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Robust Predictive Control for Constrained Uncertain Piecewise Linear Systems using Linear Matrix Inequalities
Third International Conference on Advances in Control and Optimization of Dynamical Systems March 13-15, 2014. Kanpur, India
Robust Predictive Control for Constrained Uncertain Piecewise Linear Systems using Linear Matrix Inequalities Jean Thomas Faculty of Industrial Education, Automatic Control Department, Beni-Sueif University, Egypt (Tel: +20-122-5263476; e-mail: [email protected]). Abstract: This paper considers discrete-time, uncertain Piecewise Linear (PWL) systems affected by polytopic parameter variations. Classical robust controller for uncertain PWL systems is known to be a complex problem, where the on-line computation becomes computationally burdensome and inapplicable. In this paper we present a new technique to solve robust constrained infinite horizon model predictive control for uncertain PWL systems using Linear Matrix Inequalities (LMI). The controller objective is to regulate the system states to a constant set-point that could be in general different from the origin, despite the uncertainties. Constraints over the control and output signals are taken into account. The proposed controller guarantees the system stability and reduces the computation load. To further reduce the computation load, an algorithm to calculate off-line solutions based on the system state location is proposed; where a pre-computed state-feedback control law is applied once the system states enter the region of PWL system containing the shifted origin. This algorithm reduces considerably the computation time while offering a suboptimal solution. A numerical example to validate the efficiency of the developed techniques is presented. Keywords: Piecewise Linear Systems, Model Predictive Control, Robust Control, Linear Matrix Inequalities, Uncertain Systems, Hybrid Systems. variations and bounded disturbances, two different approaches were presented in (Lin and Antsaklis, 2003) and (Thomas, 2011). These approaches share the same philosophy as they propose a robust solution for the considered problem in two steps. In the first step, a reachability and attainability technique is applied to find the state space regions for which there is a robust control and a feasible mode sequence that can drive the system states to the desired region. In the second step, suitable control algorithms are proposed to drive the system states from those feasible regions to the desired region/point. The authors of those two techniques showed the success of their technique in offering a simple robust control with a low on-line computation load. However, for PWL systems with many subsystems and for long backward horizon, applying the reachability technique would imply the exploration of a large number of regions (exponential complexity), even if these calculations are made off-line. This enormous number of regions could affect the simplicity of the proposed controller, and may become inapplicable in real time. In (Zou and Li, 2007) a RMPC for uncertain PWA systems with polytopic parameter variations was presented, where the case of unconstrained PWA systems was addressed. The authors showed how to transfer the min-max control problem to a Linear Matrix Inequality (LMI) problem. The developed technique in (Zou and Li, 2007) was based on driving the system states to the origin.
1. INTRODUCTION Piecewise Linear (PWL) Systems (Sontag, 1981) is a powerful framework that can model a broad class of hybrid systems (systems, including both continuous and discrete dynamics) and nonlinear systems where nonlinearities can be represented by a set of linear models around different operating modes or different state conditions such as saturation or dead zones. Uncertainties could arise from different sources like model simplification, limited system knowledge, and changes of the components value. Thus, control robustness becomes mandatory, so that performances of the systems are preserved in spite of these different causes of uncertainty. Robust controllers for uncertain systems are commonly presented in the literature through a min-max control problem that is known as a complex problem and computationally burdensome. Robust Model Predictive Control (RMPC) is proposed also in the literature as an affective technique for constrained uncertain discrete-time linear systems (Bemporad, et al., 2003) and for perturbed continuous Piecewise Affine (PWA) systems (Necoara, et al., 2004), where a control technique based on minimizing the worst-case cost function (min-max problem, to try to counteract the worst disturbance) is proposed. Some new techniques using parametric programming for linear systems are proposed in the literature to reduce the computation load (Rossiter, et al., 2005) and (Sakizlis, et al., 2004).
In this paper we extend the results presented in (Zou and Li, 2007) by considering the constant set-point tracking problem where the set-point in general could be different from the origin. Also, a generalization to a constrained UPWL system
Considering Uncertain Piecewise Linear (UPWL) systems, where the uncertainties are coming from polytopic parameter 978-3-902823-60-1 © 2014 IFAC
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2014 ACODS March 13-15, 2014. Kanpur, India
is considered, where constraints on control signals and measured outputs are taken into account. The proposed controller based on LMI technique. The LMI technique is known as a tractable problem. However to further reduce the computation load; we propose an algorithm that significantly reduces the on-line computation while offering a suboptimal solution. The proposed algorithm applies off-line explicit solution once the system states enter the region of the PWL system that includes the target point or the shifted origin.
Thus based on the current state location x(t ) , the active mode i is determined. Exact state measurement x(t ) is supposed to be available. System dynamics (1) depends on the uncertainty w , in the following we describe this dependence through the vertices A ij , B ij , Cij , the parameter w plays the role of weighting between them leading in a mathematical sense to a convex combination:
(A
i =1
)
, B ij , Cij is the j-th vertex of the i-th model, v being the
(
) }
3.1 Robust IH-MPC using LMI Technique The control objective is to drive the system output to the target point by moving the system to the set-point x s , u s that in general could be different from the origin. We assume that the constant set-point is feasible, i.e. satisfying the imposed constraints and is equilibrium, where:
is the polyhedral coverage of the state space X , s
j χ i = X , and χ i χ = ∅ for i ≠ j
j =1
The original concepts of applying robust control using LMI technique were presented by Kothare in (Kothare, et al., 1996), where robust Infinite Horizon Model Predictive Control (IH-MPC) using LMI technique for uncertain linear systems has been developed. In (Zou and Li, 2007), the robust MPC of Kothare is extended to PWA systems with polytopic uncertainty for guaranteeing robust stability. The case of unconstrained PWA systems is considered in (Zou and Li, 2007), and the technique developed there was to drive the system states to the origin. In this section, we extend these results by considering the constant set-point tracking problem such that the set-point in general could be different from the origin. Also, a generalization to a constrained PWL system is considered, where constraints on control signals and measured outputs are taken into account. Also, an algorithm including a suboptimal explicit solution is proposed to reduce the on-line computation. The proposed controller ensures the system stability.
being the number of subsystems (modes). Each region χ i is given by:
where:
∑w =1
where : w ≥ 0,
3. ROBUST CONTROL FOR UNCERTAIN PWL SYSTEMS USING LMI TECHNIQUE
where x(t ) ∈ X n , u(t ) ∈ U m , y (t ) ∈ Y p and wt ∈ W denote the system state, the control input, the outputs and the uncertainty, respectively, at time instant t (for the i -th mode) with X, U, Y, W assigned polytopes. n , m and p are the dimensions of state, input and output vectors respectively.
