4. A Physical Model of a Steam Heated Cylinder
5.1 A design method based on optimization
The word optimize comes from the Latin word optimus which means the best (maximize comes from maximus which means the highest).
Optimization methods are an important tool in nearly all engineering domains, and the control field is indeed no exception. Many control methods are based on optimization of some vital criteria, subject to one or a few constraints. However, it is important to consider how the optimization problem is set up, since this obviously very much affects the result and it is not necessarily a good controller that comes out of an optimal solution. If the optimization problem is not correctly formulated, the optimal controller might not even be stable. The formulation of the problem also affects how easily it is solved. There exist very powerful numerical tools that solve linear or quadratic programming problems, while non-convex optimization is much more demanding due to less effective numerics and the existence of several local optimal solutions.
The derivation of the IPZ-tuning rule is based on a design proposal in [Åström, et al, 1998] and [Åström and Hägglund, 2004]. The idea is to maximize the integral gain, defined as ki = kc/ Ti, subject to a robustness constraint. Due to the form of the constraint, the optimization problem is non-convex. Therefore, much effort is put on finding a simple relation between the process parameters and the optimal solution of controller parameters, so that the user is relieved from the optimization issue.
Instead, he or she will get an approximate solution from just a few button pushes on a pocket calculator.
By maximizing the integral gain, the absolute value of the integrated error (IE) of a step load disturbance is minimized. This can easily be seen by using the nomenclature in Figure 3.6 and writing the error as
).
) ( ( ) ( 1
) ) (
) ( ( ) ( 1
) ( ) 1 (
) ) ( ( ) ( 1
) ( ) ) (
) ( ( ) ( 1
) ) (
(
) ( ) ( ) (
s s D C s P
s s P
s R C s P
s C s P
s s R C s P
s C s s P
s D C s P
s s P
R
s Y s R s E
c c
ff
c ff c
¸¸
¹
·
¨¨
©
§
¸¸
¹
·
¨¨
©
§
(5.1)
Assume a step unit load disturbance and that the set point is zero. Evaluate the integral of the error by the final value theorem
) . ( ) 1
(
) lim (
1 ) 1
( 1
) lim (
1 ) ( ) ( 1
) lim (
) ( IE
0 2 0 2 0 0
c i d
i i c i
i s
i d i i c s
s c
k T s P s T T s T k s T
s P T
s s
T s T T s k T s P
s P
s s C s P
s dt P
t e
o o
o f
³
(5.2)
By maximizing the integral gain, this integrated error is minimized if the sign is disregarded. However, to only maximize the integral gain is not sufficient since there is no guarantee that the control loop is stable, see Figure 5.1. Therefore, an additional constraint is needed. The robustness constraint used here is characterized by a circle with its centre at the point í1 in the Nyquist diagram, see Figure 5.2. By avoiding the point of instability with a certain distance, R0, stability is guaranteed. The radius of the circle will then be the design parameter, and the smaller R0 is the more aggressive the controller will be. This can also be expressed in terms of the sensitivity function, defined as
Time
Control error
Negative area Positive area
Figure 5.1 A characteristic response to a load disturbance on the IPZ process, controlled by a PID controller. It is crucial add a constraint to the optimization problem to avoid an oscillatory solution, since equally large positive and negative areas cancel each other by the IE criteria.
), ( 1 ) 1
(s L s
S (5.3)
where L(s) is the loop transfer function. The maximum sensitivity, Ms, is then given by
). ( 1 max 1 ) (
max Z Z
Z
Z S i L i
Ms
(5.4) Since |1+L(iZ)| is the distance from a point on the Nyquist curve to the
critical point í1, the shortest distance from the Nyquist curve to the point í1 is thus 1/Ms. Therefore, we get
1 , R0
Ms (5.5)
and we can then use Ms as our design parameter, when deriving the tuning rule. The nice thing about the maximum sensitivity function is that it connects the open loop Nyquist curve with a closed loop property. It is also dimensionless which is a nice property. The disadvantage is that it can be difficult to relate to for the unfamiliar user. A common question, e.g., can be what Ms= 1.2 means in practice. Simulations and familiarization is one answer to that issue.
Im
Re R0
–1
Figure 5.2 The robustness constraint of the design method. By varying the radius of the circle, the degree of robustness is changed.
The shape of the Nyquist curve is changed by varying the controller parameters, in order to avoid the Ms-circle while achieving a high integral gain. The derivative time Td, affects the curve close to the origin (high frequencies), the integral time Ti affects low frequencies, while the gain kc
affects all frequencies equally. However, a few properties of the Nyquist curve are independent of the controller parameters. Since both the IPZ-process and a PID controller contain one integrator each, the Nyquist curve starts at the phase lag íS. At high frequencies the phase is íS/2, if we assume no dead time. The complete Nyquist plot then appears as in Figure 5.3.
Note that this design method does not give any suggestion about the feedforward parameters E or Ȗ of the controller, see (3.19), since these parameters have no influence on the sensitivity function or the disturbance rejection.
The design method puts emphasis on disturbance rejection. In [Slätteke, et al, 2002] it was shown that for a certain value of the maximum sensitivity (Ms=1.2), maximizing the integral gain or maximizing the bandwidth of the closed loop system (from set point to measurement) gave essentially the same controller parameters. That means that emphasis is put on both the regulator and servo problem. This does not always hold though, even if the resulting controllers from the two criteria prove to be close.
Figure 5.3 The complete Nyquist curve for the IPZ-process and a PID controller if the dead time is assumed to be zero. Otherwise, there will be the typical circular appearance in the origin.
Remark
Apart from IE, a few other minimization criteria have been proposed in different contexts, e.g. the integrated absolute error (IAE), integrated square error (ISE), and integrated absolute time weighted error (IATE). A disadvantage with ISE is that it penalizes large errors and gives a small but long-term error. IAE have the advantage to avoid oscillatory responses. However, if the system is well damped, IE and IAE give similar results. Keep in mind that IE = IAE, if the error is positive. This is also noted in [Åström, et al, 1998]. IE is therefore chosen as the optimization criterion, since it also is computationally effortless.