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Analysis of piecewise linear systems via convex optimization - A unifying approach
ANALYSIS OF PIECEWISE LINEAR SYSTEMS VIA CONVEX OP ...
14th World Congress ofIFAC
Copyright .© 1999 IF AC 14th Triennial World Congress, Beijing, P.R. China
D-2c-03-6
ANALYSIS OF PIECEWISE LINEAR SYSTEMS VIA CONVEX OPTIMIZATION - A UNIFYING APPROACH Mikael Johansson Department of Automatic Control Lund Institute of Technology, Box 118.22100 Lund, Sweden. Erna,il:mikael}@control.lth.se
Abstract: The recently developed technique for computation of piecewise quadratic Lyapunov functions is specialized to L1yapunov functions that are piece wise linear. This establishes a unified framework for computation of quadratic, piece wise quadratic, piece wise linear and polytopic Lyapunov functions. The search for a piecewise linear Lyapunov function is formulated as a linear programming problem, and duality is used to address the non-trivial issue of partition refinements. Copyright ID 19991FAC
L INTRODUCTION
Lyapunov functions can be computed via linear programming. This has some computational advantages over the LMI-based computations of piecewise quadratic Lyapunov functions. More importantly, this establishes a unified framework for computation of piecewise quadratic [51, globally quadratic 12]. piecewise linear [8] and polytopic Lyapunov functions [1, 10]. Finally, this approach gives valuable insight in how duality can be used to devise algorithms for automatic partition refinements.
Recent advances in hardware and software has created a strong interest for computational analysis of control systems. A large research effort has focused on quadratic stability, and it has been shown how many interesting analysis problems for linear uncertain systems can be cast as convex optimization problems, hence solved efficiently using numerical computations [2]. When applied to nonlinear systems, however, these results are often found too conservative (see [5] for some simple examples). This raises a question whether convex optimization can be used for analysis of more powerful system models, using more powerful Lyapunov functions. This question found a positive answer in the recently developed technique for analysis of piecewise linear (or rather affine) systems, See [5, 7). It was shown how the search for piecewise quadratic Lyapunov functions for piecewise linear systems could be cast as a convex optimization problem. In [11], the procedure was extended to system analysis, providing one possible generalization of the celebrated analysis oflinear systems with quadratic constraints.
Here, {XihEI :;;; Rn is a partition of the state space into a number of (possibly unbounded) polyhedral cells, as illustrated in Figure 1. We may think of these cells as operating regimes. The local behavior within each cell is described by a state space model which is affine in x.
In this paper we show how the parameterization of piecewise quadratic Lyapunov functions used in [7, 11) can be specialized to functions that are piecewise linear. We show how piecewise linear
The index set of the cells is denoted I. We let 10 E I be the set of cells that contain the origin, and h ~ I be the set of cells that do not contain the origin. It is assumed that ai = Cl = 0 for
2. PIECEWISE LINEAR SYSTEMS
We consider systems on the form {
X = ai +AiX + B,. u = Ci + Gix + D,u
y
x EX•.
(1)
2089
Copyright 1999 IF AC
ISBN: 008 0432484
ANALYSIS OF PIECEWISE LINEAR SYSTEMS VIA CONVEX OP ...
Admittedly, piecewise linear Lyapunov functions have been considered before (8J, but from a very different perspective and for a particular class of partition. This section presents several new formulations of the analysis computations and establishes a link between the piecewise linear Lyapunov functions and the piecewise quadratic Lyapunov functions described above.
4.1 A Matrix Parameterization of Continuous and Piecewise Linear Functions One may view the matrix format for continuous piecewise quadratic Lyapunov functions as a quadratic form in the coordinates z obtained by the continuous piecewise linear mapping z = Fix. If one rather considers linear forms in z, one obtains the following parameterization of continuous piecewise linear functions V(x)=t'F/i:=P~x
XEXj
Once again, we have obtained a separation of free parameters (now collected in the vector t) from the constraints imposed by the partition. One may also see the above fOrIIlat as a direct restriction of the piecewise quadratic functions as follows. Let Fi be constraint matrices satisfYing (2), and define
_ [Fi0 fi] 1 '
~. F, -
1[0 t]
T="2 t'0·
Note that Fi also satisfies the continuity condition (2), and that the piecewise quadratic function V(x) of Proposition 1 now evaluates to
14th World Congress ofIFAC
satisfY
{o = P~Ai + u)Et o=
Eh
(8)
i
p~ - WiEi
The search for the free variables t, Ui and Wi is a linear programming problem. If the system does not have any attractive sliding modes, a solution to this problem guarantees that
Vex) = P~x
for x E Xi
is a Lyapunov function for the system. It can be established, similar to Farkas' lemma, that the above conditions are both necessary and sufficient for the Lyapunov function candidate to be positive and decreasing on UiE1Xi , see [4]. If we consider polytopic partitions (where all cells are bounded), it is sufficient to check the Lyapunov conditions at the vertices of the polytopes, i.e., to establish that for all
Vk
# O. We have the following theorem.
Theorem 2. Let {XihEI be a partition ofIRn into convex polytopes with vertices Vk, and let Fi be the associated continuity matrices satisfYing (2) and (4). If there exists a vector t such that
Pi
= F:t
P; = FIt
for i E 10 foriEl
satisfY
The above parameterization will now be used in the search for continuous piecewise linear Lyapunov functions for piecewise linear systems. We state results for polytopic and polyhedral partitions, and for systems with and without attractive sliding modes. All results can be verified via linear programming. The first result considers piecewise linear Lyapunovfunction on (possibly unbounded) polyhedral partitions.