s
j
χ i , the model is affected by polytopic uncertainty.
i i x(t + 1) = A ( w)x(t ) + B ( w)u(t ) Si : , for [x(t )] ∈ χ i (1) y (t ) = Ci ( w)x(t )
}
(4)
v
j
for each mode i ∈ I . The coefficients w j are unknown and possibly time varying. In this way, for each polyhedral region
Piecewise linear systems are powerful tools for describing or approximating both nonlinear and hybrid systems, and representing a straightforward extension from linear to hybrid systems (Sontag, 1981). This paper focuses on the class of uncertain discrete-time piecewise linear systems subject to polytopic parameter variations, defined as:
{
ij
ij
{
S ( x) > 0 is equivalent to the matrix inequality: R( x)
χ i = x(t ) Q i x(t ) ≤ q i ⊂ ℜ n
j
number of vertices. The matrices A i ( w), B i ( w), Ci ( w) represent the model subject to uncertainty, described by the polytopic set Ωi = Convex _ hull ( A ij , B ij , Cij ), j = 1, , v
2. UNCERTAIN PIECEWISE LINEAR SYSTEMS
s i =1
j
j
R( x) > 0, Q( x) − S ( x) R( x) −1 S ( x) T > 0 .
i
j
C ( w) = ∑ w C
Schur Complements: let Q( x) = Q( x)T , R( x) = R( x)T , and S (x) depends affinely on x . Then the LMI:
{χ }
v
v
i
Notation: The matrix inequality A > B ( A ≥ B ) means that A and B are square symmetric and A − B is positive (semi) definite.
Q( x) S ( x) T
v
A i (w ) = ∑ w j A ij , B i ( w) = ∑ w j B ij ,
The rest of the paper is organized as follows. In section 2, a brief description of PWL systems and the considered class is given. A robust MPC controller for constrained uncertain PWL system using LMI technique is presented in section 3 together with an algorithm including off-line solution to reduce the on-line computation. A numerical example to show the success of the developed technique is presented in section 4. Finally conclusions and some remarks are given in section 5.
(2)
x s (t + 1) = A i x s (t ) + B i u s (t ) ,
y s (t ) = Ci x s (t )
(5)
for the i-th mode which contains the set-point. Analogous to the familiar approach from robust control for linear systems,
(3)
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2014 ACODS March 13-15, 2014. Kanpur, India
Theorem 1. Considering feedback control law ~ ~ u (t + k t ) = F x (t + k t ), k ≥ 0 for the UPWL systems (1)-(4), the on-line robust IH-MPC problem (6) with the robust stability constraints -or equivalently (12)- is equivalent to the following minimization problem:
the objective function is a min-max function to minimize the worst-case objective function. We consider the following infinite-horizon objective function:
min
max
(
)
u (t + k t ), k = 0,1,, ∞ A ij (t + k ), B ij (t + k ), C ij (t + k ) ∈Ω i , k ≥ 0
where : J ∞
J ∞ (t )
(C (x(t + k t ) − x )) S (C (x(t + k t ) − x ))(6) (t) = ∑ i
∞
k =0 +
s
T
i
(u(t + k t ) − u s )T R(u(t + k t ) − u s )
γ , Qk , Qk +1 ,Y
subject to :
The maximization is to consider the worst-case of model uncertainty on the value of J ∞ . This worst-case value is
Qk Aij Qk + B ij Y 1/ 2 ij S C Qk R1 / 2Y
minimized over present and future steps. S and R are positive definite weighting matrices. To simplify the notation, we let x(t + k t ) = x(k ) . Defining a ~ shifted state a shifted input x ( k ) = x( k ) − x s , ~ ~ u (k ) = u(k ) − u and a shifted output y (k ) = y (k ) − y , then s
s
the cost function can be simplified as follows:
(
∞ T ~ (k ) )T R (u ~ (k ) ) (7) J ∞ (t ) = ∑ Ci ~x (k ) S Ci ~x (k ) + (u k =0
) (
(8)
)
ij
+ B ij Y Qk +1 0 0
k
1 / 2 ij
C Qk 0 γI 0
) (R Y ) T
1/ 2 T
0 0 γI
From (9), (1) and the state feedback control law ~ ( k ) = F~ u x (k ) :
(
~ x T (k ) Ai + B i F
) P (A + B F )− P T
i
k +1
+C
iT
i
k
+ (17)
SC − F RF ~ x (k ) ≤ 0 i
T
Substituting Pk +1 = γ Q −k 1+1 , Pk = γ Q k−1 , Y = FQk , and pre-
and post-multiplying by Q k :
(
(10)
)
(
T
)
γ Q k − Ai Qk + B i Y γ Q −k1+1 Ai Qk + B i Y − iT
Thus: ij
max
ij
]
(t + k ), B (t + k ) , C (t + k ) ∈Ω
i
− C Qk T SQk C i − Y T RY ≥ 0
x (t )) J ∞ (t ) ≤ V (~
(11)
γ , Pi
x (t ) T Pi ~ x (t ) ≤ γ , x(t ) ∈ χ i subject to : ~
(18)
From the other side, the matrix inequality (15), by Schur, is equivalent to:
This gives an upper bound on the robust performance objective. Thus problem (6) -with (8) and (9)- is equivalent to:
min γ
≥ 0 (15)
1 ~x (t )T ~ ≥ 0 , Q k 0 which establish (14). x (t ) Q k
]
− V (~x (t )) ≤ − J ∞ (t )
[A
T
k
Summing (9) from k = 0 to k = ∞ , we get:
ij
) (S
complements is equivalent to the following LMIs:
T V (~ x (k + 1)) − V (~ x (k )) ≤ − C i ~ x (k ) S C i ~ x (k ) (9) T ~ ~ − (u (k ) ) R (u (k ) ), ∀ A i , B i , C i ∈ Ω i
[
(A Q
Proof. Let Pk = γ Q −k 1 , then the constraints ~ x (t )T Pi ~ x (t ) ≤ γ T −1~ ~ in (12) becomes x (t ) Q x (t ) ≤ 1 which by Schur
For PWL systems, the state at different sampling instants may belong to different regions i, to reduce the conservativeness, we consider different Pi values at different sampling instants. Denoting the values of Pi at time k and k+1 respectively by Pk and Pk +1 . In order to obtain an upper bound on the robust objective function, let V (~ x (∞)) = 0 as we assume that the system state x(∞) reaches the set-point x s , then ~ x (∞) = 0 . The following robust stability constraint is considered:
(
~ 1 x (t )T ~ ≥ 0, Qk > 0, Qk +1 > 0, (14) x (t ) Qk
s
Consider a PWA quadratic function: x T Pi ~ x , x(t ) ∈ χ i , i ∈ s V ( x) = ~
(13)
for j = 1,2,, v , i ∈ s . where F = Y Qk−1 , Pk = γ Qk−1 , and Qk , Qk +1 are symmetric matrices and mode dependent. And the state feedback control is given by: (16) u(t ) = F~ x (t ) + u
)
) (
γ
min
s
(
Q k − A i Qk + B i Y
) (S T
1/ 2
Qk +1 0 γI * 0 0 0
(12)
Which is equivalent to (18) 167
C i Qk
0 0 γI
−1
) (R Y ) * (A Q + B Y ) (19) ( ) S C Q ≥ 0 (R Y ) T
T
1/ 2
i
k 1/ 2
i
i
1/ 2
k
2014 ACODS March 13-15, 2014. Kanpur, India
{
Solution of (13)-(15), if it exists, minimizes the upper bound V (~ x (t )) on the worst case MPC objective function J ∞ (t ) .
