Analysis and Control of Linear Systems - Chapter 9 pot

Part 2 System Control This page intentionally left blank Chapter 9 Analysis by Classic Scalar Approach 9.1. Configuration of feedback loops 9.1.1. Open loop – closed loops The block diagram of any closed loop control system (Figure 9.1) consists of an action chain and of a reaction (or feedback) chain which makes it possible to elaborate an error signal ),(tε the difference between the input magnitude )(te and the measured output magnitude )(tr . The output of the system is ).(ts Figure 9.1. Block diagram of a feedback control When the system is subjected to interferences b(t), its general structure is represented by Figure 9.2 by supposing that its working point in the direct chain is known. We designate by )(),( pRpE , ε (p) and )( pS the Laplace transforms of the input, the measurement, difference and the output respectively (see Figure 9.2). Chapter written by Houria SIGUERDIDJANE and Martial DEMERLÉ. 254 Analysis and Control of Linear Systems Figure 9.2. General block diagram The open loop transfer function of this chain is the product of transfer functions of all its elements; it is the ratio: () () ()() 12 () Rp ppp p µµ β ε = [9.1] The closed loop transfer function of this chain is the ratio )( )( pE pS with: ε(p)= )()( pRpE − [9.2] We have: 21 () ()(() ()Sp p Bp p µ µ =+ε(p)) [9.3] and: )()()( pSppR β = [9.4] by using equation [9.2] and by eliminating )( pR and ε (p) of equations [9.3] and [9.4], we obtain: () () () 12 2 () () () 1 () ()() 1 () ()() 12 12 pp p Sp Ep Bp ppp ppp µ µµ µµ β µµ β =+ ++ [9.5] Analysis by Classic Scalar Approach 255 When the interferences are zero, 0)( =pB , the transfer function of the looped system is then: () () () 12 () 1 () ()() 12 pp Sp Ep p p p µ µ µµ β = + [9.6] or simply: () () () 1 ()() Sp p Ep p p µ µβ = + [9.7] by supposing that 12 () () ()ppp µ µµ = . The transfer function with respect to the interference input is obtained by having 0)( =pE in equation [9.5]: () () 2 () 1 () ()() 12 p Sp Bp p p p µ µµ β = + [9.8] 9.1.2. Closed loop harmonic analysis Bandwidth The bandwidth of a system is the interval of angular frequencies for which the module of open loop harmonic gain is more than 1 in arithmetic value: ()() 1jj µωβω >> [9.9] Approximate trace In order to simplify the determination of closed loop transfer functions, we can use the approximations [9.10] and [9.11]: If 1)()( >> ωβωµ jj then )( 1 )( )( ωβω ω jjE jS ≈ [9.10] If 1)()( << ωβωµ jj then )( )( )( ωµ ω ω j jE jS ≈ [9.11] 256 Analysis and Control of Linear Systems Figure 9.3 shows, in Bode plane, the approximate trace (in full line), of the gain curve of the closed loop frequency response. Figure 9.3. Approximate trace of the closed loop harmonic response Point A, in particular for integrator systems, is often rejected at 0= ω . 9.1.2.1. Black-Nichols diagram The Black-Nichols diagram makes it possible to graphically pass from the open loop system to the closed loop system. This diagram corresponds to a unitary feedback. The chart in Figure 9.4 is usable for open loop gains going from –40 dB to +40 dB and a phase difference between 0° and –360°. Figure 9.4. Black-Nichols diagram Analysis by Classic Scalar Approach 257 When the feedback is not unitary, the transfer function can be re-written as a unitary feedback gain transfer function that is divided by the return gain β because: 1 () 11 Fp µµβ µ βµββ == ++ [9.12] 9.1.2.2. Estimation of closed loop time performances from the harmonic analysis The closed loop (CL) frequency response is characterized by the quality factor r Q , also called magnification Q, i.e. the passage from the module through a maximum to an angular frequency r ω called resonance angular frequency. The time response is characterized, for a step function input, by the time of the first maximum m t and the overflow D, as indicated in Figure 9.5a, i.e. ∞∞ −= sssD /)( max where max s represents the maximum value obtained from the output to instant m t and ∞ s that obtained in permanent state. The overflow is expressed as a percentage. Figure 9.5. (a) Time response in CL and (b) frequency response in CL 258 Analysis and Control of Linear Systems When the system has good damping, let ξ be the value of the damping coefficient delimited between 0.4 and 0.7, we have the relation 3≈ mc t ω , where c ω represents the gap angular frequency. It is the angular frequency for which the open loop arithmetic gain is equal to the