Theorem 1. Let {XihEl ~ lR n be a polyhedral partition with continuity matrices Fi, satisfYing (2) and (4), and cell boundings Ei , satisfYing (3) and (4). Assume furthermore that EiX # 0 for every x E Xi such that x -I o. If there exist vectors t, Ui and Wi such that Ui and Wi have positive entries, while
Pi = F[t, Pi = F[t,
(7)
then every trajectory x(t) E UiE1Xi satisfying (1) for t 2:: 0 tends to zero exponentially. 0
~Ax=~~T~i=f~£=~~ 4.2 Piece wise Linear Stability
i E 10
i E 10
i E Io i E 10
{ o > P~AiVk
o < p~ Vk
Vk
E Xi
Vk
E
Xi
i Ell Vk E Xi
i E
I1 Vk
(9)
(10)
E Xi
for Vk # 0, then every trajectory x(t) E UiEIXi satisfYing (1) for t 2:: 0 tends to zero exponentially. 0 Note that all relaxation terms have vanished and that the vector inequalities of Theorem 1 are reduced to a number of scalar inequalities. The basic stability computations can be extended in several useful ways. On example is computation of decay rate, T, which can be estimated from the modified Lyapunov inequality
Vex) + TV(X) < 0
i Eh
Vx#O.
2090
Copyright 1999 IF AC
ISBN: 008 0432484
ANALYSIS OF PIECEWISE LINEAR SYSTEMS VIA CONVEX OP ...
Admittedly, piecewise linear Lyapunov functions have been considered before (8J, but from a very different perspective and for a particular class of partition. This section presents several new formulations of the analysis computations and establishes a link between the piecewise linear Lyapunov functions and the piecewise quadratic Lyapunov functions described above.
4.1 A Matrix Parameterization of Continuous and Piecewise Linear Functions One may view the matrix format for continuous piecewise quadratic Lyapunov functions as a quadratic form in the coordinates z obtained by the continuous piecewise linear mapping z = Fix. If one rather considers linear forms in z, one obtains the following parameterization of continuous piecewise linear functions V(x)=t'F/i:=P~x
XEXj
Once again, we have obtained a separation of free parameters (now collected in the vector t) from the constraints imposed by the partition. One may also see the above fOrIIlat as a direct restriction of the piecewise quadratic functions as follows. Let Fi be constraint matrices satisfYing (2), and define
_ [Fi0 fi] 1 '
~. F, -
1[0 t]
T="2 t'0·
Note that Fi also satisfies the continuity condition (2), and that the piecewise quadratic function V(x) of Proposition 1 now evaluates to
14th World Congress ofIFAC
satisfY
{o = P~Ai + u)Et o=
Eh
(8)
i
p~ - WiEi
The search for the free variables t, Ui and Wi is a linear programming problem. If the system does not have any attractive sliding modes, a solution to this problem guarantees that
Vex) = P~x
for x E Xi
is a Lyapunov function for the system. It can be established, similar to Farkas' lemma, that the above conditions are both necessary and sufficient for the Lyapunov function candidate to be positive and decreasing on UiE1Xi , see [4]. If we consider polytopic partitions (where all cells are bounded), it is sufficient to check the Lyapunov conditions at the vertices of the polytopes, i.e., to establish that for all
Vk
# O. We have the following theorem.
Theorem 2. Let {XihEI be a partition ofIRn into convex polytopes with vertices Vk, and let Fi be the associated continuity matrices satisfYing (2) and (4). If there exists a vector t such that
Pi
= F:t
P; = FIt
for i E 10 foriEl
satisfY
The above parameterization will now be used in the search for continuous piecewise linear Lyapunov functions for piecewise linear systems. We state results for polytopic and polyhedral partitions, and for systems with and without attractive sliding modes. All results can be verified via linear programming. The first result considers piecewise linear Lyapunovfunction on (possibly unbounded) polyhedral partitions.
Theorem 1. Let {XihEl ~ lR n be a polyhedral partition with continuity matrices Fi, satisfYing (2) and (4), and cell boundings Ei , satisfYing (3) and (4). Assume furthermore that EiX # 0 for every x E Xi such that x -I o. If there exist vectors t, Ui and Wi such that Ui and Wi have positive entries, while
Pi = F[t, Pi = F[t,
(7)
then every trajectory x(t) E UiE1Xi satisfying (1) for t 2:: 0 tends to zero exponentially. 0
~Ax=~~T~i=f~£=~~ 4.2 Piece wise Linear Stability
i E 10
i E 10
i E Io i E 10
{ o > P~AiVk
o < p~ Vk
Vk
E Xi
Vk
E
Xi
i Ell Vk E Xi
i E
I1 Vk
(9)
(10)
E Xi
for Vk # 0, then every trajectory x(t) E UiEIXi satisfYing (1) for t 2:: 0 tends to zero exponentially. 0 Note that all relaxation terms have vanished and that the vector inequalities of Theorem 1 are reduced to a number of scalar inequalities. The basic stability computations can be extended in several useful ways. On example is computation of decay rate, T, which can be estimated from the modified Lyapunov inequality
Vex) + TV(X) < 0
i Eh
Vx#O.
2091
Copyright 1999 IF AC
ISBN: 008 0432484
ANALYSIS OF PIECEWISE LINEAR SYSTEMS VIA CONVEX OP ...