ε = x ∈ R n x T Q −1 x ≤ 1
3.2 Robust Constrained IH-MPC Constraints over control signals and outputs can be taken into account as following: Considering peak bounds on control elements given by the following relation: (20)
These constraints are equivalent to the following LMI equation:
U max T Y
Y max ~2 ≥ 0 , with U rr ≤ u r , max Q1
Property 1: It was shown in (Kothare, 1997) that for the nominal linear system, the feedback matrix F computed from (13)-(15) is constant; independent of the state of the system. However, in the presence of uncertainty and also constraints, F depends strongly on the states. In such a case, re-computing F at each sampling time shows a significant improvement in the system performance as opposed to using a static feedback matrix F .
(21)
Also the peak bounds constraints on the output signals: ~ yr (t + k t ) ≤ ~ yr , max
, k ≥ 1 , r = 1,, p
(22)
can be presented by the following LMI equation:
( (
(
)
r
Property 2: the initial feasibility of the robust controller (13)-(15) at instant t , guarantees the feasibility at future instants ( t + k , for k ≥ 1 ).
))
T
Q1 C ijr Aij Q1 + B ij Y ij ij ≥ 0, ij ~y 2 I + C A Q B Y 1 r r , max x (k ), ∀j = 1, , v where : ~y (k ) = C ij ~
(23)
Invariant ellipsoid, suppose that there is a feasible solution for (13)-(15), and a state feed-back controller x (t )Q −1~ x (t ) ≤ 1 , u(t + k t ) = F~ x (t + k t ) + u , k ≥ 0 , then if: ~
r
Additional LMIs (21) and (23) are added to (13)-(15) to guarantee the constraints satisfactions.
⇔
k
s
Then:
max
(A (t +k ),B (t +k ),C
Note: The classical problem of asymmetrical constraints due to the shifted variables may exist in this case: u r ≤ u r , max
is said to
In (Wan and Kothare, 2003), a simple and efficient off-line solution for the robust constraint MPC problem for uncertain linear systems was proposed. The algorithm was based on providing a sequence of explicit control law corresponding to a sequence of asymptotically stable invariant ellipsoids constructed off-line one inside another in state space. This technique is simple and leads to limited number of regions, thus reduces significantly the computation load with minor loss in performance. This technique can be applied for the uncertain PWL system (1)-(4) with some modifications; taking into account the active mode i and the system states converge to a constant set-point rather than to the origin. In the following we briefly present this off-line approach, but first we define some properties of the robust controller (13)(16), and the invariant ellipsoid.
Proof. A proof for the previous lemma can be found in (Zou and Li, 2007).
, k ≥ 0 , r = 1,, m
n
be an asymptotically stable invariant ellipsoid, if it has the property that, whenever x(k1 ) ∈ ε , then x(k ) ∈ ε for all times k ≥ k1 and x(k ) → 0 as k → ∞ .
Lemma 1: Assume that (13)-(15) is feasible, then it holds that the robust MPC given by (16) asymptotically stabilizes the uncertain PWL system (1)-(4).
u~r (t + k t ) ≤ u~r , max
} of the state space ℜ
ij
ij
{
ij
)
(t + k ) ∈Ωi
~x (t + k t )T Q −1~x (t + k t ) ≤ 1 (25) k
}
Thus ε = x xT Q k−1x ≤ 1 is an invariant ellipsoid for the
− u r , max − u s ≤ u~r ≤ u r , max − u s (24)
predicted states of the uncertain system.
3.3 Suboptimal Explicit off-line Solution
By adding asymptotically stable invariant ellipsoid one inside another around the set-point, we have more freedom to adopt varying feedback matrices based on the distance between the state and the set-point. Next algorithm, computes off-line the feedback matrices ( F ) for different invariant ellipsoids which can be applied once the system states enter the PWL region that contains the set-point.
In general, LMIs with linear optimization problem, is known as a tractable problem and easier to solve than a classical min-max problem. However, the computation burden for relatively large systems may be still high for many real plants. In this section, we propose a suboptimal explicit offline solution based on the concept of the asymptotically stable invariant ellipsoid. This technique reduces the on-line computations considerably.
Algorithm 1: off-line, given an initial feasible state x 0 within the region that contains the set-point, generate a sequence of minimizers Yz , Qkz for z = 1 → N , where N is the number of required ellipsoids, as follows:
Definition 1 (Wan and Kothare, 2003): Given a discrete a subset dynamical system x(k + 1) = f (x(k ) ) ,
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2014 ACODS March 13-15, 2014. Kanpur, India
x z by solving(13)-(15) with 1. compute γ z , Yz , Qkz at ~ the constraints of (21),(23), and also with additional −1 Qkz −1
−1 .(ignored < Qkz
constraints the ellipsoids number.
(
set-point x s = [0.5 − 0.5]T , u s = 0 , taking into account the control constraints u ≤ 2 .
at z = 1 ) where z is
This system has two modes each has two vertices because of the uncertainty of β as following:
)
−1 −1 in a look-up table. 2. store Qkz , Fz = Yz Qkz
z
3. if
choose
a
new state
xz
For mode one x ∈ χ 1 : satisfying
− 0.6928 − 0.6928 0.4 0.4 A11 = , A12 = 0.4 0.7 0.6928 0.6928
−1 x Tz Qkz x z ≤ 1 . Let z = z + 1 , and go to step 1.
On line, given the current state x(t ) , if the current state
And for mode two x ∈ χ
outside the region χ i that contains the set-point, solve online the LMIs problem (13)-(15) with the given constraints of (21),(23) and apply the control law of (16).
