Multi Input Multi Output Systems
Dr.D.ANGELINE VIJULA,
Associate Professor,
Department of I&CE,
PSG College of Technology,
Coimbatore-04
Apr-21
Topics to be discussed - Overview
➢ System and Representation
➢ Control strategies
➢ Design Procedure for MIMO systems
• Process Interactions Controller Configurations
• Pairing of Controlled and Manipulated Variables
(CN and RGA)
3 Apr-21
System
A complex assembly consisting of a number of elements or
components that are connected in a sequence to perform a
specific function
Types of systems
●
SISO (Single Input Single Output)
●
SIMO (Single Input Multiple Output)
●
MISO (Multiple Input Single Output)
●
MIMO (Multiple Input Multiple Output)
4 Apr-21
1) SISO: Single input, single output. Simplest to design, uses data from one sensor to
control one thing.
PID
S1 v1
controller
2) SIMO: Single input, multiple output. Uses data from one sensor to control multiple
things.
#1 PID #2 PID
S1 v1 S1 v2
controller controller
3) MISO: Multiple input, single output. More complex as it uses data from multiple
sensors to control one thing. E.g. cascade control
#1 PID S2
S1 controller #2 PID v1
Set point
controller
5 Apr-21
4) MIMO: Multiple Input, Multiple Output. Hardest to design as it integrates multiple
sensor data to coordinate multiple actuators
6 Apr-21
SISO
➢ A single-input and single-output (SISO) system is
a simple single variable system with one input and
one output
➢ SISO systems are typically less complex than
(MIMO) systems.
7 Apr-21
SISO Example
A tank with inflow qi and outflow q1
8 Apr-21
MIMO
➢ Multi-Input Multi-Output system is a system
with more than one input and more than one
output
➢ In a MIMO system input and the output are vectors,
rather than scalars.
➢ This control system is composed of several
interacting control loops.
9 Apr-21
MIMO Systems
The number of feasible alternative configurations of
control loops is very large.
10 Apr-21
Example: Concentration control.
11 Apr-21
Representation of a System
➢ Transfer Function Model
➢ State-Space Model
12 Apr-21
Transfer Function Model
The transfer function H(S) is the linear mapping of the
Laplace transform of the output,, to the
Laplace transform of the input, :
or
13 Apr-21
Limitations of TF Model
➢ Applicable to Linear systems
➢ Applicable to time-invariant systems
➢ Restricted to SISO systems
➢ Reveals only the system output for a given input and
provides no information about the internal behavior
of the system
14 Apr-21
State-Space Model
➢ MIMO systems can be described easily with
state- space equations.
➢ To represent multiple inputs we expand the input
u(t) into a vector U(t) with the desired number of
inputs.
➢ To represent a system with multiple outputs,
we expand y(t) into Y(t), which is a vector of
all the outputs.
➢ The outputs must be linearly dependent on the
input vector and the state vector.
15 Apr-21
State-Space Model
For a continuous time-invariant system, the state-space
representation is given by
Where,
●X is input vector
●Y is output vector
●U is control vector
●A,B,C and D are gain matrices
16 Apr-21
Advantages of State-Space Model
➢ Applicable to Linear and Non-linear systems
➢ Applicable to Time-variant and Time-Invariant
Systems
➢ Can be applied to MIMO systems
➢ It provides information about the internal variables
of a system
17 Apr-21
Control Strategies
Multi loop Control:
Each output variable is controlled using a single input variable.
Multivariable Control:
Each output variable is controlled using more than one input
variable.
18 Apr-21
Multi loop control strategy
Typical industrial approach Consists of using n
standard FB controllers (e.g., PID), one for each
controlled variable.
Control system design
– Select controlled and manipulated variables.
– Select pairing of controlled and manipulated
variables.
– Specify types of FB controllers.
19 Apr-21
Multi Variable Control Strategy
Use Model Predictive Controllers/ Neural Networks
20 Apr-21
Design Procedure For
MIMO Systems
21 Apr-21
Study the physical process to understand its behavior
Develop a model of the process
Calculate the transfer function Matrix
Calculate Condition Number (CN)
No (CN>50) Yes (CN<50)
(Multivariable Control) Can the (Multi loop Control)
system be decoupled based
on CN?
