Kevickzy / Banyasz Two-Degree-of-Freedom Control Systems

The Youla Parameterization Approach
1. Auflage 2015
ISBN: 978-0-12-803346-3
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 536 Seiten

This book covers the most important issues from classical and robust control, deterministic and stochastic control, system identification, and adaptive and iterative control strategies. It covers most of the known control system methodologies using a new base, the Youla parameterization (YP). This concept is introduced and extended for TDOF control loops. The Keviczky-Banyasz parameterization (KP) method developed for closed loop systems is also presented. The book is valuable for those who want to see through the jungle of available methods by using a unified approach, and for those who want to prepare computer code with a given algorithm. - Provides comprehensive coverage of the most widely used control system methodologies - The first book to use the Youla parameterization (YP) as a common base for comparison and algorithm development - Compares YP and Keviczky-Banyasz (KB) parameterization to help you write your own computer algorithms

Professor Keviczky has a PhD in design of regression experiments and a Doctor of Sciences Degree from the Hungarian Academy of Sciences (HAS). He was the founding member of the Hungarian Academy of Engineering and was appointed as a Foreign Member of the Royal Swedish Academy of Engineering Sciences. He was Director of the Computer and Automation Research Institute (CARI) from 1986-1993 and is still a Research Professor there and a Director at the Multidisciplinary Doctoral School at Sz‚chenyi Istv n University, Gy“r.He has worked with IFAC in various positions since 1981 and was Associate Editor of IFAC's Journal Automatica for six years. He was also General Chair of ECC'2009 and the president of the European Union Control Association (EUCA) from 2010-2012.Keviczky was the founding member of the Steering Committee of the COSY European Science Foundation project and initiated the launch of the EU project DYCOMANS.He has written c-400 papers and has c-701 citations, placing him as the number one expert in this area in Hungary.

Autoren/Hrsg.

Kevickzy, László

Banyasz, Cs.

Leseproben

1.2. Closed-Loop Control

The application of a negative feedback is not necessary to realize a control system. If there is enough a priori knowledge about the process then the open control loop can be used as shown in Figure 1.2.1, where P means the transfer function of the process and C is the transfer function of the regulator. In the figure, the reference signal is r, y is the output signal (controlled signal), u is the actuating signal, and yni and yno are the input and output noises, respectively. This scheme is called open-loop control.

The reference signal tracking would be ideal if the regulator could perform the inverse of the process transfer function, but the perfect inverse usually cannot be realized. The control can provide the reference signal tracking, but it cannot eliminate the effects of disturbances.

The control is performed via negative feedback, if the input signal (manipulated variable or actuating signal) of the process is influenced by the so-called error signal e, which is the deviation between the measured and desired value of the process output signal. Based on the error signal e, the regulator C generates the manipulated variable u, which modifies the output of the process P. The output signal of the process changes according to the dynamics of the process until the output signal reaches its prescribed value. The negative feedback-based regulation is called closed-loop control. The block scheme of such a control system is given in Figure 1.2.2. In many cases the reference signal is filtered via a term of transfer function F (denoted by the dashed line in the figure).

Figure 1.2.1 The block scheme of open-loop control.

Figure 1.2.2 The block scheme of the closed-loop control system.

The relationships between the signals in Figure 1.2.2 can be given by the following expressions:

(s)=F(s)C(s)P(s)1+C(s)P(s)R(s)+11+C(s)P(s)Yno(s)+P(s)1+C(s)P(s)Yni(s)

(1.2.1)

(s)=F(s)1+C(s)P(s)R(s)-11+C(s)P(s)Yno(s)-P(s)1+C(s)P(s)Yni(s)

(1.2.2)

(s)=F(s)C(s)1+C(s)P(s)R(s)-C(s)1+C(s)P(s)Yno(s)-C(s)P(s)1+C(s)P(s)Yni(s)

(1.2.3)

If F(s)=1, it is called the one-degree-of-freedom (ODOF) control; if F(s) is a given transfer function, it is called the TDOF control. In the case of ODOF, four basic transfer functions and in the case of TDOF, six basic transfer functions determine the signal transmissions between the output signals y and u, and the input signals: the reference signal and the output noise. Since the input noise yni can always be transferred to an equivalent output disturbance yn (see Figure 1.2.3), so hereafter it is sufficient to investigate the following four transfer functions (the signs can be disregarded):