0 with B1 = B 2 = B = 1
point, do a bisection search over Q z−1 in the look-up table to −1~ find the smallest ellipsoid ε z satisfying ~ x (t )T Qkz x (t ) ≤ 1 , ~ and apply the control law u(t ) = F x (t ) + u .
The on-line robust controller according to (13)-(16), and (21) is applied with Q = I 2 and R = 0.0001 . Figure 1 shows the system state evaluations (upper) and the robust controller value (lower), solid-lines. Figure 1 shows also the system response for the case of static feedback matrix F (dashlines), which is calculated only once based on the initial state, to be compared with the response of the dynamic feedback matrix (solid-lines), i.e. recomputing F at each sampling time. It is clear that recomputing F at each sampling time improves the system performance. This indicates that the proposed robust constraint MPC controller using LMI technique successes in regulating the state to the desired setpoint while respecting the control constraints. Random uncertainty of β is considered in the simulation.
s
To construct a continuous feedback matrix over the state space using the look-up table, apply the following control law: x (t ) + u s (α F + (1 − α z )Fz +1 )~ u(t ) = z z ~ FN x (t )
for z ≠ N for z = N
(26)
where α z is computed from the following equation:
αz =
1− ~ x (t )T Q −z 1+1~ x (t ) T −1 −1 ~ ~ x (t ) Q − Q x (t )
(
z
z +1
)
(27)
System States Control Signal
Consider the following uncertain piecewise system:
x∈χ
=: {x [1 0]x < 0}
that
starting
from
the
initial
20
25
with Static state-feedback Matrix (F)
0.2 0
with Dynamic state-feedback Matrix (F)
-0.2 5
10
15
Sampling Instants
20
25
Algorithm 1, the first state x1 is considered and discretized into 10 points, approaching gradually the set-point 0.5. The following points are chosen: x1 = { 2,1.7,1.5,1.3,1, 0.75, 0.6, 0.55, 0.51, 0.501} which ~ x = {1.5,1.2,1.0, 0.8, 0.5, 0.25, 0.1, 0.05, 0.01, 0.001} . implies
,
condition
1
x 0 = [− 1 1] the controller should regulate the states to the T
15
10
To apply the partially explicit off-line solution according to
and 0 ≤ β ≤ 0.3 Assuming
x2
-1
Figure 1. State response and control signal for both Dynamic (recomputed) State feedback matrix (F)-solid line, and for Static F-dash-line.
−d 0.4 0 x(k + 1) = x(k ) + 1u(k ) β + d 0 . 4
2
x1
0
-0.4 0
4. NUMERICAL EXAMPLE
where:
1
5
This partially off-line approach sacrifices the optimality to some extent while significantly reducing the on-line computational burden.
x ∈ χ 1 =: {x [1 0]x ≥ 0}
x1 Dynamic x2 Dynamic x1 Static x2 Static
2
and 0 ≤ α z ≤ 1 . To ease the implementation, we can choose one dimensional subspace of the state space, and then discretize this set and construct a set of discrete points in this direction. Then construct a limited number of invariant ellipsoids according to the number of points.
0.6928 d = - 0.6928
:
0.6928 0.6928 0.4 0.4 22 A 21 = , A = − 0.6928 − 0.7 4 0 . 0 . 6928
Else, if x(t ) inside the region χ i that contains the set-
z
2
Figure 2 shows the computed ellipsoids for previous points. 169
2014 ACODS March 13-15, 2014. Kanpur, India
differ from the origin point. Input constraints as well as output constraints are taken into account. To further reduce the controller complexity, a technique based on invariant ellipsoids is proposed, where once the system states enter the PWL region that contains the desired set-point, a precomputed off-line solution based on the current state location is applied. The proposed technique leads to a suboptimal solution but it reduces significantly the computation time. The proposed controller asymptotically stabilizes the uncertain PWL system to the desired set-point in spite of the considered uncertainty.
1
0.5
x2
0
-0.5
-1
-1.5
REFERENCES -2 -1.5
-1
-0.5
0
0.5
1
x1
1.5
2
2.5
Figure 2. The ellipsoids defined by Qk for the 10 discrete points over x1. Figure 3 shows the system state response using the suboptimal explicit solution as proposed in Algorithm 1 (dash-lines) with the continuous feedback control law of (26). For the sake of comparison, Figure 3 shows also the system response for the on-line solution (13)-(16) (solid-lines). It is clear that the suboptimal response according to Algorithm 1 is very close to the on-line solution, but with much lower computation load. x1 : On-line solution x2 : On-line solution x1 : with Off-line solution x2 : with Off-line solution
System States
1
x1
0.5 0
x
-0.5
2
-1
Control Signal
0.1 0
5
10
15
20
25
15
20
25
On-line solution
-0.1
Algorithm 1 (with off-line solution)
-0.2 -0.3 -0.4
5
10
Sampling Instants
Figure 3. State response and control signal using the suboptimal explicit solution, dash-line, and using the on-line solution, solid line. On a standard desktop simulation computer (2 GHz PC with 1 Gb of ram), the average computation time per step for the on-line solution in (13)-(15) is estimated to be 0.8 s, and the whole simulation takes around 19s, while for the off-line solution, the average time is 0.003s and using Algorithm 1, the whole simulation is completed in 1.8s. The modeling language YALMIP (Löfberg, 2004) has been used with the SeDuMi solver to solve the considered LMIs and to compute the invariant ellipsoids. 5. CONCLUSIONS In this paper we examined a class of uncertain discrete-time piecewise linear systems affected by polytopic parameter variations. Robust Model Predictive Controller using Linear Matrix Inequalities is proposed to drive the uncertain PWL system states to a constant set-point which in general could 170
Bemporad A., F. Borrelli and M. Morari (2003). Min-max control of constrained uncertain discrete-time linear systems. IEEE Transactions on Automatic Control, 48(9):1600-1606. Kothare, M. V.; V. Balakrishnan, and M. Morari, (1996). “Robust Constrained Model Predictive Control Using Linear Matrix Inequalities”. Automatica, 32, 10, 13611379. Kothare M. V., “Control of Systems Subject to Constraints”. PhD Thesis 1997. Lin H. and P.J. Antsaklis (2003). Robust tracking and regulation control of uncertain piecewise linear hybrid systems. ISIS Technical report. http://www.nd.edu/~isis/tech.html. Löfberg J., YALMIP: A Toolbox for Modeling and Optimization in MATLAB, In Proceedings of the CACSD Conference, Taipei, Taiwan, 2004. Necoara I., B. De Schutter, T.J.J. van den Boom and J. Hellendoorn (2004). Model predictive control for perturbed continuous piecewise affine systems with bounded disturbances. Proceedings of the 43rd IEEE conference on Decision and Control, Paradise Island, The Bahamas, pp. 1848-1853. Rossiter J.A., Pluymers B., Suykens J.A.K., De Moor B., ``A Multi Parametric Quadratic Programming Solution To Robust Predictive Control'', in Proc. of the IFAC World Congress (IFAC05), Prague, Czech Republic, Jul. 2005. Sontag E.D. (1981). Nonlinear regulation: the piecewise linear approach. IEEE Trans. on Automatic Control, 26(2):346-358, April. SeDuMi solver: http://sedumi.ie.lehigh.edu. Thomas J. (2011). Complexity Reduction of Robust Model Predictive Controller for Uncertain Piecewise Affine Systems. American Control Conference ACC’11, San Francisco, California, USA, June 29- July 1, 2011. Vassilis Sakizlis, Nikolaos M. P. Kakalis, Vivek Dua, John D. Perkins, Efstratios N. Pistikopoulos: Design of robust model-based controllers via parametric programming. Automatica 40(2): 189-201 (2004). Wan Z. and M. V. Kothare, "An efficient off-line formulation of robust model predictive control using linear matrix inequalities", Automatica, vol. 39, no. 5, pp. 837 846, 2003. Zou Y., and S. Li: Robust Model Predictive Control for Piecewise Affine Systems; Circuits, Systems, and Signal Processing, Springer, Volume 26, Number 3, June 2007, pp. 393-406(14).
