Course support: Control Engineering (TBKRT05E)

Transcription

1 Course Support This course support package is offered to you by TBV Lugus, the study association for Industrial Engineering and Manageent. Although this package is coposed with great care, the association takes no responsibility with respect to the use of the package in preparing for the exa. While thoroughly coposed by experienced teacher s assistants, this inforation ay give the ipression to be a coplete and balanced reflection of the subjects being treated in the course. However, it can only be regarded as an additional aid in preparation for the exa. It is in no way a substitute to the course literature or the lectures and tutorials offered in the course. Ephasizing the iportance of active participation by students in the lectures, we decided to provide this inforation shortly before the final exa of the course. After the reset the package will be taken off line. Each year, the packages are revised and adapted to the course schedule and literature for the course of that particular year. We strongly discourage to use packages of the previous years. All rights reserved. No part of this publication ay be reproduced, stored in a retrieval syste or transitted in any for or by any eans, electronic, echanical, photocopying, recording or otherwise without prior consent of the association TBV Lugus.

2 Course support: Control Engineering (TBKRT05E Contents Introduction. Control engineering Control design procedure Exaple Modeling 3 2. Block diagras Equations of otion (Euler-Lagrange State space representation Transfer functions Linearization Analogies different doains Tie doain analysis and design 9 3. Qualitative analysis Stability Input/output response Step response State feedback Frequency doain analysis and design 4 4. Introduction Stability Standard second order systes Bode plots Nyquist criterion PID control Lead and lag filters Introduction. Control engineering Control engineering is a branch of engineering, that applies the theoretical concepts of control theory to design syste with desired behaviors. A control syste typically consists of four actions: easure a variable of interest (for exaple velocity of a car, 2 copare it with the desired value (the desired velocity, 3 copute the action needed to drive the syste fro it s current value to the desired one (how uch gas should be added to the engine, and 4 actuate the coputed action (push gas pedal. A sensor deals with the easuring, the controller copares and coputes, and finally the actuator actuates the syste. Many control syste are closed-loop (see Figure : the easured output signal y of the syste is copared with the reference signal r to obtain the error signal e. Based on this error signal e the controller calculates the needed control, which is actuated by actuation signal u, which is the input to the syste. In the setup shown in Figure, the sensor is part of the syste, while the actuator is part of the controller.

3 .2 Control design procedure INTRODUCTION The other type of control systes, i.e., open-loop controllers, do not easure the output of the to-be-controlled syste. Therefore, this type of control cannot correct for possible errors or disturbances entering the syste. (a (b Figure : Closed loop control (a and open loop control (b.2 Control design procedure The ultiate goal of control engineering is to design controllers that achieve the desired behavior of the to-be-controlled syste. Exaples are achieving a desired teperature using a therostate, or achieving a desired velocity using a cruise controller in a car. There are any ways to design control systes. One step-by-step design procedure, that is applicable to all the controller in this docuent, consists of the following five steps:. Derive a (dynaical odel, that allows for analysis of the syste under consideration. 2. Check the desired behavior and reforulate (if needed the specification in ters of the odel derived in step. 3. Analyze the odel and check whether the specifications are et. If so, there is no need to control the syste and you are done. If not, continue to step Choose an appropriate controller structure, and design/tune the controller paraeters. 5. Ipleent the controller and evaluate whether the controlled syste eets the specification..3 Exaple Throughout this docuent all theoretical concepts are illustrated by the sae linear ass spring daper syste depicted in Figure 2. For each section a worked-out exaple is given in a fraed box below the description of the theory. For this exaple the syste paraeters are taken as ass = 2kg, daping constant b = 0.8Ns/, and spring constant k = 2.4N/. Unless otherwise stated, it is assued that all coponents in the exaple are linear. Figure 2: Mass spring daper syste which is used as an exaple throughout this docuent, with = 2kg, b = 0.8Ns/, k = 2.4N/. 2

