σ ¯
< 0
(64)
A
P − σI
By redefining P as 1 σ P, we can obtain the result.
Remark 8. Inequality (51) is also equivalent to
KT
MTp P Mp − σNTp −
K − I N
I
p < 0
(65)
for some positive scalar σ. Hence, we have
KT
MT ˜
p P Mp − NT
p
−
K − I N
I
p < 0
(66)
where ˜
P = − ˜ P 0 , ˜ P = σ−1 P. Using the fact that ( K − K
0
˜
P
0) T ( K − K 0) ≥ 0 , we may obtain an
iterative solution from initial condition K 0 , where K 0 may be gotten from Lemma 5.
5. Comparison and examples
We shall note that the comparisons of some existing methods (Bara & Boutayeb, 2005; Crusius
& Trofino, 1999; Garcia et al., 2001) with the case of ε = 0 in Theorem 1 has been given in
(Bara & Boutayeb, 2006), where it states that there are many numerical examples for which
Theorem 1 with ε = 0 works successfully while the methods in (Bara & Boutayeb, 2005;
Crusius & Trofino, 1999; Garcia et al., 2001) do not and vice-versa. It also stands for our
conditions. Hence, in the section, we will only compare these methods introduced above. The
LMI solvers used here are SeDuMi (v1.3) Sturm et al. (2006) and SDPT3 (v3.4) Toh et al. (2006)
with YALMIP Löfberg (2004) as the interface.
In the first example, we will show the advantage of the scaling LMI with ε compared with the
non-scaling ones. In the second example, we will show that different scaling LMI approaches
have different performance for different situations. As a by-product, we will also illustrate the
different solvability of the different solvers.
154
Discrete Time Systems
Example 1. Consider the unstable system as follows.
⎡
⎤
0.82 0.0576 0.2212 0.8927 0.0678
⎢
⎢ 0.0574 0.0634 0.6254 0.0926 0.9731 ⎥
⎥
A
⎢
⎥
o = ⎢ 0.0901 0.7228 0.5133 0.2925 0.9228
⎣
⎥
0.6967 0.0337 0.5757 0.8219 0.9587 ⎦
0.1471 0.6957 0.2872 0.994 0.5632
⎡
⎤
0.9505 0.2924
⎢
⎢ 0.3182 0.4025 ⎥
⎥
B
⎢
⎥
o = ⎢ 0.2659 0.0341
⎣
⎥
0.0611 0.2875 ⎦
0.3328 0.2196
Co = 0.5659 0.255 0.5227 0.0038 0.3608
0.8701 0.5918 0.1291 0.3258 0.994
This example is borrowed from (Bara & Boutayeb, 2006), where output feedback controllers
have been designed. For A 22 from A, it has stable eigenvalue. In this paper, we compare the
design problem with the maximum decay rate, i.e.,
max ρ s. t. ˜
ATP ˜
A − P < − ρP
Note that in this example, m < n − m. With ε = 0, i.e., using the method in (Bara & Boutayeb,
2006), we obtain the maximum ρ = 0.16, while Theorem 1 gives ρ = 0.18 with ε = −0.09.
However, Theorem 5 only obtains a maximum ρ = 0.03 with a choice of ˆ Z = [ I 2 I 2 0] T.
Note that the solvability heavily depends on the choice of ε. For example, when ε = 0.09 for
Theorem 1, the LMI is not feasible.
Now we consider a case that A 22 has an unstable eigenvalue. Consider the above example
with slight changes on Ao
⎡
⎤
0.9495 0.12048 0.14297 0.19192 0.019139
⎢
⎢ 0.8656 0.28816 0.67152 0.01136 0.38651 ⎥
⎥
A
⎢
⎥
o = ⎢ 0.5038 0.46371 0.9712 0.93839 0.42246
⎣
⎥
0.13009 0.76443 0.47657 0.54837 0.4089 ⎦
0.34529 0.61187 0.15809 0.46639 0.53536
We can easily verify that A 22 from A has one unstable eigenvalue 1.004. Hence, the method
in (Bara & Boutayeb, 2006) cannot solve it. However, Theorem 1 generates a solution as
K =
−0.233763 −0.31506
−
. Meanwhile, Theorem 5 also can get a feasible solution for
3.61207 0.376493
ε = −0.1879 and K = 0.9373 −0.4008 . Theorem 4 via a standard SVD without scaling
1.5244 −0.7974
can also obtain K = −0.3914 −0.3603
−
using (43) or K =
1.4813
0.5720
using (44).