Given a fixed value of 't, the above condition can be verified using a slight modification of the previous theorellls (where Ai has been replaced by Ai + r I in the decreasing conditions). The optimal value of T can then be found by bisection. Another possibility is to prove stability for piecewise linear inclusions,
14th World Congress ofIFAC
can be ruled out a priori, one should rather use Theorems 1 and 2. If sliding modes can not be ruled out, it is preferable to try to detect on what surfaces sliding will occur and use the augmented conditions only on these surfaces. Apart from decreasing the computational burden, information of where sliding modes appear gives valuable engineering insight.
In this case, one need to simultaneously solve several decreasing conditions in each region (one for each k E K{i)), see [6, 10].
The previous sections unify several approaches to numerical Lyapunov function construction for (possibly) uncertain piecewise linear systems.
4.3 Systems with Attractive Sliding Modes
Cell boundaries of piecewise linear systems may in general have regions where the trajectories in adjacent cells are directed towards the boundary. With our definition, the trajectories are well-defined only until the moment when they reach this region. The theorems stated so far avoids this problem by only considering trajectories defined for all t 2: o. It is possible to extend the trajectory definition
further using the concept of sliding mode [3]. We then say that a piecewise Cl function x(t) E UiE1Xi is a generalized motion of the system (1) if, for every t such that the derivative x(t) is defined, the differential inclusion x(t) E convjEJ{Ajx
+ aj}
(11)
holds. Here, J is the set of all indices j such that x(t) E Xj. The following theorem guarantees stability in the presence of sliding modes. Theorem 3. Let {XihEI be a partition of1Rn into convex polytopes with vertices Vk, and let Pi be the associated constraint matrices, satisfYing (2) and (4). If there exists a vector t such that Pi = Fit for i E lo and Pi = FIt for i E I satisfy i E 10 , Vk E Xi' Xj ::1 i E 10, v" E Xi
v"
(12)
E Xi, Xj 3 E Xi
v"
(13)
i E 11 , i E 10 •
Vk Vk
5. A UNIFYING VIEW
for every Vii #- 0, then every generalized motion of (1) in Ui€lX, tends to zero exponentially. 0 The conditions of Theorem 3 should be read as follows. For each cell Xi, the index k should range over all vertices of that cell, and for each vertex Vk, the indexj should range over all cells to which this vertex belongs. The number of inequalities that need to be satisfied increases drastically when we take sliding modes into account. If attractive sliding modes
Most versatile are the piecewise quadratic Lyapunov functions [4, 7, 11]
Vex)
= ilF;TFix
X
E Xi
Piecewise quadratic Lyapunov functions can be computed via convex optimization in terms of LMls. The conditional analysis (that inequalities are only required to hold for those x such that x E XL) can be done using the S-procedure, which appears to work well in practice but is only a sufficient condition. The quadratic Lyapunov functions [2] are special instances of the piecewise quadratics, obtained by letting Fi = I. Quadratic Lyapunov functions can be computed via LMI optimization, and conditional analysis can be done using the S-procedure. Also the piecewise affine Lyapunov /itnctions[8]
Vex)
=
t'Fix
x E Xi
can be seen as a special case of the piecewise quadratics. They can be computed via linear programming, and the conditional analysis can be formulated without loss (using slight modifications of the Farkas' lemma). Poly topic Lyapunov functions [9, 10, 1Jare a special case of the piecewise affine Lyapunov functions. The polytopic Lyapunov functions can be obtained from the piecewise affine by considering partitions that consist of convex cones with base in the origin. The computations can be done using linear programming and the conditional analysis is performed without loss.
The choice of Lyapunov function candidate involves several trade-offs. For example, the linear matrix inequalities in Section 3 are more demanding to solve than the linear programming problems in Section 4. Moreover, welldeveloped linear programming software exists for large scale problems that exploits sparsity and admits systems with several thousands of cells to be analyzed in a matter of seconds. On the other hand, one may need many more 2092
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress oflFAC
ANAL YSIS OF PIECEWISE LINEAR SYSTEMS VIA CONVEX OP ...
segments when constructing a piecewise linear Lyapunov function than one would need in the case of piecewise quadratics. As always, it is advisable to try the simplest things first, and use more powerful Lyapunov function candidates only when necessary. The unifying framework established in this paper makes it simple to move from one function class to the other in order to find the most appropriate Lyapunov function candidate. Once the constraint matrices Fi and Ei for a given partition are fixed, the unifying parameterization allows seamless transfer between piecewise linear and piecewise quadratic Lyapunov function computations. Another issue appears when Lyapunov-like functions are used in system analysis and optimal control problems. Different problem formulations then call for different classes of loss functions. While energy-related problems (such as computation of the induced L 2 -gain) are conveniently expressed as quadratic integrals, piecewise linear functions have been useful in analysis of systems with absolute constraints.
6. AUTOMATIC PARTITION REFINEMENTS For piecewise linear systems with an initial cell partition for the dynamics it is natural to use the same partition for the Lyapunov function. There are however many examples where a more refined partition is needed for the analysis. Partition refinements increase the flexibility of the LyapwlOv function candidate at the cost of more computations. Successive partition refinements then allow a direct tradeoff between the precision in the analysis and the computations involved. In order to limit the computational cost it is important that partition refinements are made only where increased flexibility is really needed. In this section we will show how linear programming duality can be used for automated partition refinements.
the strongest constraint on the optimization problem, and to subdivide this cell in order to increase the flexibility of the Lyapunov function candidate. The computations can then be repeated, proving stability or suggesting further partition refinements. In the case oflinear programming, it is particularly simple to obtain the information about to what degree a certain constraint restricts the optimal value r. This sensitivity information is obtained as the solution to the associated dual problem. (Many LP solvers solve the primal and the dual problem simultaneously, so this step does not require any additional computations.) Since there are several constraints associated to each cell, we compute the total constraint cost for each cell as the sum of the dual variables associated to it. The cell with the largest constraint cost is then subdivided in order to increase the fiexibilityofthe Lyapunov function. We propose the following algorithm.