4 2 MODELING 2 Modeling An scheatic overview of all the odeling ethods discussed below is shown in Figure 3. Figure 3: Scheatic overview of the odeling ethods discussed in the Control Engineering course. 2. Block diagras Block diagras provide a systeatic way to represent systes. Each block in the diagra corresponds to a (subsyste or coponent of the syste that is odelled. Arrows show the relationships between the blocks. The cobination of block diagras and transfer functions (see Section 2.4 provide a powerful way to represent control systes. Exaple Block diagra of the ass spring daper syste depicted in Figure Equations of otion (Euler-Lagrange The Euler-Lagrange ethod is a systeatic step-by-step procedure to derive the equations of otion for a given syste. First the Lagrangian L(q, q is calculated fro the kinetic co-energy T (q, q and the potential energy V (q as L(q, q = T (q, q V (q, ( where q = (q,..., q n denotes the n-diensional state of the syste. Furtherore, the Rayleigh dissipation function is derived for all eleents that dissipate energy (e.g. dapers, resistors. Table denotes the kinetic co-energies, potential energies and Rayleigh dissipation function for linear systes in the echanical (translation and rotation and electrical doain. For linear systes the kinetic co-energy equals the kinetic energy. 3

5 2.3 State space representation 2 MODELING Kinetic coenergy Potential energy Rayleigh dissipation function Translation Rotation Electric 2 dt 2 dt 2 T (ẋ = 2 dx T ( θ = 2 J dθ T ( q = 2 L dq dt V (x = 2 kx2 V (θ = 2 kθ2 V (q = 2C q2 2 dt 2 D(ẋ = 2 b dx D( θ = 2 b dθ dt D( q = 2 R dq dt 2 Table : The kinetic coenergy, potential energy, and Rayleigh dissipation function for linear systes in the echanical and electrical doain. After deriving the Lagrangian L(q, q and the Rayleigh dissipation function D( q, the equations of otion are given for i =,..., n by d L L = D + Fi e. (2 dt q i q i q i Note that (2 result in n second order differential equations. For linear systes the equations of otion (2 boil down to M q + Qq = C q + Bu, (3 while for nonlinear syste (2 boils down to M(q q + C(q, q + k(q = D( q + Bu. (4 For the ass spring daper exaple the kinetic coenergy for the ass is given by T ( q = 2 q2. The potential energy corresponding to the spring is given by V (q = 2 kq2, while that Rayleigh dissipation function for the daper is given by D( q = 2 b q2. The Lagrangian of the syste follows fro ( and is given by L(q, q = 2 q2 2 kq2. The equations of otion follow fro (2, with external force F e = F : 2.3 State space representation q + kq = d q + F. (5 The Euler-Lagrange ethod described the dynaics of the syste by n second order differential equations. One can rewrite these n second order differential equations into 2n first order differential equations to obtain the dynaics of the syste in a state space representation. A state space representation is a atheatical odel of a physical syste consisting of an input u, an output y, and a state x related by first order differential equations. The ost general (nonlinear state space representation of (4 is given by ẋ = f(x, u y = g(x, u, (6 where f(x, u describes how the tie derivative of the state ẋ is related to the state x and input u, and g(x, u describes how the output y follows fro the state x and input u. The first line in (6 is called the state equation, the second line is called the output equation. For a linear systes of the for (3, the state space representation can be rewritten as ẋ = Ax + Bu y = Cx + Du, (7 where A R N N, B R N M, C R P N, D R P M, with N the nuber of states, M the nuber of inputs, and P the nuber of outputs. The state space respresentation provides a convenient and copact way to odel and analyze systes with ultiple inputs and outputs. The saller the nuber of states, the easier it is to analyze the behavior 4

