RTII Vorlesung 8
19.04.2016
1
MIMO systems
Effects (signals)
Causes (signals)
𝑢1
𝑢2
𝑢𝑚
𝑥 ∈ ℝ𝑛
MIMO system
𝑦1
𝑦2
𝑦𝑝
Most MIMO systems that we analyze are 2 × 2 systems. Almost all are square.
2
c
3
4
5
Definition of MIMO transfer
functions
Loop gain: 𝐿𝑢 𝑠 = 𝐶 𝑠 ⋅ 𝑃 𝑠
Loop gain: 𝐿𝑒 𝑠 = 𝑃 𝑠 ⋅ 𝐶 𝑠
Return difference: 𝐷𝑒 𝑠 = 𝐼 + 𝐿𝑒 (𝑠)
Sensitivity: 𝑆𝑒 𝑠 = 𝐼 + 𝐿𝑒 𝑠
Complementary sensitivity:
−1
𝑇𝑒 s = 𝐼 + 𝐿𝑒 𝑠
⋅ 𝐿𝑒 (s)
with 𝑇𝑒 s + 𝑆𝑒 s = 𝐼 𝑝×𝑝
−1
Return difference: 𝐷𝑢 𝑠 = 𝐼 + 𝐿𝑢 (𝑠)
Sensitivity: 𝑆𝑢 𝑠 = 𝐼 + 𝐿𝑢 𝑠
−1
Complementary sensitivity:
−1
𝑇𝑢 s = 𝐼 + 𝐿𝑢 𝑠
⋅ 𝐿𝑢 (s)
with 𝑇𝑢 s + 𝑆𝑢 s = 𝐼 𝑚×𝑚
6
𝑧
c
7
Vorlesung 8
Thema
Singulärwerte
Lernziele



Sie können die Pole und Nullstellen eines MIMO Systems berechnen.
Sie können die RGA Matrix berechnen (von Hand 2 × 2) und korrekt
interpretieren.
Sie können die Singulärwerte einer komplexen Matrix berechnen und
daraus Eigenschaften der Matrix ablesen.
Skript: Kapitel 2
Ablauf
Für nächste Woche: Kapitel 3
15’
Recap
15’
Pole und Nullstellen
30’
RGA Matrix
30’
Singulärwertzerlegung
Vorlesungsplan
23.02.
01.03.
08.03.
15.03.
Lektion
Lektion
Lektion
Lektion
SISO Reglersynthese
1 – Loop shaping II
2 – Prüfungsnachbesprechung
3 – Robuste Regelgüte, Prädiktive Regelung
4 – Kaskadierte Regelung, Ball on Wheel (BoW)
Reglerimplementation
22.03.
05.04.
Lektion 5 – Echter PID, Gain scheduling, Anti-reset windup
Lektion 6 – Emulation, z-Operator, Aliasing
12.04
19.04.
26.04.
MIMO Systemanalyse
Lektion 7 – MIMO Einführung
Lektion 8 – Singulärwerte
Lektion 9 – Frequenzgang
MIMO Reglersynthese
03.05.
10.05.
17.05.
24.05.
31.05.
Lektion 10 – Zustandsrückführung (LQR), BoW
Lektion 11 – Finite horizon LQR, Model Predictive Control (MPC)
Lektion 12 – Zustandsbeobachter
Lektion 13 – Ausgangsrückführung (LQG, LTR)
Lektion 14 – Fallstudien, Prüfungsvorbereitung
9
Recap: poles and zeros of SISO
systems
Poles:
𝑃 𝑠=𝜋 →∞

e𝜋⋅𝑡 describes the natural dynamics of the system, i.e.,
it describes how the system evolves when the input is
zero.