Algorithm 1. -Automated Partition Refinements (1) Solve the linear program associated to Theorem 2, modified as in (14). (2) If T < 0, the procedure has terminated and asymptotic stability has been proven. Otherwise, refine the cell with the highest constraint cost and return to 1.
6.2 Cell Splitting After deciding what cell to subdivide, one must also decide how this subdivision should be carried out. This appears to be a delicate issue, since not every splitting operation increases the flexibility of the Lyapunov function. The simple idea of splitting a cell by introducing a new vertex in its center has the disadvantage that the faces of the cells are never refined (see Figure 2a). We therefore suggest to split a cell
6.1 Introducing Flexibility where Needed
a.
The feasibility problem of Theorem 2 can be solved by the linear programming problem min -r
Fig. 2. Procedures for subdividing a simplex.
subject to
t.T
r
r
> p~Aivk > ji~AiVk
i E
10 Vk
i E It
vk
E Xi
(14)
E Xi
In this formulation, exponential stability according to Theorem 2 is obtained when r < O. If the optimal value of the linear program is positive, no piecewise linear Lyapunov function exists on the current partition. In this case it is reasonable to find the cell which imposes
by introducing a new vertex in the center of its largest face (see Figure 2b). Note that this operation induces a subdivision also of one neighboring cell. Since the vector field in cells containing the origin are homogeneous, we propose to split these cells by introducing a new vertex in the face facing the origin.
Example 1. Consider the following system 2093
Copyright 1999 IF AC
ISBN: 0 08 043248 4
ANAL YS1S OF PIECEWTSE LINEAR SYSTEMS VIA CONVEX OP .. .
{
Xl
X2
= Xz = -5Xl -
X2 -
sat(xl
+ X2)
(15)
where sat(x) denotes the unit saturation. This system is piecewise linear and has oscillatory dynamics in both the linear and the saturated operating regions. It is well-known that piecewise linear Lyapunov functions may require a rather fine partition of the state space in order prove stability of oscillatory systems [9] . Indeed, for the coarse initial partition shown in Figure 3 (left), no piecewise linear Lyapunov function can be found. Based on this initial partition, however, the automatic refinement procedure terminates with the partition shown in Figure 3 (right). The corresponding Lyapunov function, shown in Figure 4 (left), guarantees exponential decay of all trajectories within the estimated region of attraction, shown as the outermost level set in Figure 4 (right).
Fig. 3. Initial partition(left) and automatically refined partition(right) of Example 1.
Fig. 4. Lyapunov function(left) and guaranteed region attraction(right) in Example 1.
7. CONCLUSIONS The recently developed procedure for computation of piecewise quadratic Lyapunov functions was specialized to Lyapunov functions that are piecewise linear. It was shown how such Lyapunov functions can be computed via linear programming. The results were given for both polyhedral and polytopic partitions, and for systems with and without sliding modes. This established a unifying framework for computation of piecewise quadratic, globally quadratic, piecewise linear and polytopic Lyapunov functions. The unifYing framework makes it easy to move from one function class to the other in order to find the most appropriate Lyapunov function candidate. Another novel development was the automatic procedure for partition refinements based on
14th World Congress ofIFAC
linear programming duality. This part leaves many important problems open. Is it for example possible to prove that the refinement procedure terminates? If so, under what conditions? How does the procedure extend to the LMI case? Piecewise linear analysis of hybrid systems and extensions to control design problems are also interesting venues for future investigations.
Acknowledgments. This work was partially supported by the Swedish Research Council for Engineering Sciences under grant 95-759. The paper was written while the author visited SUPELEC within the Trident program. 8. REFERENCES (1] F. BLANCHlNI. "Nonquadratic Lyapunov functions for robust control." Automatica, 31:3, pp. 451-461, 1995. [2] S. BOYD, L. E. GHAOUI, E . FERON, and v. BALAKRISHNAN. Linear Matrix Inequalities in Systems and Control Theory. Siam, 1994. [3} A. FILIPPOV. Differential Equations with Discontinuous Righthand Sides. Kluwer, Dordrecht, 1988. [4) M. JOHANSSON. "Piecewise linear control systems." PhD Thesis in preparation, 1999. [5} M. JOHANSSON and A. RANTZER. "Computation of piecewise quadratic Lyapunov functions for hybrid systems." 'Thchnical Report, Department of Automatic Control, Lund Institute of Technology, June 1996. [6] M. JOHANSSON and A. RANTZER. "Computation of piecewise quadratic Lyapunov functions for hybrid systems." In Proc. of the 1997 ECC, Brussels, Belgium, July 1997. [7J M. JOHANSSON and A. RANTZER "Computation of piecewise quadratic Lyapunov functions for hybrid systems." IEEE TAC, 43:4, pp. 555-559, April 1998. [8J H. KIENDL and J. RDGER. "Stabilityanalysis of fuzzy control systems using facet functions." Fuzzy Sets and Systems, No 70, pp. 275-285,1995. [9] A. MOLCHANOV and E. FYATNITSKII. "Lyapunov functions that specifY necessary and sufficient conditions for absolute stability of nonlinear systems Hi." Automation and Remote Control, 47, pp. 620-630, 1986. [1OJ Y. OHTA, H. lMANISHI, L. GoNG, and H . HANEDA. "Computer generated Lyapunov functions for a class of nonlinear systems." IEEE Transactions on Circuits and Systems - I, 40:5, pp. 343-354, 1993. [11] A. RANTZER and M. JOHANSSON. "Piecewise linear quadratic optimal control." In Proc. of the 1997 A CC, Albuquerque, USA, June 1997. 2094
Copyright 1999 IF AC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
Copyright .© 1999 IF AC 14th Triennial World Congress, Beijing, P.R. China
D-2c-03-6
ANALYSIS OF PIECEWISE LINEAR SYSTEMS VIA CONVEX OPTIMIZATION - A UNIFYING APPROACH Mikael Johansson Department of Automatic Control Lund Institute of Technology, Box 118.22100 Lund, Sweden. Erna,il:mikael}@control.lth.se
Abstract: The recently developed technique for computation of piecewise quadratic Lyapunov functions is specialized to L1yapunov functions that are piece wise linear. This establishes a unified framework for computation of quadratic, piece wise quadratic, piece wise linear and polytopic Lyapunov functions. The search for a piecewise linear Lyapunov function is formulated as a linear programming problem, and duality is used to address the non-trivial issue of partition refinements. Copyright ID 19991FAC
L INTRODUCTION
Lyapunov functions can be computed via linear programming. This has some computational advantages over the LMI-based computations of piecewise quadratic Lyapunov functions. More importantly, this establishes a unified framework for computation of piecewise quadratic [51, globally quadratic 12]. piecewise linear [8] and polytopic Lyapunov functions [1, 10]. Finally, this approach gives valuable insight in how duality can be used to devise algorithms for automatic partition refinements.