6 2.4 Transfer functions 2 MODELING of (7. The inial state space representation is that state space representation, where the nuber of states is as sall as possible. The following rule-of-thub gives the iniu nuber of states needed to odel the syste dynaics: iniu nuber of states = nuber of energy storing eleents. For the linear ass spring daper syste given by (5 define the state as x = q, x 2 = q, the input u = F, and the output y = q. The state space representation of the for (7 then follows as ( ẋ ẋ 2 = ( 0 k b y = ( 0 ( x x 2. ( x x 2 + ( 0 u (8 Since there are two energy storing eleents (one ass and one spring, (8 is a inial state space representation. 2.4 Transfer functions The Transfer function describes input-output behavior of linear tie-invariant (LTI systes with zero initial conditions and an equilibriu at x = 0. The internal dynaics of the syste are not considered by the transfer function, which is a ajor difference with the state space representation. In the Control Engineering course only single input single output (SISO transfer functions are considered. The transfer function fro input u to the output y is denoted by H yu (s, where s = iω denotes the coplex arguent of the Laplace transfor. Transfer functions can be obtain in roughly three ways: Fro block diagras using block diagra algebra, 2 Fro the equations of otion using the Laplace transfor, and 3 Fro the state space representation. f(t F (s Step input s Rap input s 2 e at s+a sin ωt cos ωt ω s 2 +ω 2 s s 2 +ω 2 f (k (t = dk f dt k (t s k F (s s k f(0 s k 2 f ( (0... f (k (0 Table 2: Iportant Laplace transfors pairs Transfer function fro block diagra The transfer function can be obtained fro a block diagra using block diagra algebra. Block diagra algebra provides a systeatic way to relate output signals to the intput signals. Starting fro the output signal, one works backward in the block diagra, until only the output and input of interest reain (see exaple below. Transfer function fro equations of otion The transfer function can also be obtained directly fro the equations of otion (3 of a linear tie-invariant syste by applying the Laplace transfor, with zero initial conditions. Table 2 gives the ost iportant Laplace transfor pairs. Transfer function fro state space representation Fro the linear state space representation (7 the transfer function fro the input u to the output y can be directly coputed using the relation H yu (s = C (si A B + D. (9 5

7 2.5 Linearization 2 MODELING Transfer function fro block diagra In the Laplace doain s the block diagra in Figure 2. can be redrawn as corresponds to integration, thereby Using block diagra algebra, the transfer function H yu (s fro the input u to the output y follows as Y (s = s 2 E(s = (U(s bv (s ky (s s2 = (U(s bsy (s ky (s s2 ( s 2 + bs + k Y (s = U(s H yu (s = Y (s U(s = s 2 + bs + k Transfer function fro equations of otion Take input u = F and output y = q and apply the Laplace transfor to (5 with zero initial conditions (i.e., q(0 = 0, q(0 = 0 to obtain s 2 Y (s + ky (s = sby (s + U(s. Cobining all Y (s ters and dividing by U(s gives the transfer function H uy (s as H uy (s = Y (s U(s = s 2 + bs + k. Transfer function fro state space representation The last way to obtain the transfer function H uy (s is obtained by applying (9 to (8 resulting in 2.5 Linearization H uy (s = ( 0 ( s k s + b = s 2 + b s + k = s 2 + b s + k ( 0 ( ( s + b 0 k s = s 2 + bs + k ( 0 A nonlinear syste ay be approxiated by a linear linear one using a linearization procedure. One should keep in ind that a linearization is only locally valid and essential nonlinear phenoena (e.g. finite escape ties, liit cycles are lost in the linear odel. To linearize the state space representation (6 around the point x, u let A = f (x, u x, B = f (x, u x,u u, C = g (x, u x,u x, D = g (x, u x,u u. x,u (0 6

8 2.6 Analogies different doains 2 MODELING Define x = x x, ū = u u, then the linear approxiation of (6 can be rewritten as a linear state space representation (7 as x = A x + Bū y = C x + Dū. Assue that the linear daper is replaced by a nonlinear one with corresponding daping force F d = b q3 2. The corresponding state space representation is given by ( ẋ ẋ 2 = ( x 2 k x b x3 2 + u. Furtherore, suppose the desired output is given by y = x ( x 2 2. Linearizing the nonlinear syste around the equilibriu point (x x 2 = (0, 0 for u = 0 gives A = ( 0 k 3b x2 2 (0,0,0 = ( 0 k 0 ( ( 0 0 B = = (0,0,0 C = ( ( x 2 2 2x ( x 2 (0,0,0 = ( 0 D = 0 (0,0,0 = 0 So the linearized dynaics around the equilibriu point (x x 2 = (0, 0 for input u = 0 are given by ( x x Analogies different doains = ( 0 k 0 y = ( 0 ( x x 2. ( x x 2 + ( 0 The odeling in [3] is based on the idea that there exist ore general relationships, that hold in different doains. Table 3 shows the variables for the different odeling doains for the echanical, electrical, hydraulical, and therodynaical doain. ū Effort Flow Generalized displaceent Generalized oentu Translation force F [N] velocity ẋ = v [/s] displaceent x oentu p [Ns] Rotation torque τ [N] angular velocity ω = θ [rad/s] anglular displaceent θ rotational oentu h [Ns] Electric voltage u [V ] current i = q [A] charge q flux φ [V s] Hydraulic pressure p [N/ 2 ] volue flow Q [ 3 /s] volue V [ 3 ] oentu of flow tube Γ [Ns/ 2 ] Therodynaic teperature T [K] entropy flow f T [W/K] entropy S [J/K] - Table 3: Doains and variables (based on Table. fro [3]. Furtherore, Table 4 gives the corresponding coponents and their relationships for linear systes in the echanical and electrical doain. Using these relationships it is possible to obtain the equivalent 7