unit. For a well damped system, the quality factor r Q has a value of less than 3 dB. 9.2. Stability A looped system is called stable if its transfer function: () () 1()() p Fp pp µ µβ = + [9.13] does not have poles of positive or zero real part. In other words, the necessary and sufficient condition of stability of such a system is that )( pF has all its poles with a negative real part. When the denominator of )( pF is a polynomial of order higher than 3 and does not reveal any obvious root, the analytical calculation of the roots may be fastidious. To study the stability, we then use either the geometrical criterion called Nyquist, where we reason only on the open loop in order to determine the stability of the closed loop or the so-called Routh algebraic criterion where we reason on the )( pF specific equation without calculating its roots. We can firstly show that the stability of a linear closed loop control system is connected to the diagrams of its open loop frequency response. The transfer functions )( p µ and )( p β are in general in the form of polynomials in p : )( 1 )( 1 )( pD pN p = µ and )( 2 )( 2 )( pD pN p = β [9.14] then: )( 2 )( 1 )( 2 )( 1 )( 2 )( 1 )( pNpNpDpD pDpN pF + = [9.15] Analysis by Classic Scalar Approach 259 The system’s characteristic equation is: 0)( 2 )( 1 )( 2 )( 1 =+ pNpNpDpD [9.16] or: 0)()(1 =+ pp β µ [9.17] The system is stable if equation [9.17] does not have zeros of positive or zero real part. NOTE 9.1.– the methods presented in this chapter are valid when the open loop transfer function ()()pp µ β does not result from a set of transfer functions presenting simplifications of the poles-positive real part zeros type. 9.2.1. Nyquist criterion This criterion is based on the traditional property of analytical functions and it makes it possible to predict the behavior of a looped system by only knowing the open loop. To do this, we use the following Cauchy’s theorem. When a point M of affix p describes in the complex plane a closed contour C (Figure 9.6a), clockwise, surrounding P poles and Z zeros of a function )( pA of the complex variable p, then the image of the point M through application A surrounds PZN −= times the origin in the same direction. We suppose that there is no singularity on C. If we take, for example, 2= Z and 3=P , then 1−=N , the point M makes 1 tour around the origin, in counterclockwise direction (Figure 9.6b). Figure 9.6. Plane of the complex variable p and plane of )( pA 260 Analysis and Control of Linear Systems The application to the Nyquist criterion leads to consider the transformation )( pA as being the denominator of the transfer function of the looped system. We want this transfer function not to have poles of positive real part and hence its denominator: )()(1)( pppA β µ += [9.18] not to have positive real part zeros. We then choose as contour C a semicircle of infinite radius in the complex semi-plane on the right of the imaginary axis. C is called the Nyquist contour. The image of C through )( pA transformation must thus surround the origin, in a counterclockwise direction, as many times as the number of unstable poles of equation [9.18] and hence of ()()pp µ β . Figure 9.7. Contour and image of the Nyquist curve Contour C is chosen in such as way as to surround the poles of possible zeros of )()(1 pp β µ + with strictly positive real part. If C contains, for example, 1= Z zero and 3=P poles (Figure 9.7a), the Nyquist diagram will make 2−=N circuits around the origin in a clockwise direction and it will go around twice in a counterclockwise direction (Figure 9.7b). To be stable in closed loop, it is necessary that z = 0, the image of C must then make N = – P circuits around the origin. If the open loop transfer function )()( pp β µ is stable, we have 0=P , the image of C through the transformation )()(1 pp β µ + should not surround the origin. However, the number of circuits made around the origin in the transformation )()(1 pp β µ + is equal to the number of circuits made around the critical point –1 in the transformation )()( pp β µ . In what follows, we will deal only with this latter transformation and we will study the case of open loop stable systems, that of integrator systems and finally the case of open loop unstable systems. [...]... necessary and sufficient condition so that any n degree polynomial has all its roots of strictly negative real part We re-write the characteristic equation [9. 