Robust Predictive Control for Constrained Uncertain Piecewise Linear Systems using Linear Matrix Inequalities Jean Thomas Faculty of Industrial Education, Automatic Control Department, Beni-Sueif University, Egypt (Tel: +20-122-5263476; e-mail: [email protected]). Abstract: This paper considers discrete-time, uncertain Piecewise Linear (PWL) systems affected by polytopic parameter variations. Classical robust controller for uncertain PWL systems is known to be a complex problem, where the on-line computation becomes computationally burdensome and inapplicable. In this paper we present a new technique to solve robust constrained infinite horizon model predictive control for uncertain PWL systems using Linear Matrix Inequalities (LMI). The controller objective is to regulate the system states to a constant set-point that could be in general different from the origin, despite the uncertainties. Constraints over the control and output signals are taken into account. The proposed controller guarantees the system stability and reduces the computation load. To further reduce the computation load, an algorithm to calculate off-line solutions based on the system state location is proposed; where a pre-computed state-feedback control law is applied once the system states enter the region of PWL system containing the shifted origin. This algorithm reduces considerably the computation time while offering a suboptimal solution. A numerical example to validate the efficiency of the developed techniques is presented. Keywords: Piecewise Linear Systems, Model Predictive Control, Robust Control, Linear Matrix Inequalities, Uncertain Systems, Hybrid Systems. variations and bounded disturbances, two different approaches were presented in (Lin and Antsaklis, 2003) and (Thomas, 2011). These approaches share the same philosophy as they propose a robust solution for the considered problem in two steps. In the first step, a reachability and attainability technique is applied to find the state space regions for which there is a robust control and a feasible mode sequence that can drive the system states to the desired region. In the second step, suitable control algorithms are proposed to drive the system states from those feasible regions to the desired region/point. The authors of those two techniques showed the success of their technique in offering a simple robust control with a low on-line computation load. However, for PWL systems with many subsystems and for long backward horizon, applying the reachability technique would imply the exploration of a large number of regions (exponential complexity), even if these calculations are made off-line. This enormous number of regions could affect the simplicity of the proposed controller, and may become inapplicable in real time. In (Zou and Li, 2007) a RMPC for uncertain PWA systems with polytopic parameter variations was presented, where the case of unconstrained PWA systems was addressed. The authors showed how to transfer the min-max control problem to a Linear Matrix Inequality (LMI) problem. The developed technique in (Zou and Li, 2007) was based on driving the system states to the origin.
1. INTRODUCTION Piecewise Linear (PWL) Systems (Sontag, 1981) is a powerful framework that can model a broad class of hybrid systems (systems, including both continuous and discrete dynamics) and nonlinear systems where nonlinearities can be represented by a set of linear models around different operating modes or different state conditions such as saturation or dead zones. Uncertainties could arise from different sources like model simplification, limited system knowledge, and changes of the components value. Thus, control robustness becomes mandatory, so that performances of the systems are preserved in spite of these different causes of uncertainty. Robust controllers for uncertain systems are commonly presented in the literature through a min-max control problem that is known as a complex problem and computationally burdensome. Robust Model Predictive Control (RMPC) is proposed also in the literature as an affective technique for constrained uncertain discrete-time linear systems (Bemporad, et al., 2003) and for perturbed continuous Piecewise Affine (PWA) systems (Necoara, et al., 2004), where a control technique based on minimizing the worst-case cost function (min-max problem, to try to counteract the worst disturbance) is proposed. Some new techniques using parametric programming for linear systems are proposed in the literature to reduce the computation load (Rossiter, et al., 2005) and (Sakizlis, et al., 2004).
In this paper we extend the results presented in (Zou and Li, 2007) by considering the constant set-point tracking problem where the set-point in general could be different from the origin. Also, a generalization to a constrained UPWL system
Considering Uncertain Piecewise Linear (UPWL) systems, where the uncertainties are coming from polytopic parameter 978-3-902823-60-1 © 2014 IFAC
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2014 ACODS March 13-15, 2014. Kanpur, India
is considered, where constraints on control signals and measured outputs are taken into account. The proposed controller based on LMI technique. The LMI technique is known as a tractable problem. However to further reduce the computation load; we propose an algorithm that significantly reduces the on-line computation while offering a suboptimal solution. The proposed algorithm applies off-line explicit solution once the system states enter the region of the PWL system that includes the target point or the shifted origin.