9 2.6 Analogies different doains 2 MODELING odel of one odel into another doain. An equivalent odel is defined in a different doain, but has the sae dynaical behavior. Doain Coponent / relationship Translation Rotation Electric ass F = d2 x dt 2 inertia J τ J = J d2 θ dt 2 inductor L u L = L d2 q dt 2 spring k F k = kx rotational spring k τ k = kθ capacitor C u C = C q daper b F b = b dx dt rotational daper b τ b = b dθ dt resistor R u R = R dq dt Table 4: Doains, coponents, and relationships for linear systes For the ass spring daper syste depicted in Figure 2 the equivalent coponents in the electrical doain are the inductor (ass, capacitor (spring and resistor (daper. To obtain the sae dynaical behavior for the equivalent circuit take L =, C = k, R = b. The input force F corresponds to a voltage source u in the electrical doain. Since the left end of the spring and daper are fixed, it follows that the velocity dq dt is the sae for all three coponents. Fro Table 3 it follows that in the electrical circuit the current i should be the sae for all coponents, which iplies that the inductor, capacitor and resistor are interconnected in series. Hence, the equivalent electrical circuit of the ass spring daper syste in Figure 2 is given by 8

10 3 TIME DOMAIN ANALYSIS AND DESIGN 3 Tie doain analysis and design 3. Qualitative analysis Phase portraits provide a convenient way to study the behavior of second order systes (i.e., x R 2. The phase portrait plots for a set of intitial conditions the solution of the differential equation (6. Following the arrows in the phase plot gives the trajectories of the syste. The equilibriu points of a dynaical syste are the stationary conditions for the dynaics. The equilibriu points for a dynaical syste in the state space representation are found by setting ẋ = 0 in (6 and solving for x. The equilibriu points for u = 0 for (8 are found by setting ẋ = 0, u = 0. Fro the first equation it follows that x 2 = 0, substituting into the second lines gives x = 0. So the only equilibriu point for (8 for u = 0 is given by x = (0, 0. The corresponding phase plot is shown below. Looking at the phase plot it is easily seen that fro any initial condition, the solution converges to the equilibriu point x = (0, 0 and hence the equilibriu point is assyptotically stable (see Section Stability The stability of an equilibriu point tells whether solutions nearby the equilibriu point reain close (stable, get closer (asyptotic stability, or ove away (unstable fro the equilibriu point. The stability of linear systes of the for (7 is copletely characterized by the eigenvalues of the syste atrix A. The eigenvalues of atrix A are found by solving the characteristic equation λ(s, which is given by λ(s = det (si A = s n + a s n a n s + a n = (s + λ (s + λ 2... (s + λ n = 0. ( Not only the stability, but also whether or not oscillations will arise in the step response of the linear syste can be deteined fro the eigenvalues as follows: Eigenvalues with strictly negative real part The syste (7 is asyptotically stable. Eigenvalues with zero real part The syste (7 is stable. Eigenvalues with strictly positive real part The syste (7 is unstable. Eigenvalues with nonzero coplex part The step response of syste (7 has oscillations. 9