R=FCP1+CP;YYn=P1+CPUR=FC1+CP;YYn=11+CP

(1.2.4)

In the case of F=1 the transfer function of the system is

=CP1+CP=L1+L

(1.2.5)

which is the complementary sensitivity function. Here L=CP is the so-called loop transfer function of the open-loop system. The error transfer function regarding the reference signal is

Figure 1.2.3 One-degree-of-freedom closed-loop control system.

=11+CP=11+L

(1.2.6)

and it is also called the sensitivity function. This terminology can be interpreted if the sensitivity of T is calculated for the change of the process.

Let us investigate the behavior of the control loop if the transfer function of the process changes from a nominal value P to ˆ=P+?P. This means, in addition, that instead of the ideal process only its model ˆ is known. Let the relative error of L be

L=?LL=C?PCP=?PP=l

(1.2.7)

that is, it is equal to the relative error of the process model. Then the relative error of T can be obtained by simple computation as

TT=11+CP?PP=S?PP=Sl

(1.2.8)

since

T=?T?P?P=C(1+CP)2?P

(1.2.9)

Thus the sensitivity function S shows how the relative change of the process influences the relative change of the overall transfer function. The complementary sensitivity function can be explained by the apparent relationship

+S=1

(1.2.10)

The TDOF control systems can also be given in the form shown in Figure 1.2.4.

Figure 1.2.4 Two-degrees-of-freedom closed-loop control.

Figure 1.2.5 Equivalences of the two-degrees-of-freedom closed-loop controls.

This form can be rearranged to the equivalent form given in Figure 1.2.2 with the equivalences of Figure 1.2.5.

M-a and E-ß Curves

For deeper analysis of the relationship between the frequency functions of the open- and closed-loop systems, let us analyze the following. The complementary sensitivity function of the closed system can be calculated by the expression (1.2.5). The frequency function of the closed-loop system can be expressed by its amplitude and phase angle:

(j?)=L(j?)1+L(j?)=M(?)eja(?)

(1.2.11)

The gain of the closed system approaches the constant value one for high amplification value of the open system. This conclusion can be drawn from the expression

T(j?)|=|L(j?)L(j?)+1||L|»1˜1

(1.2.12)

Since the control, in general, ensures high gain in the low frequency region, the gain of the closed loop is close to one. Low gain values of the closed loop belongs to the low amplification values of the open loop

T(j?)|=|L(j?)L(j?)+1||L|«1˜|L(j?)|

Since the amplification of physical systems decreases at high frequencies, this expression can be interpreted as indicating that the amplification values of the open- and closed-loop systems are almost the same at high frequencies, that is, the negative feedback does not change the open loop.

During the control design, the relationship between the transfer functions of the closed loop T(s) and of the open loop L(s)=C(s)P(s) is taken into account. This relatively simple relationship actually means conform nonlinear mapping from a complex plane L(s) to T(s). This nonlinearity is the reason why the control design cannot always be performed unambiguously with simple methods.

Every mapping point (complex vector) can be determined for each point of the complex plane by the expression (1.2.5). First, investigate what kinds of curves belong to the values T|=M, where M is a constant. The points of closed systems with the same amplitude correspond to circles on the complex plane. This can easily be verified by solving the equation

=|L(j?)1+L(j?)|=|u+jv1+u+jv|=u2+v21+2u+u2+v2

(1.2.13)

The equation of the curves belonging to the constant amplitude M can be obtained by rearranging the above equation as

u-M21-M2)2+v2=(M1-M2)2

(1.2.14)

This gives the equation of a circle with radius r and center point (uo,vo):

=|M1-M2|;uo=M21-M2;vo=0

(1.2.15)

The curves belonging to different constant, closed-loop amplitudes M are shown in Figure 1.2.6.

The curve at M=1 is a vertical line at u=-0.5. For M>1, the curves are to the left of the line; for M<1, they are to the right of the line. If M tends to infinity, the curves shrink to the point...