Develop a MIMO Controller using methods such Calculate the relative gain array (RGA)
as Model Predictive Control (MPC) or neural
networks
Choose pairings of inputs and outputs
Design decoupler
Tune SISO Controllers to achieve desired behavior
22 Apr-21
Example: Concentration Control
23 Apr-21
Effects of Control-loop Interactions
➢ Process interactions may induce undesirable
interactions between two or more control loops.
➢ Problems arising from control loop interactions
• Closed-loop system may become destabilized
• Controller tuning becomes more difficult.
24 Apr-21
25 10/26/13 Apr-21
26 10/26/13 Apr-21
27 10/26/13 Apr-21
Controller Configuration
A process with N controlled outputs and N manipulated
variables there are N! different ways to form the control
loops.
28 Apr-21
Example (2X2 System)
➢ Two possible controller pairings:
U1 with Y1, U2 with Y2 (1-1/2-2 pairing) or
U1 with Y2, U2 with Y1 (1-2/2-1 pairing)
➢ Note: For n x n system, n! possible pairing configurations.
29 Apr-21
Transfer Function Model (2x2 system)
Two controlled variables and two manipulated variables
Thus, the input-output relations for the process can be written
as:
30 Apr-21
In vector-matrix notation as,
Where Y(s) and U(s) are vectors,
U
And Gp(s) is the transfer function matrix for the
process
31 Apr-21
1-1/2-2 Control pairing
Ysp1 E1 U1 Y1
+ Gc1 Gp11 ++
Gp12
Gp21
Ysp2 E2 U2 + Y2
+ Gc2 Gp22 +
32 Apr-21
1-2/2-1 Control pairing
Ysp1 E1 U1 Y1
+ Gc1 Gp11 ++
Gp12
Gp21
Ysp2 U2 + Y2
+ Gc2 Gp22 +
E2
33 Apr-21
The Hidden Feedback Control
Loop(in Dark Lines)
Ysp1 E1 U1 Y1
+ Gc1 Gp11 ++
Gp12
Gp21
Ysp2 E2 U2 + Y2
+ Gc2 Gp22 +
For A 1-1/2-2 Control Pairing
34 Apr-21
Simplified Process schematic
CAO Process CA
Q T
Simplified Process Model
C A a11 a12 C A0
T = a Q
a22
21
Y(s) = G(s) U(s)
35 Apr-21
To find the gain array
➢ Change each single input at a time and observe outputs
➢ Calculate local change in outputs and these are entries in
the gain array
Elements of the array:
CA CA
a11 = a12 =
C AO Q
Q C Ao
T T
a21 = a22 =
C AO Q
Q C Ao
36 Apr-21
Intuitive evaluation of gain arrays:
C A k1 0 C Ao
T =
0 k 2 Q
Good or bad?
37 Apr-21
Intuitive evaluation of gain arrays:
C A k11 0 C Ao
= k Q
T 0 22
Good or bad?
Best case because each manipulated variable exactly
controls only one output.
Here CA0 controls CA and Q controls T.
38 Apr-21
Intuitive evaluation of gain arrays:
C A k11 k12 C Ao
T = k
21 k 22 Q
Good or bad?
39 Apr-21
Intuitive evaluation of gain arrays:
C A k11 k12 C Ao
T = k
21 k 22 Q
Good or bad?
Worst case. Manipulated variables can’t
individually control outputs. System is fully
coupled.
How can we measure the degree of coupling?
40 Apr-21
Study the physical process to understand its behavior
Develop a model of the process
Calculate the transfer function matrix
Calculate Condition Number (CN)
No (CN>50) Yes (CN<50)
(Multivariable Control) Can the (Multi loop Control)
system be decoupled based
on CN?
Develop a MIMO Controller using methods such Calculate the relative gain array (RGA)
as Model Predictive Control (MPC) or neural
networks
Choose pairings of inputs and outputs
Design decoupler
Tune SISO Controllers to achieve desired behavior
41 Apr-21
Condition Number
42 Apr-21
Condition Number
➢ The Condition number is measure of how much the
output value of a function can change for a small
change in the input argument.
➢ A problem with a low condition number is said to be
well-conditioned.
➢ A problem with high condition number is said to be ill-
conditioned.
➢ Tells how near to a linearly dependent system we are
based on a single value decomposition (SVD)
43 Apr-21
Singular Value Decomposition
It is a matrix technique that determines if a system is able to be decoupled.
Two input two output system
• The SVD method starts with the steady state gain matrix,
• Using [G], we obtain the Eigen values for the system.