11 3.3 Input/output response 3 TIME DOMAIN ANALYSIS AND DESIGN For the ass spring daper syste, the characteristic equation is given by. λ(s = s 2 + b s + k = s s +.2 = 0 (2 Solving λ(s = 0 using quadratic forula (Dutch: ABC-forule the eigenvalues are given by λ,2 = 0.2 ±.077i. Hence the ass spring daper syste is assyptotically stable and the the step response will have oscillations (see exaple in Section Input/output response The tie response y(t to a input u(t for a linear syste (7 is given by y(t = Ce At x(0 + ˆ t 0 Ce A(t τ Bu(τdτ + Du(t, where x(0 denotes the initial condition and e At denotes the atrix exponential. The atrix exponential is defined as e At = k! Ak t k = I + At + 2 A2 t A3 t Step response k=0 The step response gives the tie behavior of a syste when it s input is changed fro zero to one in a very short tie. There are four specifications which characterize the step response: Steady state value y ss The output value of the syste for t (y ss =.0 in Figure 4. Rise tie t r (s The tie taken by the output to change fro 0% to 90% (T r in Figure 4. Overshoot M p (% The axiu percentage that the output exceeds the steady state value y ss. Settling tie t s (s The tie taken by the output to reach and stay within a certain range of the steady state value y ss (range indicated by δ in Figure 4. Figure 4: Step response characteristics 0

12 3.5 State feedback 3 TIME DOMAIN ANALYSIS AND DESIGN For the ass spring daper syste the step response is shown below. The four step response specifications for this exaple syste are given by y ss = 0.47, t r =.09s, M p = 55.6%, t s = 8.3s. 3.5 State feedback State feedback provides a ethod to assign the eigenvalues of the closed-loop syste to desired positions. A necessary condition for eigenvalue assignent is reachability. Reachability addresses whether it is possible to reach all points in the state space using the inputs available. A linear syste of the for (7 is reachable if and only if the reachability atrix W r is invertible, where the reachability atrix is defined as W r = ( B AB... A n B. The reachability atrix W r is invertible if and only if its deterinant is nonzero det W r 0 (i.e., W r is full rank. Reachability atrix W r for (8 given by ( 0 W r = b 2 Since det W r = = W r is invertible, and hence the syste (8 is reachable. When the syste (7 is reachable, it can be rewritten in the so-called reachable canonical for. In this for, the state equation has a special structure, which akes the analysis the syste uch ore easy. To obtain the reachable canonical fro, define the atrix W r as a a 2... a n 0 a a n 2 W r = a Then define a transforation atrix T = W r W r and define the new state z of the syste as z = T x. Then, the linear syste (7 can be rewritten in reachable canonical for as ż = Ãz + Bu (3 y = Cz + Du, (4

13 3.5 State feedback 3 TIME DOMAIN ANALYSIS AND DESIGN where à = T AT, B = T B, C = CT, D = D. Syste atrix à and input atrix B have a special structure, given by a a 2 a n à = , B = where a,..., a n are the coefficients of the characteristic equation (. Note that even though the analysis ight be easier, the physical interpretation of the state variables ight be lost, as the following exaple shows. For (8 the inverse of the reachability atrix is given by Matrix W r follows as W r = ( 0 b 2 W r = = 2 ( b ( b 0 and the transforation atrix T is given by ( b T = 0 = ( b ( b 0 = =. 0 0 ( 0 0, ( b 0. (5, (6 The reachable canonical for, with new state z = q, z 2 = q follows fro (4 and is given by. ( b à = ( k, 0 B =, 0 C = ( 0, D = 0 The goal of state feedback is to design a control input u = Kx + k r r to assign the closed-loop eigenvalues to a desired position. Let π... π n denote the desired closed-loop eigenvalues, such that the closed-loop characteristic equation is given by p(s = (s + π (s + π 2... (s + π n = s n + p s n p n s + p n. The feedback gain atrix K is given by K = ( p a p 2 a 2... p n a n Wr W r, (7 where a,..., a n denote the coefficients of the open loop characteristic equation. The reference gain k r is given by ( k r = / C (A BK B. (8 The reference gain k r ensures a zero steady state error to the new reference input r. Substituting u = Kx + k r r into (7 gives closed-loop dynaics. ẋ = Ax BKx + Bk r r = (A BK x + Bk r r y = Cx DKx + Dk r r = (C DK x + Dk r r, where the eigenvalues of the closed-loop syste atrix (A BK are now in the desired position. 2

