those i,j that correspond to feasible transitions in the dynamics. Similar to the partition refinements in our previous analysis, it can sometimes be useful to introduce additional discrete states to obtain more flexibility in the Lyapunov function candidate.
There is a strong relation between Theorem 7.1 and Theorem 4.1. By allowing non-strict inequalities in(7.15)and(7.16), Theorem 4.1 can be seen as a special case of Theorem 7.1 where ¯fij f¯ji,∀i,j ∈I. However, a formulation with non-strict inequalities is numerically very sensitive and most LMI solvers can not treat non-strict inequalities as they stand.
Inherent algebraic constraints must first be eliminated. Theorem 4.1 can be seen as the outcome of such an elimination.
Theorem 7.1 can be applied directly to the system of Example 7.1.
EXAMPLE7.2
Consider again the switching system (7.9). To illustrate the use of en-abling conditions, we let
E1
0 0 0 0
, E2 −10 −1 2 −1
.
The LMI conditions of Theorem 7.1 have a feasible solution P1
17.9 −0.89
−0.89 179
, P2
739 −38.1
−38.1 91.8
.
A simulated trajectory of the system and the corresponding value of the computed Lyapunov function are shown in Figure 7.9. The discontinuities in the Lyapunov function concur with changes in the discrete state.
Chapter 7. Extensions
−3 −2 −1 0
−3
−2
−1 0
x1
x2
0 0.05 0.1 0.15 0.2 0.25
0 500 1000 1500 2000
t V(t)
Figure 7.9 Sample trajectory of the hybrid system(left)and the corresponding value of the Lyapunov function(right).
Several extensions can be made to the above result. It is for example straightforward to extend the class of systems to allow differential inclu-sions rather than differential equations in modeling the continuous dy-namics. System with alternative transition rules would also be possible.
From a computational viewpoint, the most important development is the observation that the search for a Lyapunov function with specified discon-tinuities can be formulated as a convex optimization problem. A similar development could be done for analysis via piecewise linear Lyapunov functions.