Recent advances in hardware and software has created a strong interest for computational analysis of control systems. A large research effort has focused on quadratic stability, and it has been shown how many interesting analysis problems for linear uncertain systems can be cast as convex optimization problems, hence solved efficiently using numerical computations [2]. When applied to nonlinear systems, however, these results are often found too conservative (see [5] for some simple examples). This raises a question whether convex optimization can be used for analysis of more powerful system models, using more powerful Lyapunov functions. This question found a positive answer in the recently developed technique for analysis of piecewise linear (or rather affine) systems, See [5, 7). It was shown how the search for piecewise quadratic Lyapunov functions for piecewise linear systems could be cast as a convex optimization problem. In [11], the procedure was extended to system analysis, providing one possible generalization of the celebrated analysis oflinear systems with quadratic constraints.
Here, {XihEI :;;; Rn is a partition of the state space into a number of (possibly unbounded) polyhedral cells, as illustrated in Figure 1. We may think of these cells as operating regimes. The local behavior within each cell is described by a state space model which is affine in x.
In this paper we show how the parameterization of piecewise quadratic Lyapunov functions used in [7, 11) can be specialized to functions that are piecewise linear. We show how piecewise linear
The index set of the cells is denoted I. We let 10 E I be the set of cells that contain the origin, and h ~ I be the set of cells that do not contain the origin. It is assumed that ai = Cl = 0 for
2. PIECEWISE LINEAR SYSTEMS
We consider systems on the form {
X = ai +AiX + B,. u = Ci + Gix + D,u
y
x EX•.
(1)
2089
Copyright 1999 IF AC
ISBN: 008 0432484
ANALYSIS OF PIECEWISE LINEAR SYSTEMS VIA CONVEX OP ...
Admittedly, piecewise linear Lyapunov functions have been considered before (8J, but from a very different perspective and for a particular class of partition. This section presents several new formulations of the analysis computations and establishes a link between the piecewise linear Lyapunov functions and the piecewise quadratic Lyapunov functions described above.
4.1 A Matrix Parameterization of Continuous and Piecewise Linear Functions One may view the matrix format for continuous piecewise quadratic Lyapunov functions as a quadratic form in the coordinates z obtained by the continuous piecewise linear mapping z = Fix. If one rather considers linear forms in z, one obtains the following parameterization of continuous piecewise linear functions V(x)=t'F/i:=P~x
XEXj
Once again, we have obtained a separation of free parameters (now collected in the vector t) from the constraints imposed by the partition. One may also see the above fOrIIlat as a direct restriction of the piecewise quadratic functions as follows. Let Fi be constraint matrices satisfYing (2), and define
_ [Fi0 fi] 1 '
~. F, -
1[0 t]
T="2 t'0·
Note that Fi also satisfies the continuity condition (2), and that the piecewise quadratic function V(x) of Proposition 1 now evaluates to
14th World Congress ofIFAC
satisfY
{o = P~Ai + u)Et o=
Eh
(8)
i
p~ - WiEi
The search for the free variables t, Ui and Wi is a linear programming problem. If the system does not have any attractive sliding modes, a solution to this problem guarantees that
Vex) = P~x
for x E Xi
is a Lyapunov function for the system. It can be established, similar to Farkas' lemma, that the above conditions are both necessary and sufficient for the Lyapunov function candidate to be positive and decreasing on UiE1Xi , see [4]. If we consider polytopic partitions (where all cells are bounded), it is sufficient to check the Lyapunov conditions at the vertices of the polytopes, i.e., to establish that for all
Vk
# O. We have the following theorem.
Theorem 2. Let {XihEI be a partition ofIRn into convex polytopes with vertices Vk, and let Fi be the associated continuity matrices satisfYing (2) and (4). If there exists a vector t such that
Pi
= F:t
P; = FIt
for i E 10 foriEl
satisfY
The above parameterization will now be used in the search for continuous piecewise linear Lyapunov functions for piecewise linear systems. We state results for polytopic and polyhedral partitions, and for systems with and without attractive sliding modes. All results can be verified via linear programming. The first result considers piecewise linear Lyapunovfunction on (possibly unbounded) polyhedral partitions.