14 3.5 State feedback 3 TIME DOMAIN ANALYSIS AND DESIGN Suppose the eigenvalues should be assigned to, 0.6, such that the closed-loop characteristic equation is given by p(s = (s + (s = s 2 +.6s Substituting the inverse of the reachability atrix Wr obtain K = ( p a p 2 a 2 ( b 0 ( b 0 The reference gain follows fro (8 and is given by k r = ( (( 0 0 k b ( 0 fro (5 and atrix W r fro (6 into (7 we = ( (p 2 a 2 (p a = ( (9 ( (p2 a 2 (p a ( 0 =.2. (20 To check that the closed-loop eigenvalues are now in the desired position, calculate the characteristic equation using the closed-loop syste atrix (A BK as ( s λ cl (s = det (si (A BK = det = s 2 +.6s = (s + (s p 2 s + p The corresponding step response is shown below. As expected, the steady state error is zero and there are no oscillations. 3

15 4 FREQUENCY DOMAIN ANALYSIS AND DESIGN 4 Frequency doain analysis and design 4. Introduction Figure 5 shows the setup of negative unity feedback. The transfer function of the syste is denoted by P (s, while C(s denotes the transfer function of the controller. Figure 5: Negative unity feedback setup Poles and zeros The poles of a transfer function are the roots of the denoinator, while the zeros of the transfer function are the roots of the nuerator. I.e., for a transfer of the for G(s = (s + n (s + n 2... (s + n (s + d (s + d 2... (s + d n, the n poles are given by d,..., d n, while the zeros are given by n,..., n. The denoinator of a transfer function equals the characteristic equation of the corresponding syste represented by the transfer function. Hence the poles of a transfer function are equal to the eigenvalues of the corresponding state space representation atrix A. Fro the ass spring daper syste s transfer function it is easily seen that it has no zeros, since the nuerator does not depend on s. The poles of the transfer function are found by solving the characteristic equation of the open loop syste which gives λ,2 = b ± b 2 2 λ(s = s 2 + b s + k = 0, 4 k = 0.2 ±.077i. Final Value Theore The Final Value Theore provides a relation between steady state signals in the tie doain and signals in the Laplace doain for s 0. Therefore it can be used to deterine the steady state value of the ouput y(t or error e(t for a given reference signal r(t or disturbance signal d(t using the corresponding signals Y (s, E(s, R(s, D(s in the Laplace doain. The Final Value Theore states that li f(t = li sf (s. (2 t s 0 Consider a negative feedback loop of Figure 5 where P (s equals the transfer function of the ass spring daper syste (0 and the controller C(s = K p (i.e., a proportional controller. The closed-loop transfer function for this syste is given by H yr (s = Y (s R(s = C(sP (s + C(sP (s = K p s 2. + bs + k + K p The steady state output y ss for a step input 0 s Theore (2 and is given by and control gain K p = 0.6 follows fro the Final Value y ss = li y(t = li sy (s = li sh yr (sr(s = 0K p = 2. t s 0 s 0 k + K p Hence, this proportional controller gives a steady state error of % = 80% (i.e., the output signal y(t is 80% lower that the reference input r(t. 4

16 4.2 Stability 4 FREQUENCY DOMAIN ANALYSIS AND DESIGN Loop transfer function The loop transfer function L(s is a special transfer function that is used quite often during the analysis and design in the frequency doain (e.g. Bode plots, Nyquist criterion. For a unity feedback syste like the one shown in Figure 5 it is siply the open-loop transfer function L(s = C(sP (s. With a proportional controller C(s = K p the loop transfer function L(s for the ass spring daper syste with unity feedback is siply given by 4.2 Stability K p / L(s = C(sP (s = s 2 + b s + k. The poles of a transfer function are equal to the eigenvalues of the corresponding state space representation atrix A. Therefore, the stability of a transfer function is copletely characterized by its poles in the sae way as the eigenvalues characterize the stability of the state space representation: Poles with strictly negative real part The syste is asyptotically stable. Poles with zero real part The syste is stable. Poles with strictly positive real part The syste is unstable. Poles with nonzero coplex part The step response of the syste has oscillations. As shown in the previous section the poles of (0 are given by λ,2 = b ± b 2 4 k 2 = 0.2 ±.077i. Hence, the syste is assyptotically stable and the step response will have oscillations. 4.3 Standard second order systes Standard second order systes have a transfer function of the for ω 2 0 H yu (s = s 2 + 2ζω 0 + ω0 2, (22 where ω 0 denotes the natural frequency and ζ denotes the daping ratio. Table 5 gives the specifications of the step response (steady state value y ss, rise tie t r, overshoot M p, settling tie t s for a standard second order syste in ters of the natural frequency ω 0 and daping ration ζ. Property Value ζ = 0.5 ζ = 2 ζ = Steady-state value k k k k Rise tie t r = ω 0 e arccos ζ/ tan arccos ζ.8/ω 0 2.2/ω 0 2.7/ω 0 Overshoot M p = e πζ/ ζ 2 6% 4% 0% Settling tie (2% t s 4/ζω 0 8/ω 0 5.9/ω 0 5.8/ω 0 Table 5: Properties of the step response for a standard second-order syste with 0 < ζ <.[2] The poles of (22 are found by solving the characteristic equation λ(s = s 2 + 2ζω 0 + ω 2 0 = 0 and are given by λ,2 = ζω 0 ± ω 0 ζ2. The daping ratio has the following influence on the syte: Daping ratio 0 < ζ < The syste (22 is underdaped, both poles have a coplex part. 5

