4. Lyapunov Stability
4.6 Piecewise Quadratic Lyapunov Functions
Chapter 4. Lyapunov Stability
Hence, V(x) >0 for x60, and the assertion follows.
The technique used in Lemma 4.4 is known as the S-procedure, see[1, 153, 19]. Intuitively, the point with this approach is that if ¯xTE¯iTWiE¯i¯x<0 for x6∈Xi, the inequality(4.20)may be simpler to satisfy than the corre-sponding LMI without relaxation term. Although this approach is conser-vative in general it is very useful in practice. For some situations, the S-procedure is both a necessary and a sufficient way to account for quadratic constraints, see[153, 19]. The specific way we construct a relaxation term from affine constraint can be seen as a special case of Shor’s relaxation used in non-convex and combinatorial optimization, see[128, 112].
4.6 Piecewise Quadratic Lyapunov Functions satisfy
(0>ATi Pi+PiAi+EiTUiEi
0<Pi−EiTWiEi
i∈I0
(0>A¯Ti P¯i+P¯iA¯i+E¯iTUiE¯i
0<P¯i−E¯iTWiE¯i
i∈I1
Then every trajectory x(t) ∈ ∪i∈IXisatisfying(2.3)with u0 for all t≥0 tends to zero exponentially.
Proof: Consider the Lyapunov function candidate V(t) V(x(t)) defined by(4.21). Since trajectories x(t)in the sense of Definition 2.1 are contin-uous and piecewise
C
1, Lemma 4.2 implies that V(t)constructed in this way is continuous and piecewiseC
1. Moreover, according to Lemma 4.2, there exists β > 0 such that the upper bound of (4.2) in Lemma 4.1 holds. A solution to the inequalities above implies that there existsα >0 andγ >0 such thatαttx(t)tt22< V(t) and dV(t)/dt< −γttx(t)tt22. Hence, exponential convergence follows from Lemma 4.1 withtt⋅tt tt⋅tt2and p2.Since Theorem 4.1 only considers trajectories defined for all t≥0, no conclusion can be drawn about trajectories that end up in attractive slid-ing modes. In the absence of attractive slidslid-ing modes, the above conditions assure that(4.21)is a Lyapunov function for the system. The first LMI condition for each region assures that the Lyapunov function decreases along system trajectories, ˙V(x(t)) <0, while the second condition assures positivity, V(x(t)) > 0. Any level set of V(x) that is fully contained in the partition∪i∈IXi is a region of attraction for the equilibrium x0. In particular, if the partition covers the whole state space then the system is globally exponentially stable. If no solution can be found to the conditions of Theorem 4.1 for a given partition, it is natural to refine the partition and to try again. Such partition refinements increase the flexibility of the Lyapunov function candidate(4.21).
With the piecewise quadratic stability theorem at hand, we can re-turn to the motivating examples where the standard LMI conditions for quadratic stability failed.
EXAMPLE4.6—PIECEWISEQUADRATIC WHERE QUADRATIC FAILS
Consider the system(4.9)in Example 4.2. Let Eidenote the cell bounding used in quadrant i. Setting
E1 −E3
1 0 0 1
, E2 −E4 −1 0
0 1
,
Chapter 4. Lyapunov Stability
and Fi [EiT In]T we invoke Theorem 4.1 and find a feasible solution V(x) xTx x∈Xi,i∈I.
Hence, stability can indeed be proved using a quadratic Lyapunov function but one needs to account for the composition of the state space partition.
The level curves of the computed Lyapunov function are shown together with a simulation in Figure 4.7(left).
−1 0 1
−1 0 1
−2 0 2
−2 0 2
Figure 4.7 Level surfaces(dashed)for the systems in Example 4.2 and Exam-ple 4.3 computed using Theorem 4.1. In both cases, the standard conditions for quadratic stability fail while Theorem 4.1 verifies stability.
As a second example, consider the system with flower-like trajectories used in Example 4.3. Similarly as above, we let
E1 −E3 −1 1
−1 −1
, E2 −E4 −1 1
1 1
,
and Fi [EiT In]T. From the conditions of Theorem 4.1 we find the piecewise quadratic Lyapunov function V(x) xTPix with
P1P3
5 0 0 1
, P2P4
1 0 0 5
.
As seen in Figure 4.7(right), the level surfaces of the computed Lyapunov function are neatly tailored to the system trajectories.
Theorem 4.1 only treats systems that do not have attractive sliding modes.
Thus, in order to be able to conclude stability for every possible behavior of the model(2.3), we must rule out the possibility of attractive sliding modes. A direct application of Proposition 3.4 verifies the absence of at-tractive sliding modes for the systems in Example 4.6. For systems with attractive sliding modes one has to extend the analysis conditions. Such extensions will be made in Section 4.11.
4.6 Piecewise Quadratic Lyapunov Functions Analysis of a Min-Max Selector System
The examples analyzed so far have been small examples in two dimen-sions, constructed to illustrate the shortcomings of quadratic and merits of piecewise quadratic Lyapunov functions. Our next example is motivated by industrial applications, has higher state dimension and a nonlinearity that is not easily treated with absolute stability techniques.