Thus based on the current state location x(t ) , the active mode i is determined. Exact state measurement x(t ) is supposed to be available. System dynamics (1) depends on the uncertainty w , in the following we describe this dependence through the vertices A ij , B ij , Cij , the parameter w plays the role of weighting between them leading in a mathematical sense to a convex combination:
(A
i =1
)
, B ij , Cij is the j-th vertex of the i-th model, v being the
(
) }
3.1 Robust IH-MPC using LMI Technique The control objective is to drive the system output to the target point by moving the system to the set-point x s , u s that in general could be different from the origin. We assume that the constant set-point is feasible, i.e. satisfying the imposed constraints and is equilibrium, where:
is the polyhedral coverage of the state space X , s
j χ i = X , and χ i χ = ∅ for i ≠ j
j =1
The original concepts of applying robust control using LMI technique were presented by Kothare in (Kothare, et al., 1996), where robust Infinite Horizon Model Predictive Control (IH-MPC) using LMI technique for uncertain linear systems has been developed. In (Zou and Li, 2007), the robust MPC of Kothare is extended to PWA systems with polytopic uncertainty for guaranteeing robust stability. The case of unconstrained PWA systems is considered in (Zou and Li, 2007), and the technique developed there was to drive the system states to the origin. In this section, we extend these results by considering the constant set-point tracking problem such that the set-point in general could be different from the origin. Also, a generalization to a constrained PWL system is considered, where constraints on control signals and measured outputs are taken into account. Also, an algorithm including a suboptimal explicit solution is proposed to reduce the on-line computation. The proposed controller ensures the system stability.
being the number of subsystems (modes). Each region χ i is given by:
where:
∑w =1
where : w ≥ 0,
3. ROBUST CONTROL FOR UNCERTAIN PWL SYSTEMS USING LMI TECHNIQUE
where x(t ) ∈ X n , u(t ) ∈ U m , y (t ) ∈ Y p and wt ∈ W denote the system state, the control input, the outputs and the uncertainty, respectively, at time instant t (for the i -th mode) with X, U, Y, W assigned polytopes. n , m and p are the dimensions of state, input and output vectors respectively.
s
j
χ i , the model is affected by polytopic uncertainty.
i i x(t + 1) = A ( w)x(t ) + B ( w)u(t ) Si : , for [x(t )] ∈ χ i (1) y (t ) = Ci ( w)x(t )
}
(4)
v
j
for each mode i ∈ I . The coefficients w j are unknown and possibly time varying. In this way, for each polyhedral region
Piecewise linear systems are powerful tools for describing or approximating both nonlinear and hybrid systems, and representing a straightforward extension from linear to hybrid systems (Sontag, 1981). This paper focuses on the class of uncertain discrete-time piecewise linear systems subject to polytopic parameter variations, defined as:
{
ij
ij
{
S ( x) > 0 is equivalent to the matrix inequality: R( x)
χ i = x(t ) Q i x(t ) ≤ q i ⊂ ℜ n
j
number of vertices. The matrices A i ( w), B i ( w), Ci ( w) represent the model subject to uncertainty, described by the polytopic set Ωi = Convex _ hull ( A ij , B ij , Cij ), j = 1, , v
2. UNCERTAIN PIECEWISE LINEAR SYSTEMS
s i =1
j
j
R( x) > 0, Q( x) − S ( x) R( x) −1 S ( x) T > 0 .
i
j
C ( w) = ∑ w C
Schur Complements: let Q( x) = Q( x)T , R( x) = R( x)T , and S (x) depends affinely on x . Then the LMI:
{χ }
v
v
i
Notation: The matrix inequality A > B ( A ≥ B ) means that A and B are square symmetric and A − B is positive (semi) definite.
Q( x) S ( x) T
v
A i (w ) = ∑ w j A ij , B i ( w) = ∑ w j B ij ,
The rest of the paper is organized as follows. In section 2, a brief description of PWL systems and the considered class is given. A robust MPC controller for constrained uncertain PWL system using LMI technique is presented in section 3 together with an algorithm including off-line solution to reduce the on-line computation. A numerical example to show the success of the developed technique is presented in section 4. Finally conclusions and some remarks are given in section 5.
(2)
x s (t + 1) = A i x s (t ) + B i u s (t ) ,
y s (t ) = Ci x s (t )
(5)
for the i-th mode which contains the set-point. Analogous to the familiar approach from robust control for linear systems,
(3)
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Theorem 1. Considering feedback control law ~ ~ u (t + k t ) = F x (t + k t ), k ≥ 0 for the UPWL systems (1)-(4), the on-line robust IH-MPC problem (6) with the robust stability constraints -or equivalently (12)- is equivalent to the following minimization problem:
the objective function is a min-max function to minimize the worst-case objective function. We consider the following infinite-horizon objective function:
min
max
(
)
u (t + k t ), k = 0,1,, ∞ A ij (t + k ), B ij (t + k ), C ij (t + k ) ∈Ω i , k ≥ 0
where : J ∞
J ∞ (t )
(C (x(t + k t ) − x )) S (C (x(t + k t ) − x ))(6) (t) = ∑ i
∞
k =0 +
s
T
i
(u(t + k t ) − u s )T R(u(t + k t ) − u s )
γ , Qk , Qk +1 ,Y
subject to :
The maximization is to consider the worst-case of model uncertainty on the value of J ∞ . This worst-case value is
Qk Aij Qk + B ij Y 1/ 2 ij S C Qk R1 / 2Y
minimized over present and future steps. S and R are positive definite weighting matrices. To simplify the notation, we let x(t + k t ) = x(k ) . Defining a ~ shifted state a shifted input x ( k ) = x( k ) − x s , ~ ~ u (k ) = u(k ) − u and a shifted output y (k ) = y (k ) − y , then s
s
the cost function can be simplified as follows:
(
∞ T ~ (k ) )T R (u ~ (k ) ) (7) J ∞ (t ) = ∑ Ci ~x (k ) S Ci ~x (k ) + (u k =0
) (
(8)
)
ij
+ B ij Y Qk +1 0 0
k
1 / 2 ij
C Qk 0 γI 0
) (R Y ) T
1/ 2 T
0 0 γI
From (9), (1) and the state feedback control law ~ ( k ) = F~ u x (k ) :
(
~ x T (k ) Ai + B i F
) P (A + B F )− P T
i
k +1
+C
iT
i
k
+ (17)
SC − F RF ~ x (k ) ≤ 0 i
T
Substituting Pk +1 = γ Q −k 1+1 , Pk = γ Q k−1 , Y = FQk , and pre-
and post-multiplying by Q k :
(
(10)
)
(
T
)
γ Q k − Ai Qk + B i Y γ Q −k1+1 Ai Qk + B i Y − iT
Thus: ij
max
ij
]
(t + k ), B (t + k ) , C (t + k ) ∈Ω
i
− C Qk T SQk C i − Y T RY ≥ 0
x (t )) J ∞ (t ) ≤ V (~
(11)
γ , Pi
x (t ) T Pi ~ x (t ) ≤ γ , x(t ) ∈ χ i subject to : ~
(18)
From the other side, the matrix inequality (15), by Schur, is equivalent to:
This gives an upper bound on the robust performance objective. Thus problem (6) -with (8) and (9)- is equivalent to:
min γ
≥ 0 (15)
1 ~x (t )T ~ ≥ 0 , Q k 0 which establish (14). x (t ) Q k
]
− V (~x (t )) ≤ − J ∞ (t )
[A
T
k
Summing (9) from k = 0 to k = ∞ , we get:
ij
) (S
complements is equivalent to the following LMIs:
T V (~ x (k + 1)) − V (~ x (k )) ≤ − C i ~ x (k ) S C i ~ x (k ) (9) T ~ ~ − (u (k ) ) R (u (k ) ), ∀ A i , B i , C i ∈ Ω i
[
(A Q
Proof. Let Pk = γ Q −k 1 , then the constraints ~ x (t )T Pi ~ x (t ) ≤ γ T −1~ ~ in (12) becomes x (t ) Q x (t ) ≤ 1 which by Schur
For PWL systems, the state at different sampling instants may belong to different regions i, to reduce the conservativeness, we consider different Pi values at different sampling instants. Denoting the values of Pi at time k and k+1 respectively by Pk and Pk +1 . In order to obtain an upper bound on the robust objective function, let V (~ x (∞)) = 0 as we assume that the system state x(∞) reaches the set-point x s , then ~ x (∞) = 0 . The following robust stability constraint is considered:
(
~ 1 x (t )T ~ ≥ 0, Qk > 0, Qk +1 > 0, (14) x (t ) Qk
s
Consider a PWA quadratic function: x T Pi ~ x , x(t ) ∈ χ i , i ∈ s V ( x) = ~
(13)
for j = 1,2,, v , i ∈ s . where F = Y Qk−1 , Pk = γ Qk−1 , and Qk , Qk +1 are symmetric matrices and mode dependent. And the state feedback control is given by: (16) u(t ) = F~ x (t ) + u
)
) (
γ
min
s
(
Q k − A i Qk + B i Y
) (S T
1/ 2
Qk +1 0 γI * 0 0 0
(12)
Which is equivalent to (18) 167
C i Qk
0 0 γI
−1
) (R Y ) * (A Q + B Y ) (19) ( ) S C Q ≥ 0 (R Y ) T
T
1/ 2
i
k 1/ 2
i
i
1/ 2
k
2014 ACODS March 13-15, 2014. Kanpur, India
{
Solution of (13)-(15), if it exists, minimizes the upper bound V (~ x (t )) on the worst case MPC objective function J ∞ (t ) .
ε = x ∈ R n x T Q −1 x ≤ 1
3.2 Robust Constrained IH-MPC Constraints over control signals and outputs can be taken into account as following: Considering peak bounds on control elements given by the following relation: (20)
These constraints are equivalent to the following LMI equation:
U max T Y
Y max ~2 ≥ 0 , with U rr ≤ u r , max Q1
Property 1: It was shown in (Kothare, 1997) that for the nominal linear system, the feedback matrix F computed from (13)-(15) is constant; independent of the state of the system. However, in the presence of uncertainty and also constraints, F depends strongly on the states. In such a case, re-computing F at each sampling time shows a significant improvement in the system performance as opposed to using a static feedback matrix F .
(21)
Also the peak bounds constraints on the output signals: ~ yr (t + k t ) ≤ ~ yr , max
, k ≥ 1 , r = 1,, p
(22)
can be presented by the following LMI equation:
( (
(
)
r
Property 2: the initial feasibility of the robust controller (13)-(15) at instant t , guarantees the feasibility at future instants ( t + k , for k ≥ 1 ).
))
T
Q1 C ijr Aij Q1 + B ij Y ij ij ≥ 0, ij ~y 2 I + C A Q B Y 1 r r , max x (k ), ∀j = 1, , v where : ~y (k ) = C ij ~
(23)
Invariant ellipsoid, suppose that there is a feasible solution for (13)-(15), and a state feed-back controller x (t )Q −1~ x (t ) ≤ 1 , u(t + k t ) = F~ x (t + k t ) + u , k ≥ 0 , then if: ~
r
Additional LMIs (21) and (23) are added to (13)-(15) to guarantee the constraints satisfactions.
⇔
k
s
Then:
max
(A (t +k ),B (t +k ),C
Note: The classical problem of asymmetrical constraints due to the shifted variables may exist in this case: u r ≤ u r , max
is said to
In (Wan and Kothare, 2003), a simple and efficient off-line solution for the robust constraint MPC problem for uncertain linear systems was proposed. The algorithm was based on providing a sequence of explicit control law corresponding to a sequence of asymptotically stable invariant ellipsoids constructed off-line one inside another in state space. This technique is simple and leads to limited number of regions, thus reduces significantly the computation load with minor loss in performance. This technique can be applied for the uncertain PWL system (1)-(4) with some modifications; taking into account the active mode i and the system states converge to a constant set-point rather than to the origin. In the following we briefly present this off-line approach, but first we define some properties of the robust controller (13)(16), and the invariant ellipsoid.
Proof. A proof for the previous lemma can be found in (Zou and Li, 2007).
, k ≥ 0 , r = 1,, m
n
be an asymptotically stable invariant ellipsoid, if it has the property that, whenever x(k1 ) ∈ ε , then x(k ) ∈ ε for all times k ≥ k1 and x(k ) → 0 as k → ∞ .
Lemma 1: Assume that (13)-(15) is feasible, then it holds that the robust MPC given by (16) asymptotically stabilizes the uncertain PWL system (1)-(4).
u~r (t + k t ) ≤ u~r , max
} of the state space ℜ
ij
ij
{
ij
)
(t + k ) ∈Ωi
~x (t + k t )T Q −1~x (t + k t ) ≤ 1 (25) k
}
Thus ε = x xT Q k−1x ≤ 1 is an invariant ellipsoid for the
− u r , max − u s ≤ u~r ≤ u r , max − u s (24)
predicted states of the uncertain system.
3.3 Suboptimal Explicit off-line Solution
By adding asymptotically stable invariant ellipsoid one inside another around the set-point, we have more freedom to adopt varying feedback matrices based on the distance between the state and the set-point. Next algorithm, computes off-line the feedback matrices ( F ) for different invariant ellipsoids which can be applied once the system states enter the PWL region that contains the set-point.
In general, LMIs with linear optimization problem, is known as a tractable problem and easier to solve than a classical min-max problem. However, the computation burden for relatively large systems may be still high for many real plants. In this section, we propose a suboptimal explicit offline solution based on the concept of the asymptotically stable invariant ellipsoid. This technique reduces the on-line computations considerably.
Algorithm 1: off-line, given an initial feasible state x 0 within the region that contains the set-point, generate a sequence of minimizers Yz , Qkz for z = 1 → N , where N is the number of required ellipsoids, as follows:
Definition 1 (Wan and Kothare, 2003): Given a discrete a subset dynamical system x(k + 1) = f (x(k ) ) ,
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2014 ACODS March 13-15, 2014. Kanpur, India
x z by solving(13)-(15) with 1. compute γ z , Yz , Qkz at ~ the constraints of (21),(23), and also with additional −1 Qkz −1
−1 .(ignored < Qkz
constraints the ellipsoids number.