17 4.4 Bode plots 4 FREQUENCY DOMAIN ANALYSIS AND DESIGN Daping ratio ζ = The syste (22 is critically daped, both poles are real and at the sae location λ,2 = ω 0. Daping ratio ζ > The syste (22 is overdaped, both poles are real. Figure 6 shows the influence of the daping ratio on the step response. Figure 6: Step respnse for an under-, overdaped, and critically daped standard second order syste. The transfer function of the ass spring daper syste can be rewritten as a standard second order syste as H yu (s = s 2 + bs + k = αω0 2 s 2 + 2ζω 0 + ω0 2, b 2ω 0 = b where ω n = k/ =.095rad/s and ζ = 2 = Constant α = k k has no effect on the characteristic equation, but will have a (sall effect on the approxiations in Table 5, since these approxiations are based on α =. Furtherore, since ζ < we know that the syste is underdaped, which can also been seen fro the step response in Section Bode plots The frequency response of a linear syste is calculated fro the transfer function G(s by setting s = iω. I G(iω The frequency response is characterized by its agnitude G(iω and phase G(iω = arctan Re G(iω in degrees. Usually the agnitude is easured in db calculated as 20 log G(iω. For a general transfer function of the for G(s = d (sd 2 (s n (sn 2 (s, the agnitude (in db is given by 20 log G(iω = 20 log d (iω + 20 log d 2 (iω 20 log n (iω 20 log n 2 (iω. (23 while the phase (degrees is given by G(iω = d (iω + d 2 (iω n (iω n 2 (iω. (24 Fro (23 and (24 it can be seen that both the agnitude and phase of G(iω follow by adding and substracting the corresponding ters in the nuerator and denoinator. Plotting the agnitude and phase of the loop transfer function using a log/log scale and a log/linear scale gives Bode plot. The Bode plot not only characterizes the frequency response of the loop transfer function, it also provides a useful tool in the design of controllers. To see why, consider a plant with transfer function P (s = N p (s/d p (s and a controller C(s = N c (s/d c (s. The corresponding loop transfer function is given by L(s = C(sP (s = N c(sn p (s D c (sd p (s. And hence the agnitude of the loop transfer function is obtained by adding the agnitude of the plant (20 log N p (iω 20 log D p (iω and the agnitude of the controller (20 log N c (iω 20 log D c (iω. Likewise the phase of the loop transfer function is obtained by adding the phase of the plant ( N p (iω 6