Theorem 1. Let {XihEl ~ lR n be a polyhedral partition with continuity matrices Fi, satisfYing (2) and (4), and cell boundings Ei , satisfYing (3) and (4). Assume furthermore that EiX # 0 for every x E Xi such that x -I o. If there exist vectors t, Ui and Wi such that Ui and Wi have positive entries, while
Pi = F[t, Pi = F[t,
(7)
then every trajectory x(t) E UiE1Xi satisfying (1) for t 2:: 0 tends to zero exponentially. 0
~Ax=~~T~i=f~£=~~ 4.2 Piece wise Linear Stability
i E 10
i E 10
i E Io i E 10
{ o > P~AiVk
o < p~ Vk
Vk
E Xi
Vk
E
Xi
i Ell Vk E Xi
i E
I1 Vk
(9)
(10)
E Xi
for Vk # 0, then every trajectory x(t) E UiEIXi satisfYing (1) for t 2:: 0 tends to zero exponentially. 0 Note that all relaxation terms have vanished and that the vector inequalities of Theorem 1 are reduced to a number of scalar inequalities. The basic stability computations can be extended in several useful ways. On example is computation of decay rate, T, which can be estimated from the modified Lyapunov inequality
Vex) + TV(X) < 0
i Eh
Vx#O.
2090
Copyright 1999 IF AC
ISBN: 008 0432484
ANALYSIS OF PIECEWISE LINEAR SYSTEMS VIA CONVEX OP ...
Admittedly, piecewise linear Lyapunov functions have been considered before (8J, but from a very different perspective and for a particular class of partition. This section presents several new formulations of the analysis computations and establishes a link between the piecewise linear Lyapunov functions and the piecewise quadratic Lyapunov functions described above.
4.1 A Matrix Parameterization of Continuous and Piecewise Linear Functions One may view the matrix format for continuous piecewise quadratic Lyapunov functions as a quadratic form in the coordinates z obtained by the continuous piecewise linear mapping z = Fix. If one rather considers linear forms in z, one obtains the following parameterization of continuous piecewise linear functions V(x)=t'F/i:=P~x
XEXj
Once again, we have obtained a separation of free parameters (now collected in the vector t) from the constraints imposed by the partition. One may also see the above fOrIIlat as a direct restriction of the piecewise quadratic functions as follows. Let Fi be constraint matrices satisfYing (2), and define
_ [Fi0 fi] 1 '
~. F, -
1[0 t]
T="2 t'0·
Note that Fi also satisfies the continuity condition (2), and that the piecewise quadratic function V(x) of Proposition 1 now evaluates to
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satisfY
{o = P~Ai + u)Et o=
Eh
(8)
i
p~ - WiEi
The search for the free variables t, Ui and Wi is a linear programming problem. If the system does not have any attractive sliding modes, a solution to this problem guarantees that
Vex) = P~x
for x E Xi
is a Lyapunov function for the system. It can be established, similar to Farkas' lemma, that the above conditions are both necessary and sufficient for the Lyapunov function candidate to be positive and decreasing on UiE1Xi , see [4]. If we consider polytopic partitions (where all cells are bounded), it is sufficient to check the Lyapunov conditions at the vertices of the polytopes, i.e., to establish that for all
Vk
# O. We have the following theorem.
Theorem 2. Let {XihEI be a partition ofIRn into convex polytopes with vertices Vk, and let Fi be the associated continuity matrices satisfYing (2) and (4). If there exists a vector t such that
Pi
= F:t
P; = FIt
for i E 10 foriEl
satisfY
The above parameterization will now be used in the search for continuous piecewise linear Lyapunov functions for piecewise linear systems. We state results for polytopic and polyhedral partitions, and for systems with and without attractive sliding modes. All results can be verified via linear programming. The first result considers piecewise linear Lyapunovfunction on (possibly unbounded) polyhedral partitions.
Theorem 1. Let {XihEl ~ lR n be a polyhedral partition with continuity matrices Fi, satisfYing (2) and (4), and cell boundings Ei , satisfYing (3) and (4). Assume furthermore that EiX # 0 for every x E Xi such that x -I o. If there exist vectors t, Ui and Wi such that Ui and Wi have positive entries, while
Pi = F[t, Pi = F[t,
(7)
then every trajectory x(t) E UiE1Xi satisfying (1) for t 2:: 0 tends to zero exponentially. 0
~Ax=~~T~i=f~£=~~ 4.2 Piece wise Linear Stability
i E 10
i E 10
i E Io i E 10
{ o > P~AiVk
o < p~ Vk
Vk
E Xi
Vk
E
Xi
i Ell Vk E Xi
i E
I1 Vk
(9)
(10)
E Xi
for Vk # 0, then every trajectory x(t) E UiEIXi satisfYing (1) for t 2:: 0 tends to zero exponentially. 0 Note that all relaxation terms have vanished and that the vector inequalities of Theorem 1 are reduced to a number of scalar inequalities. The basic stability computations can be extended in several useful ways. On example is computation of decay rate, T, which can be estimated from the modified Lyapunov inequality
Vex) + TV(X) < 0
i Eh
Vx#O.
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ANALYSIS OF PIECEWISE LINEAR SYSTEMS VIA CONVEX OP ...
Given a fixed value of 't, the above condition can be verified using a slight modification of the previous theorellls (where Ai has been replaced by Ai + r I in the decreasing conditions). The optimal value of T can then be found by bisection. Another possibility is to prove stability for piecewise linear inclusions,
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can be ruled out a priori, one should rather use Theorems 1 and 2. If sliding modes can not be ruled out, it is preferable to try to detect on what surfaces sliding will occur and use the augmented conditions only on these surfaces. Apart from decreasing the computational burden, information of where sliding modes appear gives valuable engineering insight.