e𝜁⋅𝑡 describes the zero dynamics of the system, i.e.,
when the system is driven by the input to evolve
according the e𝜁⋅𝑡 , the output is zero.
Zeros:
𝑃 𝑠=𝜁 =0
10
Matrix minors
Matrix minors are the determinants of all square submatrices that can be formed
from 𝑃(𝑠).
Die Minoren sind die Determinanten aller quadratischen Untermatrizen der Matrix 𝑃(𝑠).
𝑃=
𝑎
𝑑
𝑏
𝑒
𝑐
𝑓
A maximum minor is a minor that is formed using a submatrix with the largest
possible dimension. For square matrices 𝑃(𝑠), obviously, the only maximum
minor corresponds to the matrix determinant itself.
11
Poles
The poles of 𝑃(𝑠) are the roots of the least common denominator of all minors of
𝑃(𝑠).
Die Pole von P(s) sind die Nullstellen des kleinsten gemeinsamen Nenners aller Minoren von 𝑃 𝑠 .
Example:
12
Zeros
The zeros of 𝑃(𝑠) are the roots of the greatest common divisor of the numerators
of the maximum minors of 𝑃(𝑠) after normalization to have the pole polynomial of
𝑃(𝑠) as denominators.
Die Nullstellen von 𝑃(𝑠) sind die Nullstellen des grössten gemeinsamen Teilers aller maximalen
Minoren, nachdem diese normiert wurden, so dass sie das Polpolynom von 𝑃(𝑠) als Nenner haben.
Example continued:
13
Zero/pole cancellations
In MIMO systems, poles and zeros are associated with directions.
Zero/pole cancellations only take place when the frequencies and the directions
coincide.
14
Relative gain array
Heat exchanger
System representation
SISO vs. MIMO
Cross couplings
15
RGA example
16
RGA example
𝑃21 1 + 𝐶21 𝑃12 − 𝑃22 𝐶21 𝑃11
𝑦2 =
⋅ 𝑢1
1 + 𝐶21 𝑃12
17
RGA computation
For square matrices, the RGA can be computed by
Matlab: RGA = P .* inv(P).’
For general, non-square matrices, the RGA can be computed using the pseudoinverse (careful: the function pinv can not be applied to transfer functions, but
only to complex matrices)
Matlab: RGA = P .* pinv(P).’
The results is a matrix of transfer functions (or many frequency response matrices),
which can be analyzed in a bode diagram.
18
RGA interpretation
1. An input-output paring such that the diagonal elements of the RGA matrix are
close to one is preferable since this paring is related with a diagonal dominant
plant.
2. Avoid input-output parings which generate diagonal negative elements on the
RGA matrix with 𝑠 = 0. This condition is closely related to systems with lack of
integrity; i.e., a system which cannot maintain stability if one of the diagonal
closed loops is open.
3. High positive values in the diagonal elements of the RGA matrix indicate
difficulty for designing diagonal controllers.
Look at
SISO control possible iff
𝑅𝐺𝐴(𝑠 = 0)
Diagonal entries positive
Bode diagram with
|𝑅𝐺𝐴1,1 | and 𝑅𝐺𝐴1,2
RGA similar to identity around crossover
frequency 𝜔𝑐
19
RGA example: heat exchanger
𝑘 = 100
W
m2 ⋅K
𝑅𝐺𝐴 0 =
1.0076
−0.0076
𝑘 = 10′ 000
−0.0076
1.0076
𝑅𝐺𝐴 0 =
W
m2 ⋅K
5.5244
−4.5244
−4.5244
5.5244
20
Additional information
• The columns and the rows of 𝑅𝐺𝐴(𝑠) always add up to 1.
• The RGA is invariant with respect to scaling, i.e., for any regular diagonal
matrices 𝐷𝑖 the equation 𝑅𝐺𝐴[𝑃(𝑠)] = 𝑅𝐺𝐴[𝐷1 · 𝑃(𝑠) · 𝐷2 ] holds true.
• The RGA of a triangular matrix 𝑃(𝑠) is the identity matrix.
• If the diagonal entries are small, but the off-diagonal entries are similar to
1, SISO control may still be used, but the inputs and outputs need to be
paired differently.
21
Singular value decomposition
Induced matrix norm
𝑀 = 𝑈 ⋅ Σ ⋅ 𝑉𝑇
ഥ𝑇 ⋅ 𝑀
𝑀
Effective matrix rank
22
Singular value decomposition
(SVD)
Any matrix 𝑀 ∈ ℝ𝑝×𝑚 can be written using a singular value decomposition
Where 𝑈 ∈ ℝ𝑝×𝑝 and 𝑉 ∈ ℝ𝑚×𝑚 are unitary matrices, i.e.,
and Σ ∈ ℝ𝑝×𝑚 is a matrix with at most
min(𝑝, 𝑚) nonzero entries on its diagonal.
23
The induced matrix norm
The starting point is the linear transformation
where we can define an induced norm of the matrix 𝑀 as the maximum amplification
from an input vector 𝑢 to output vector 𝑦
When the norm of the vectors is the Euclidean norm 𝑥 = 𝑥 𝑇 ⋅ 𝑥, then the induced
norm is given by
where 𝜎𝑖 (𝑀) are the singular values of the matrix 𝑀.
24
Graphical interpretation
𝑀𝑇 ⋅ 𝑀 =
𝑀 = 𝑈 ⋅ Σ ⋅ 𝑉𝑇 =
−0.55
−0.83
3.94
−1.37
−1.37
1.01
0
−0.93
−0.83 2.12
⋅
⋅
0
0.68
−0.36
0.55
0.36
−0.93
25
Singular values vs. eigenvalues
26
Singular values of complex
matrices
The generalization to the complex case 𝑀 ∈ ℂ𝑝×𝑚 is not difficult. First we need to
define the Euclidean norm for complex vectors
i.e., 𝑣 = 𝑣ҧ 𝑇 ⋅ 𝑣. The induced matrix norm is then defined as
27