(
set-point x s = [0.5 − 0.5]T , u s = 0 , taking into account the control constraints u ≤ 2 .
at z = 1 ) where z is
This system has two modes each has two vertices because of the uncertainty of β as following:
)
−1 −1 in a look-up table. 2. store Qkz , Fz = Yz Qkz
z
3. if
choose
a
new state
xz
For mode one x ∈ χ 1 : satisfying
− 0.6928 − 0.6928 0.4 0.4 A11 = , A12 = 0.4 0.7 0.6928 0.6928
−1 x Tz Qkz x z ≤ 1 . Let z = z + 1 , and go to step 1.
On line, given the current state x(t ) , if the current state
And for mode two x ∈ χ
outside the region χ i that contains the set-point, solve online the LMIs problem (13)-(15) with the given constraints of (21),(23) and apply the control law of (16).
0 with B1 = B 2 = B = 1
point, do a bisection search over Q z−1 in the look-up table to −1~ find the smallest ellipsoid ε z satisfying ~ x (t )T Qkz x (t ) ≤ 1 , ~ and apply the control law u(t ) = F x (t ) + u .
The on-line robust controller according to (13)-(16), and (21) is applied with Q = I 2 and R = 0.0001 . Figure 1 shows the system state evaluations (upper) and the robust controller value (lower), solid-lines. Figure 1 shows also the system response for the case of static feedback matrix F (dashlines), which is calculated only once based on the initial state, to be compared with the response of the dynamic feedback matrix (solid-lines), i.e. recomputing F at each sampling time. It is clear that recomputing F at each sampling time improves the system performance. This indicates that the proposed robust constraint MPC controller using LMI technique successes in regulating the state to the desired setpoint while respecting the control constraints. Random uncertainty of β is considered in the simulation.
s
To construct a continuous feedback matrix over the state space using the look-up table, apply the following control law: x (t ) + u s (α F + (1 − α z )Fz +1 )~ u(t ) = z z ~ FN x (t )
for z ≠ N for z = N
(26)
where α z is computed from the following equation:
αz =
1− ~ x (t )T Q −z 1+1~ x (t ) T −1 −1 ~ ~ x (t ) Q − Q x (t )
(
z
z +1
)
(27)
System States Control Signal
Consider the following uncertain piecewise system:
x∈χ
=: {x [1 0]x < 0}
that
starting
from
the
initial
20
25
with Static state-feedback Matrix (F)
0.2 0
with Dynamic state-feedback Matrix (F)
-0.2 5
10
15
Sampling Instants
20
25
Algorithm 1, the first state x1 is considered and discretized into 10 points, approaching gradually the set-point 0.5. The following points are chosen: x1 = { 2,1.7,1.5,1.3,1, 0.75, 0.6, 0.55, 0.51, 0.501} which ~ x = {1.5,1.2,1.0, 0.8, 0.5, 0.25, 0.1, 0.05, 0.01, 0.001} . implies
,
condition
1
x 0 = [− 1 1] the controller should regulate the states to the T
15
10
To apply the partially explicit off-line solution according to
and 0 ≤ β ≤ 0.3 Assuming
x2
-1
Figure 1. State response and control signal for both Dynamic (recomputed) State feedback matrix (F)-solid line, and for Static F-dash-line.
−d 0.4 0 x(k + 1) = x(k ) + 1u(k ) β + d 0 . 4
2
x1
0
-0.4 0
4. NUMERICAL EXAMPLE
where:
1
5
This partially off-line approach sacrifices the optimality to some extent while significantly reducing the on-line computational burden.
x ∈ χ 1 =: {x [1 0]x ≥ 0}
x1 Dynamic x2 Dynamic x1 Static x2 Static
2
and 0 ≤ α z ≤ 1 . To ease the implementation, we can choose one dimensional subspace of the state space, and then discretize this set and construct a set of discrete points in this direction. Then construct a limited number of invariant ellipsoids according to the number of points.
0.6928 d = - 0.6928
:
0.6928 0.6928 0.4 0.4 22 A 21 = , A = − 0.6928 − 0.7 4 0 . 0 . 6928
Else, if x(t ) inside the region χ i that contains the set-
z
2
Figure 2 shows the computed ellipsoids for previous points. 169
2014 ACODS March 13-15, 2014. Kanpur, India
differ from the origin point. Input constraints as well as output constraints are taken into account. To further reduce the controller complexity, a technique based on invariant ellipsoids is proposed, where once the system states enter the PWL region that contains the desired set-point, a precomputed off-line solution based on the current state location is applied. The proposed technique leads to a suboptimal solution but it reduces significantly the computation time. The proposed controller asymptotically stabilizes the uncertain PWL system to the desired set-point in spite of the considered uncertainty.
1
0.5
x2
0
-0.5
-1
-1.5
REFERENCES -2 -1.5
-1
-0.5
0
0.5
1
x1
1.5
2
2.5
Figure 2. The ellipsoids defined by Qk for the 10 discrete points over x1. Figure 3 shows the system state response using the suboptimal explicit solution as proposed in Algorithm 1 (dash-lines) with the continuous feedback control law of (26). For the sake of comparison, Figure 3 shows also the system response for the on-line solution (13)-(16) (solid-lines). It is clear that the suboptimal response according to Algorithm 1 is very close to the on-line solution, but with much lower computation load. x1 : On-line solution x2 : On-line solution x1 : with Off-line solution x2 : with Off-line solution
System States
1
x1
0.5 0
x
-0.5
2
-1
Control Signal
0.1 0
5
10
15
20
25
15
20
25
On-line solution
-0.1
Algorithm 1 (with off-line solution)
-0.2 -0.3 -0.4
5
10
Sampling Instants
Figure 3. State response and control signal using the suboptimal explicit solution, dash-line, and using the on-line solution, solid line. On a standard desktop simulation computer (2 GHz PC with 1 Gb of ram), the average computation time per step for the on-line solution in (13)-(15) is estimated to be 0.8 s, and the whole simulation takes around 19s, while for the off-line solution, the average time is 0.003s and using Algorithm 1, the whole simulation is completed in 1.8s. The modeling language YALMIP (Löfberg, 2004) has been used with the SeDuMi solver to solve the considered LMIs and to compute the invariant ellipsoids. 5. CONCLUSIONS In this paper we examined a class of uncertain discrete-time piecewise linear systems affected by polytopic parameter variations. Robust Model Predictive Controller using Linear Matrix Inequalities is proposed to drive the uncertain PWL system states to a constant set-point which in general could 170
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