17] in the polynomial form: 1 + µ( p)β ( p) = an pn + an − 1 pn − 1 + … + a1 p + a0 [9. 28] 266 Analysis and Control of Linear Systems Then, we create Table 9. 1 (with n + 2 rows) 1 an an-2 an-4 … 2 an-1 an-3 an-5 … 3 b1=an-2 -an an-3 /an-1 b2=an-4 -an... an-5 /an-1 b3=an-6 -an an-7 /an-1 … 4 c1=an-3 -b2 an-1 /b1 c2=an-5 -b3 an-1 /b1 c3=an-7 -b4 an-1 /b1 … … … … … … n+1 … 0 … … n+2 0 0 0 0 Table 9. 1 Routh’s table For a regular system, the number of non-zero terms decreases with 1 every 2 rows and we stop as soon as we obtain a row consisting only of zeros The first column of Routh’s table has n + 1 non-zero elements for a characteristic equation of. .. function when µ1 ≈ K1 and µ 2 ≈ K 2 (Figure 9. 18b) whereas there is absence of error as soon as µ1 ≈ K1 / p (Figure 9. 18b) Analysis by Classic Scalar Approach Figure 9. 17 The image of responses to a step function input and a ramp input (zero interference) Figure 9. 18 The image of responses to an interference step function (zero input) 275 276 Analysis and Control of Linear Systems 9. 3.1.4 Sinusoidal... direction of increasing angular frequencies, does not go through the critical point – 1 and makes around it a number of circuits in the counterclockwise direction equal to the number of unstable poles of µ ( p) β ( p) Case 3 Unstable system in open loop EXAMPLE 9. 1.– the Nyquist trace is given in Figure 9. 10 µ ( p) β ( p) = K p(1 − 2 p ) [9. 25] 264 Analysis and Control of Linear Systems Figure 9. 10 Nyquist... gain and 0 dB to the angular frequency where the phase reaches –180° These margins, noted by ∆φ and ∆G , are represented in Bode, Nyquist and Black-Nichols planes in Figure 9. 13 268 Analysis and Control of Linear Systems Figure 9. 13 Bode plane (a), Nyquist plane (b), Black-Nichols plane (c) In Bode and Black-Nichols planes: ∆φ = φ − (−180 o ) to the angular frequency ω c for which µβ = 0 dB ∆G = 0dB... 1 + µ1 µ 2 β [9. 37] 1 + µ1 µ 2 β and: εb (p)= These expressions are calculated by considering firstly the zero interferences b(t ) = 0 ( e(t ) ≠ 0 ; see Figure 9. 16a) and then the zero input e(t ) = 0 ( b(t ) ≠ 0 , see Figure 9. 16b) 272 Analysis and Control of Linear Systems Figure 9. 16 Zero interferences b(t) = 0, e(t) ≠ 0 (a), zero input e(t) =0, b(t) ≠ 0 (b) 9. 3.1 Permanent error 9. 3.1.1 Step function... order The roots of this equation are of strictly negative real part if and only if the terms of this first column of the table have the same sign and are not zero Statement of Routh’s criterion A system is stable in closed loop if and only if the elements of the first column of Routh’s table have the same sign EXAMPLE 9. 4.– let us consider again example 9. 1 : µ( p )β( p) = K p(1 − 2 p) [9. 29] The characteristic... 0 and ω 0 : E0 = 2 vM γM and ω 0 = γM vM [9. 53] 278 Analysis and Control of Linear Systems hence the condition: 1 µ1 µ 2 β p= j γM vM ≤ ε d max 2 vM [9. 54] γM We note that this condition is necessary and, in certain cases, it may not be sufficient 9. 4 Parametric sensitivity Multiple factors, such as ageing or change of working points may lead to variations on the representative model parameters of. .. and the open loop gain is infinite Analysis by Classic Scalar Approach 263 Figure 9. 9 The choice of the contour of C excludes the origin (a), the transform of the contour C (b), Cε circle of radius ε, p = εejθ (c) The image of the semicircle of an infinitely big radius is the origin of the complex plane The transform of contour C, representing the complete Nyquist place, is represented by Figure 9. 9b... low, the unit-step response is oscillating, if ξ is high, the response is strongly damped and the transient state is long The magnification Q r , i.e the maximal gain of the closed loop module curve, which can be measured directly in the Black-Nichols plane (Figure 9. 15), is related to the damping coefficient by the relation: Qr = 1 2ξ 1 − ξ 2 [9. 33] 270 Analysis and Control of Linear Systems We note . of Linear Systems Then, we create Table 9. 1 (with n + 2 rows). 1 a n a n-2 a n-4 … 2 a n-1 a n-3 a n-5 … 3 b 1 =a n-2 -a n a n-3 /a n-1 b 2 =a n-4 -a n a n-5 /a n-1 b 3 =a n-6 . b 3 =a n-6 -a n a n-7 /a n-1 … 4 c 1 =a n-3 -b 2 a n-1 /b 1 c 2 =a n-5 -b 3 a n-1 /b 1 c 3 =a n-7 -b 4 a n-1 /b 1 … … …. … … …. n+1 … 0 … … n+2 0 0 0 0 Table 9. 1. Routh’s. φ ∆ and G∆ , are represented in Bode, Nyquist and Black-Nichols planes in Figure 9. 13. 268 Analysis and Control of Linear Systems Figure 9. 13. Bode plane (a), Nyquist plane (b), Black-Nichols