In this case, one need to simultaneously solve several decreasing conditions in each region (one for each k E K{i)), see [6, 10].
The previous sections unify several approaches to numerical Lyapunov function construction for (possibly) uncertain piecewise linear systems.
4.3 Systems with Attractive Sliding Modes
Cell boundaries of piecewise linear systems may in general have regions where the trajectories in adjacent cells are directed towards the boundary. With our definition, the trajectories are well-defined only until the moment when they reach this region. The theorems stated so far avoids this problem by only considering trajectories defined for all t 2: o. It is possible to extend the trajectory definition
further using the concept of sliding mode [3]. We then say that a piecewise Cl function x(t) E UiE1Xi is a generalized motion of the system (1) if, for every t such that the derivative x(t) is defined, the differential inclusion x(t) E convjEJ{Ajx
+ aj}
(11)
holds. Here, J is the set of all indices j such that x(t) E Xj. The following theorem guarantees stability in the presence of sliding modes. Theorem 3. Let {XihEI be a partition of1Rn into convex polytopes with vertices Vk, and let Pi be the associated constraint matrices, satisfYing (2) and (4). If there exists a vector t such that Pi = Fit for i E lo and Pi = FIt for i E I satisfy i E 10 , Vk E Xi' Xj ::1 i E 10, v" E Xi
v"
(12)
E Xi, Xj 3 E Xi
v"
(13)
i E 11 , i E 10 •
Vk Vk
5. A UNIFYING VIEW
for every Vii #- 0, then every generalized motion of (1) in Ui€lX, tends to zero exponentially. 0 The conditions of Theorem 3 should be read as follows. For each cell Xi, the index k should range over all vertices of that cell, and for each vertex Vk, the indexj should range over all cells to which this vertex belongs. The number of inequalities that need to be satisfied increases drastically when we take sliding modes into account. If attractive sliding modes
Most versatile are the piecewise quadratic Lyapunov functions [4, 7, 11]
Vex)
= ilF;TFix
X
E Xi
Piecewise quadratic Lyapunov functions can be computed via convex optimization in terms of LMls. The conditional analysis (that inequalities are only required to hold for those x such that x E XL) can be done using the S-procedure, which appears to work well in practice but is only a sufficient condition. The quadratic Lyapunov functions [2] are special instances of the piecewise quadratics, obtained by letting Fi = I. Quadratic Lyapunov functions can be computed via LMI optimization, and conditional analysis can be done using the S-procedure. Also the piecewise affine Lyapunov /itnctions[8]
Vex)
=
t'Fix
x E Xi
can be seen as a special case of the piecewise quadratics. They can be computed via linear programming, and the conditional analysis can be formulated without loss (using slight modifications of the Farkas' lemma). Poly topic Lyapunov functions [9, 10, 1Jare a special case of the piecewise affine Lyapunov functions. The polytopic Lyapunov functions can be obtained from the piecewise affine by considering partitions that consist of convex cones with base in the origin. The computations can be done using linear programming and the conditional analysis is performed without loss.
The choice of Lyapunov function candidate involves several trade-offs. For example, the linear matrix inequalities in Section 3 are more demanding to solve than the linear programming problems in Section 4. Moreover, welldeveloped linear programming software exists for large scale problems that exploits sparsity and admits systems with several thousands of cells to be analyzed in a matter of seconds. On the other hand, one may need many more 2092
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ANAL YSIS OF PIECEWISE LINEAR SYSTEMS VIA CONVEX OP ...
segments when constructing a piecewise linear Lyapunov function than one would need in the case of piecewise quadratics. As always, it is advisable to try the simplest things first, and use more powerful Lyapunov function candidates only when necessary. The unifying framework established in this paper makes it simple to move from one function class to the other in order to find the most appropriate Lyapunov function candidate. Once the constraint matrices Fi and Ei for a given partition are fixed, the unifying parameterization allows seamless transfer between piecewise linear and piecewise quadratic Lyapunov function computations. Another issue appears when Lyapunov-like functions are used in system analysis and optimal control problems. Different problem formulations then call for different classes of loss functions. While energy-related problems (such as computation of the induced L 2 -gain) are conveniently expressed as quadratic integrals, piecewise linear functions have been useful in analysis of systems with absolute constraints.
6. AUTOMATIC PARTITION REFINEMENTS For piecewise linear systems with an initial cell partition for the dynamics it is natural to use the same partition for the Lyapunov function. There are however many examples where a more refined partition is needed for the analysis. Partition refinements increase the flexibility of the LyapwlOv function candidate at the cost of more computations. Successive partition refinements then allow a direct tradeoff between the precision in the analysis and the computations involved. In order to limit the computational cost it is important that partition refinements are made only where increased flexibility is really needed. In this section we will show how linear programming duality can be used for automated partition refinements.
the strongest constraint on the optimization problem, and to subdivide this cell in order to increase the flexibility of the Lyapunov function candidate. The computations can then be repeated, proving stability or suggesting further partition refinements. In the case oflinear programming, it is particularly simple to obtain the information about to what degree a certain constraint restricts the optimal value r. This sensitivity information is obtained as the solution to the associated dual problem. (Many LP solvers solve the primal and the dual problem simultaneously, so this step does not require any additional computations.) Since there are several constraints associated to each cell, we compute the total constraint cost for each cell as the sum of the dual variables associated to it. The cell with the largest constraint cost is then subdivided in order to increase the fiexibilityofthe Lyapunov function. We propose the following algorithm.