M A X
M I N
Cmin Cmax
C
ul zmin
zmax
y uh
Process
u z
G2 G1
un
SP PV ysp
SP PV
Figure 4.8 Control system with min/max selectors, from[4].
The example is the min-max selector control system shown in Fig-ure 4.8. This scheme is common in situations where several process vari-ables have to be taken into account using a single control signal. In Fig-ure 4.8, y is the primary variable, and z is a process variable that must remain within given ranges. The controller C is designed to control the primary variable, while the controllers Cmax and Cmin are designed to keep the critical variable z within certain bounds. Designed correctly, the min-max selector chooses the controller that is most appropriate for the situation, and allows good control of the primary variable while respecting the constraints.
Consider a system characterized by G1(s) 40
0.05s3+2s2+22s+40, G2(s) s2+7s5 +5.
To control the primary variable, we design a lead-lag controller C(s) s2+3s+3
0.02s2+s+0.01,
Chapter 4. Lyapunov Stability
while Cmin and Cmaxare proportional controllers uhKh(zmax−z),
ul Kl(zl−z).
The plots in Figure 4.9 show a simulation of the system without con-straint controllers. The tracking of the primary variable is quite good, but the critical variable z exceeds it constraint limits(shown in dashed lines). The plots in Figure 4.10 show a simulation of the min-max selector strat-egy. The constraints are now respected while the tracking of the primary variable remains satisfactory.
0 10 20 30 40
−2 0 2
0 10 20 30 40
−10 0 10
0 10 20 30 40
−5 0 5 r,y
u
z
Figure 4.9 Simulation of the control system in Figure 4.8 without constraint han-dling. The tracking of the primary variable is quite good(top), but the critical vari-able exceeds it limits(bottom).
We will apply Theorem 4.1 to stability analysis of the system for a con-stant set-point yspand constant constraint limits zmin,zmaxon the critical variable. Different values of ysp,zmin and zmaxresult in different equilib-rium points. For sake of simplicity, we will let ysp zmax zmin 0, but the technique would apply similarly to any choice of these constant inputs.
For analysis purposes, it is convenient to re-write the system equations as a linear system interconnected with the static nonlinearity
umin(uh,max(un,ul)).
The full details for this step are given in Section B.1. The selector non-linearity has three input signals uh,un,ul and one output, u. Similar to
4.6 Piecewise Quadratic Lyapunov Functions
0 10 20 30 40
−2 0 2
0 10 20 30 40
−20 0 20
0 10 20 30 40
−5 0 5 r,y
u,v
z
Figure 4.10 Simulation of the min-max selector system. The constraint limits are respected(bottom), while the tracking of the primary variable is still satisfactory (top). The middle plot shows how the constraint controllers override the primary control signal(dashed)resulting in a control(full)that respects the constraints.
the simpler system used as motivating example in Section 4.4, we can reduce the number of inputs by one using a simple loop transformation.
This results in the system shown in Figure 4.11. The transformed system
m i n
Ge(s)
ϕ
vhl zmax
ysp
zmin vnl
w
Figure 4.11 Selector control system rewritten as linear system interconnected with a static multi-variable nonlinearity.
has two outputs vhland vnl, and the selector nonlinearity is now reduced to the two-dimensional mapping ϕ(vhl,vnl) shown in Figure 4.12. The
Chapter 4. Lyapunov Stability
nonlinearityϕ is piecewise linear, and has the explicit expression
ϕ
0 if vhl≥0, vnl≤0 vnl if vnl≥0, vhl≥vnl
vhl otherwise
.
Since the region whereϕ(vhl,vnl) vhlis not convex, we have to introduce an additional region, see Figure 4.12(right).
−1 0
1
−1 0
−11 0 1
vhl
vnl ϕ(vhl,vnl)
4
3 2 1
vnl vhl
Figure 4.12 Static nonlinearity in the selector control system(left). The corre-sponding non-convex state partition(right)is rendered convex by splitting one cell in two(the dashed line in rightmost figure).
While this nonlinearity fits directly in the piecewise linear framework, it is not easily dealt with using other techniques. It is easy to verify that the nonlinearity has gain less than one, which motivates an attempt to apply the small gain theorem. However, the
L
2-induced gain of the linear system is 15.8, and the small gain can not verify stability. An approach based on linear differential inclusions(Corollary 4.1)also fails.In contrast, a numerical stability analysis using Theorem 4.1 verifies system stability. In this case, the optimization routines return a Lyapunov function which is globally quadratic. Since Corollary 4.1 fails, this example shows the importance of using partition information in the analysis.
It may appear strange to let zmin zmax 0 in the analysis. This choice was made for sake of simplicity, and similar results could be ob-tained for any choice of zminand zmax. The analysis only considers stability of an equilibrium point. Such an analysis is particularly interesting for verifying that no undesired switching occurs.