2.3604 −1.1034
−3.7203 −1.8693
Example 2. We randomly generate 5000 stabilizable and detectable systems of dimension n =
4(6, 6, 6, 7, 7) , m = 2(3, 1, 5, 4, 3) and l = 2(3, 5, 1, 3, 4) .
Output Feedback Control of Discrete-time LTI Systems: Scaling LMI Approaches
155
T 1 T 3
SeDuMi 5000 4982
SDPT3 4975 5000
Table 1. Different solvability of different solvers
T 1 α T 3 4.2.2 β 6.3.3 6.1.5 6.5.1 7.4.3 7.3.4
Y
Y
4999 4999 4994 4996 4998 4998
Y
N
1
0
2
3
1
1
N
Y
0
1
4
1
1
1
N
N
0
0
0
0
0
0
Superscriptγ: Y (N) means that the problem can (not) be solved by the corresponding theorems. For
example, the value 4 of third row and third column means that in the random 5000 examples, there are 4
cases that cannot be solved by Theorem 1 while can be solved by Theorem 3.
Table 2. Comparison of Theorem 1 and Theorem 3
Hence we can use Theorem 1 and Theorem 3 with ε = 0 to solve this problem. Note that
different solvers may give different solvability. For example, given n = 6, m = 3 and l = 3,
in a one-time simulation, the result is given in Table 1. Thus in order to partially eliminate
the effect of the solvers, we choose the combined solvability result from two solvers in this
section.
Table 2 shows the comparison of Theorem 1 and Theorem 3. Some phenomenons (the
solvability of Theorem 1 and Theorem 3 depends on the l and m. When m > l, Theorem 1
tends to have a higher solvability than Theorem 3. And vise verse.) was observed from these
results obtained using LMITOOLS provided by Matlab is not shown here.
6. Extension to H∞ synthesis
The aforementioned results can contribute to other problems, such as robust control. In this
section, we extend it to H∞ output feedback control problem. Consider the following system:
x( t + 1) = Ax( t) + B 2 u( t) + B 1 w
(67)
y( t) = Cx( t) + Dw
(68)
z( t) = Ex( t) + Fw
(69)
We only consider the case that B 2 is with full rank and assume that the system has been
transferred into the form like (7). Using the controller as (3), the closed-loop system is
x( t + 1) = ˆ
Ax( t) + ˆ Bw
= (
(70)
A + B 2 KC) x( t) + ( B 1 + B 2 KD) w
We attempt to design the controller, such that the L 2 gain sup z 2 ≤ γ. It should be noted
w 2
that all the aforementioned scaling LMI approaches can be applied here. However, we only
choose one similar to Theorem 1.
156
Discrete Time Systems
Theorem 8. The discrete-time system (67)-(69) is stabilized by (3) and satisfies H∞ , if there exist a
matrix P > 0 defined in (8) and R, such that
⎧
⎨ (Θ1) < 0, m = n − m
(Θ
⎩
2) < 0, m < n − m
(71)
(Θ3) < 0, m > n − m
where ε ∈ R , Θ i is defined in Theorem 1,
(Θ i) =
⎡
⎤
− P 11 RC + [ P 11 P 12] A RD + [ P 11 P 12] B 1 0
⎢
⎢ ∗
ATΘ
⎥
i A − P
ATΘ iB 1
ET
(72)
⎣
⎥
∗
∗
BTΘ
⎦
1
iB − γ I
FT
∗
∗
∗
− γI
Proof: Following the arguments in Theorem 1, we can see that (71) implies
⎡
⎤
ˆ
ATP ˆ
A − P
ˆ
ATP ˆ
B
ET
(Θ
⎣
⎦
i) =
∗
ˆ
BT P ˆ
B − γI FT
< 0
(73)
∗
∗
− γI
Using bounded real lemma (Boyd et al., 1994), we can complete the proof.
7. Conclusion
In this paper, we have presented some sufficient conditions for static output feedback control
of discrete-time LTI systems. Some approaches require a similarity transformation to convert
B or C to a special form such that we can formulate the design problem into a scaling
LMI problem with a conservative relaxation. Based on whether B or C is full rank, we
consider several cases with respect to the system state dimension, output dimension and
input dimension. These methods are better than these introduced in (Bara & Boutayeb, 2006)
and might achieve statistical advantages over other existing results (Bara & Boutayeb, 2005;
Crusius & Trofino, 1999; Garcia et al., 2001). The other approaches apply Finsler’s lemma
directly such that the Lyapunov matrix and the controller gain can be separated, and hence
gain benefits for the design. All the presented approaches can be extended to some other
problems. Note that we cannot conclude that the approaches presented in this paper is
definitely superior to all the existing approaches, but introduce some alternative conditions
which may achieve better performance than others in some circumstances.
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