Algorithm 1. -Automated Partition Refinements (1) Solve the linear program associated to Theorem 2, modified as in (14). (2) If T < 0, the procedure has terminated and asymptotic stability has been proven. Otherwise, refine the cell with the highest constraint cost and return to 1.
6.2 Cell Splitting After deciding what cell to subdivide, one must also decide how this subdivision should be carried out. This appears to be a delicate issue, since not every splitting operation increases the flexibility of the Lyapunov function. The simple idea of splitting a cell by introducing a new vertex in its center has the disadvantage that the faces of the cells are never refined (see Figure 2a). We therefore suggest to split a cell
6.1 Introducing Flexibility where Needed
a.
The feasibility problem of Theorem 2 can be solved by the linear programming problem min -r
Fig. 2. Procedures for subdividing a simplex.
subject to
t.T
r
r
> p~Aivk > ji~AiVk
i E
10 Vk
i E It
vk
E Xi
(14)
E Xi
In this formulation, exponential stability according to Theorem 2 is obtained when r < O. If the optimal value of the linear program is positive, no piecewise linear Lyapunov function exists on the current partition. In this case it is reasonable to find the cell which imposes
by introducing a new vertex in the center of its largest face (see Figure 2b). Note that this operation induces a subdivision also of one neighboring cell. Since the vector field in cells containing the origin are homogeneous, we propose to split these cells by introducing a new vertex in the face facing the origin.
Example 1. Consider the following system 2093
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ISBN: 0 08 043248 4
ANAL YS1S OF PIECEWTSE LINEAR SYSTEMS VIA CONVEX OP .. .
{
Xl
X2
= Xz = -5Xl -
X2 -
sat(xl
+ X2)
(15)
where sat(x) denotes the unit saturation. This system is piecewise linear and has oscillatory dynamics in both the linear and the saturated operating regions. It is well-known that piecewise linear Lyapunov functions may require a rather fine partition of the state space in order prove stability of oscillatory systems [9] . Indeed, for the coarse initial partition shown in Figure 3 (left), no piecewise linear Lyapunov function can be found. Based on this initial partition, however, the automatic refinement procedure terminates with the partition shown in Figure 3 (right). The corresponding Lyapunov function, shown in Figure 4 (left), guarantees exponential decay of all trajectories within the estimated region of attraction, shown as the outermost level set in Figure 4 (right).
Fig. 3. Initial partition(left) and automatically refined partition(right) of Example 1.
Fig. 4. Lyapunov function(left) and guaranteed region attraction(right) in Example 1.
7. CONCLUSIONS The recently developed procedure for computation of piecewise quadratic Lyapunov functions was specialized to Lyapunov functions that are piecewise linear. It was shown how such Lyapunov functions can be computed via linear programming. The results were given for both polyhedral and polytopic partitions, and for systems with and without sliding modes. This established a unifying framework for computation of piecewise quadratic, globally quadratic, piecewise linear and polytopic Lyapunov functions. The unifYing framework makes it easy to move from one function class to the other in order to find the most appropriate Lyapunov function candidate. Another novel development was the automatic procedure for partition refinements based on
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linear programming duality. This part leaves many important problems open. Is it for example possible to prove that the refinement procedure terminates? If so, under what conditions? How does the procedure extend to the LMI case? Piecewise linear analysis of hybrid systems and extensions to control design problems are also interesting venues for future investigations.
Acknowledgments. This work was partially supported by the Swedish Research Council for Engineering Sciences under grant 95-759. The paper was written while the author visited SUPELEC within the Trident program. 8. REFERENCES (1] F. BLANCHlNI. "Nonquadratic Lyapunov functions for robust control." Automatica, 31:3, pp. 451-461, 1995. [2] S. BOYD, L. E. GHAOUI, E . FERON, and v. BALAKRISHNAN. Linear Matrix Inequalities in Systems and Control Theory. Siam, 1994. [3} A. FILIPPOV. Differential Equations with Discontinuous Righthand Sides. Kluwer, Dordrecht, 1988. [4) M. JOHANSSON. "Piecewise linear control systems." PhD Thesis in preparation, 1999. [5} M. JOHANSSON and A. RANTZER. "Computation of piecewise quadratic Lyapunov functions for hybrid systems." 'Thchnical Report, Department of Automatic Control, Lund Institute of Technology, June 1996. [6] M. JOHANSSON and A. RANTZER. "Computation of piecewise quadratic Lyapunov functions for hybrid systems." In Proc. of the 1997 ECC, Brussels, Belgium, July 1997. [7J M. JOHANSSON and A. RANTZER "Computation of piecewise quadratic Lyapunov functions for hybrid systems." IEEE TAC, 43:4, pp. 555-559, April 1998. [8J H. KIENDL and J. RDGER. "Stabilityanalysis of fuzzy control systems using facet functions." Fuzzy Sets and Systems, No 70, pp. 275-285,1995. [9] A. MOLCHANOV and E. FYATNITSKII. "Lyapunov functions that specifY necessary and sufficient conditions for absolute stability of nonlinear systems Hi." Automation and Remote Control, 47, pp. 620-630, 1986. [1OJ Y. OHTA, H. lMANISHI, L. GoNG, and H . HANEDA. "Computer generated Lyapunov functions for a class of nonlinear systems." IEEE Transactions on Circuits and Systems - I, 40:5, pp. 343-354, 1993. [11] A. RANTZER and M. JOHANSSON. "Piecewise linear quadratic optimal control." In Proc. of the 1997 A CC, Albuquerque, USA, June 1997. 2094
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