44 Apr-21
Theoretical Calculation for Eigen
Values
The parameters b, c and d are calculated by
+………………….(1)
+…………..(2)
d+…………………..(3)
Using (1), (2), and (3)
………………(4)
…………………………(5)
s1 and s2 are the positive square roots of the respective
eigenvalues.
45 Apr-21
46 Apr-21
Remarks
➢ The greater the CN value, it is harder for the system
to be decoupled.
➢ As a rule of thumb, a system with a CN number of
more than 50 is impossible to decouple.
➢ An ideal system would have a CN number of one,
where each control variable controls a single distinct
output variable.
47 Apr-21
LabVIEW function for CN
48 Apr-21
CN limits
49 Apr-21
Quiz
The CN, or condition number
a) Is the ratio of the smaller number (s2) to the larger
number (s1)
b) Determines the feasibility of decoupling a system.
c) Is the unit eigenvectors of the 𝑛 𝑋 𝑛 matrix 𝐴𝑇 𝐴
d) Is always less than 50.
50 Apr-21
Answer:
(b)
Determines the feasibility of decoupling
a system.
51 Apr-21
Quiz
For MIMO systems
a) Control loops are isolated
b) Each controlled variable is only manipulated by one
variable
c) Decoupling the system makes it more complicated
d) Manipulated variables may affect several controlled
variables
52 Apr-21
Answer
(d)
Manipulated variables may affect
several controlled variables
53 Apr-21
Study the physical process to understand its behavior
Develop a model of the process
Calculate the transfer function matrix
Calculate Condition Number (CN)
No (CN>50) Yes (CN<50)
(Multivariable Control) Can the (Multi loop Control)
system be decoupled based
on CN?
Develop a MIMO Controller using methods such Calculate the relative gain array (RGA)
as Model Predictive Control (MPC) or neural
networks
Choose pairings of inputs and outputs
Design Decoupler
Tune SISO Controllers to achieve desired behavior
54 Apr-21
Relative Gain Array
Provides two useful information:
➢ Measure of process interactions
➢ Recommendation about best pairing of controlled
and manipulated variables.
Requires knowledge of steady-state gains
55 Apr-21
Example of RGA Analysis: 2 x 2 system
• Steady-state process model,
y1 = K11u1 + K12u2
y2 = K21u1 + K22u2
• The RGA is defined as:
11 12
=
21
22
• where the relative gain, ij, relates the ith controlled variable and the
jth manipulated variable
56 10/26/13 Apr-21
Scaling Properties
●
ij is dimensionless
=
ij ij = 1.0
i j
For a 2 x 2 system,
11 = 1
, 12 = 1− 11 = 21
1− K12 K 21
K11K 22
Recommended Controller Pairing
It corresponds to the ij which have the largest
positive values that are closest to one.
57 Apr-21
➢ In general:
1. Pairings which correspond to negative pairings should not be selected.
2. Otherwise, choose the pairing which has ij closest to one.
RGA Different Loop Configurations
Two non-interacting loops
Two non-interacting loops
Degree of interaction remains the same
The two larger numbers (0.75) indicate the
recommended coupling
Opposite to that of the previous case
Creates difficult control problems
58 10/26/13 Apr-21
For 2 x 2 systems:
1
y1 = K11u1 + K12u2 11 = 12 = 1− 11 = 21
K12 K 21 ,
1−
y2 = K21u1 + K22u2 K11 K 22
Example 1:
K11 K12 2 1.5
K = =
K 21 K 22 1.5 2
2.29 −1.29
Λ = −1.29 2.29 Recommended pairing is Y1
and U1, Y2 and U2.
Example 2:
−2 1.5 0.64 0.36
KΛ= =
1.5 2 0.36 0.64
59
Recommended pairing is Y1 with U1 and Y2 with U2.
Apr-21
Matrix Method to Calculate RGA
60 Apr-21
In Matrix Terms
RGA = G*(G −1 )T
LabVIEW Terms
61 Apr-21
1 0 1 0
𝐺= 𝑅𝐺𝐴 =
0 1 0 1
1 1.01 −100 101
𝐺= 𝑅𝐺𝐴 =
0 1 101 −100
1 0.5 1.33 −0.33
𝐺= 𝑅𝐺𝐴 =
0.5 1 −0.33 1.33
Remarks:
➢ The nearer all of the entries are to 1 the more decoupled
the system is.
➢ The best pairing is found by taking the max of the RGA
matrix for each row.
➢ Each row and each column of the RGA sum to 1
62 Apr-21
