CONTROL SYSTEM ENGINEERING Module IV

MODULE-IV
OPTIMAL CONTROL SYSTEMS
Introduction:

There are two approaches to the design of control systems. In one approach we select the
configuration of the overall system by introducing compensators to meet the given
specifications on the performance. In other approach, for a given plant we find an overall
system that meets the given specifications & then compute the necessary compensators.

The classical design based on the first approach, the designer is given a set of specifications
in time domain or in frequency domain & system configuration. Compensators are selected
that give as closely as possible, the desired system performance. In general, it may not be
possible to satisfy all the desired specifications. Then, through a trial & error procedure, an
acceptable system performance is achieved.

The trial & error uncertainties are eliminated in the parameter optimization method. In
parameter optimization procedure, the performance specification consists of a single
performance index. For a fixed system configuration, parameters that minimize the
performance index are selected.

Parameter Optimization: Servomechanisms

The analytical approach of parameter optimization consists of the following steps:-

(i) Compute the performance index J as a function of the free parameters
K1,K2,….,Kn of the system with fixed configuration:

J=f(K1,K2,….,Kn)............................ (1)
(ii) Determine the solution set Ki of the equations
𝜕𝐽
= 0; 𝑖 = 1,2, … … . . 𝑛 … … … … … … … … … . (2)
𝜕𝐾𝑖
Equation (2) give the necessary conditions for J to be minimum.

Sufficient conditions
From the solution set of equation(2), find the subset that satisfies the sufficient conditions
which require that the Hessian matrix given below is positive definite.
𝜕2𝐽 𝜕2𝐽 … … 𝜕2𝐽
𝜕𝐾2 𝜕𝑘1𝜕𝑘2 𝜕𝑘1𝜕𝑘𝑛
21 2 2
𝐻= 𝜕 𝐽 𝜕 𝐽 𝜕 𝐽 … … … … … (3)
𝜕𝑘2𝜕𝑘1 𝜕𝐾2 … … . 𝜕𝑘2𝜕𝑘𝑛
2
……. … ….. ………
𝜕2𝐽 𝜕2𝐽 𝜕2𝐽
…… 2
𝜕𝑘𝑛𝜕𝑘1 𝜕𝑘𝑛𝜕𝑘2 𝜕𝐾𝑛

102

, 𝜕2 𝐽 𝜕2 𝐽
Since = , the matrix H is always symmetric.
𝜕𝑘 𝑖 𝜕𝑘 𝑗 𝜕𝑘 𝑗 𝜕𝑘 𝑖


(iii) If there are two or more sets of Ki satisfying the necessary as well as sufficient
conditions of minimization given by equations (2) & (3) respectively, then
compute the corresponding J for each set.
The set that has the smallest J gives the optimal parameters.

Solution of Optimization Problem

The minimization problem will be more easily solved if we can express performance index
interms of transform domain quantities.
The quadratic performance index, this can be done by using the Parseval‟s theorem which
allows us to write
∞ 𝑗∞
1
𝑥2 𝑡 𝑑𝑡 = 𝑋 𝑠 𝑋 −𝑠 𝑑𝑠 … … … … … … … … (4)
2𝜋𝑗
0 −𝑗∞


The values of right hand integral in equation(4) can easily be found from the published tables,
provided that X(s) can be written in the form

𝐵(𝑠) 𝑏0 + 𝑏0𝑠 + ⋯ + 𝑏𝑛−1𝑠𝑛−1
𝑋 𝑠 = =
𝐴(𝑠) 𝑎0 + 𝑎0𝑠 + ⋯ + 𝑎𝑛 𝑠𝑛

Where A(s) has zeros only in the left half of the complex plane.
𝑏2
𝐽1 = 0
2𝑎0𝑎1
𝑏12𝑎0 + 𝑏02𝑎2
𝐽2 =
2𝑎0𝑎1𝑎2
𝑏22𝑎0𝑎1 + 𝑏12 − 2𝑏0𝑏2 𝑎0𝑎3 + 𝑏02𝑎2𝑎3
𝐽3 =
2𝑎0𝑎3(−𝑎0𝑎3 + 𝑎1𝑎2)



Servomechanism or Tracking Problem

In servomechanism or tracking systems, the objective of design is to maintain the actual
output c(t) of the system as close as possible to the desired output which is usually the
reference input r(t) to the system.

We may define error e(t)=c(t) - r(t)

The design objective in a servomechanism or tracking problem is to keep error e(t) small. So
∞
performance index 𝐽 = 0 𝑒2(𝑡) 𝑑𝑡 is to be minimized if control u(t) is not constrained in
magnitude.



103

, EXAMPLE
Referring to the block diagram given below, consider 𝐺 𝑠 = 100 and 𝑠 = 1 .
𝑠2 𝑠
∞
Determine the optimal value of parameter K such that 𝐽 = 0
𝑒 (𝑡) 𝑑𝑡
2 is minimum.




Solution
G(s) 100 s2 100
H s = ==
100
1 + G s Ks s + 100K
Ks 1+
s2
𝐸 𝑠 1
=
𝑅 𝑠 1+𝐻 𝑠
𝑅(𝑠) 1 𝑠 𝑠 + 100𝑘
⇒𝐸 𝑠 = = = 2
1 + 𝐻 𝑠 1 + 100 𝑠 + 100𝑘𝑠 + 100
𝑠 + 100𝐾

Here b0=100K, b1=1, a0=100, a1=100K, a2=1

As E(s) is 2nd order
𝑏12𝑎0 + 𝑏02𝑎2 1 + 100𝐾2
𝐽2 = =
2𝑎0𝑎1𝑎2 200𝑘
𝜕𝐽
𝜕𝐾 = 0 gives K=0.1 (necessary condition)
2
To check the sufficient condition, the Hessian matrix is 𝜕 𝐽
> 0 (+ve definite)
𝜕𝐾2
∴ As necessary & sufficient condition satisfied, so the optimal value of the free parameter of
the system is K=0.1.
Compensator design subject to constraints

The optimal design of servo systems obtained by minimizing the performance index
∞

𝐽= 𝑒2(𝑡) 𝑑𝑡 … … … … . (5)
0
may be unsatisfactory because it may lead to excessively large magnitudes of some control
signals.

A more realistic solution to the problem is reached if the performance index is modified to
account for physical constraints like saturation in physical devices. Therefore, a more realistic
PI should be to minimize
∞

𝐽= 𝑒2(𝑡) 𝑑𝑡 … … … … . (6)
0
Subject to the constraint
𝑚𝑎𝑥 𝑢(𝑡) ≤ 𝑀 … … … … … … . (6𝑎)

The constant M is determined by the linear range of the system plant.


104