18 4.4 Bode plots 4 FREQUENCY DOMAIN ANALYSIS AND DESIGN D p (iω and the phase of the controller ( N c (iω D c (iω. Therefore, if we have a Bode plot of the open loop syste and we need a certain agnitude or phase, we should siply look for a controller structure that adds the agnitude/phase needed. Table 6 gives the effect of several coon transfer function coponents. Ter Transfer function Effect on agnitude plot Effect on phase plot Constant gain k Shift by 20 log k db none Pole at origin s Enters at slope 20dB/dec Enters at 90 deg Zero at origin s Enters at slope +20db/dec Enters at +90 deg Pole at s = a s+a Breakpoint at frequency ω = a rad/s, where slope changes fro 0dB/dec to 20dB/dec Zero at s = b s + b Breakpoint at frequency ω = b rad/s, where slope changes fro 0dB/dec to +20dB/dec Second order syste ω 2 0 s 2 +2ζω 0s+ω 2 0 Peak at frequency ω = ω 0, and then slope of 40dB/dec beyond the peak. Peak height depends on ζ Table 6: Effect of coon ters of the transfer function Breakpoint at frequency ω = a rad/s, where phase changes fro 0deg to 90deg Breakpoint at frequency ω = b rad/s, where phase changes fro 0deg to +90deg Phase decrease fro 0deg to 80deg at frequency ω = ω 0. Rate of change depends on ζ The Bode plot presents the agnitude (in db and phase (in degrees of the loop transfer function L(iω. Let gain cross-over frequency ω cg denote the frequency where 20log L(iω = 0dB (i.e., L(iω = and let phase cross-over frequency ω pc denote the sallest frequency where L(iω = 80deg. Fro the Bode plot of the open loop syste the gain argin g of the closed loop syste can now be obtained as the the separation between 20 log L(iω and the 0dB line at the frequency ω = ω pc. The phase argin ϕ follows fro the Bode plot as the separation between L(iω and the 80deg at the frequency ω = ω gc. It ight happen that the phase plot crosses the 80deg line ore than once, such that you have ultiple phase cross-over frequencies. In that case, the gain argin g is defined as the sallest separation between 20 log L(iω and the 0dB line for one of the phase cross-over frequencies. In case you have ultiple gain cross-over frequencies (see exaple below, the phase argin ϕ is defined as the sallest separation between L(iω and the 80deg line for one of the gain cross-over frequencies. 7

19 4.5 Nyquist criterion 4 FREQUENCY DOMAIN ANALYSIS AND DESIGN Consider the Bode plot of the ass spring daper syste, given below. Since the phase of the loop transfer function never crosses the 80deg line, we cannot deterine the phase cross-over frequency ω pc such that the gain argin can not be deterined fro the Bode plot (to be ore precise, g =. The phase argin ϕ is the sallest separation between L(iω and the 80deg line at one of the two gain cross-over frequencies. For the Bode plot below, the phase argin is therefore equal to ϕ = 69.7deg. 4.5 Nyquist criterion The Nyquist criterion is a graphical ethod to asses the stability of a syste. The criterion looks at the Nyquist plot of the open-loop transfer function to deterine the stability of the closed-loop syste with negative unity feedback (see Figure 5. The Nyquist criterion states that Z = N + P, where Z is the nuber of unstable closed-loop poles, P is the nuber of unstable open-loop poles and N is the nuber of clockwise encircleents of the point (, 0 by the Nyquist plot. Figure 7 shows how to obtain the gain argin g and phase argin ϕ fro the Nyquist plot. Figure 7: Obtaining the gain argin g and phase argin ϕ fro a Nyquist plot. 8

20 4.6 PID control 4 FREQUENCY DOMAIN ANALYSIS AND DESIGN The Nyquist plot for the open-loop transfer function (0 is shown below. Since the poles of the open-loop transfer function (0 have a strictly negative real part, it follows that P = 0. Fro the Nyquist plot it follows that there are no encircleents of the point (, 0, such that N = 0. It follows fro the Nyquist criterion that Z = N + P = 0 and hence the closed-loop syste (unity feedback is stable. The phase argin ϕ is given by ϕ = 69.7deg, which equal to the phase argin deterined fro the Bode plot before. 4.6 PID control A PID controller is a controller with the following transfer function C pid (s = K p + K i s + K ds = K i + K p s + K d s 2. (25 s The controller consist of a proportional ter K p, an integral ter K i and a differential ter K d. Table (7 shows the effect on the step response for increasing controller gains of the PID controller. Gain Rise tie Overshoot Settling tie Steady-state error Stability K p Decrease Increase Sall change Decrease Degrade K i Decrease Increase Increase Eliinate Degrade K d Minor change Decrease Decrease No effect Iprove (for K d sall Table 7: Effect of increasing the PID controller gains ( controller PID controller (25 has a clear physical interpretation and is therefore useful in the tuning of the step response (e.g. decreasing overshoot, eliinating steady-state error. For tuning using Bode plots (25 is harder to interpret and often an alternative for is used, naely C pid (s = k s (T ds + (s + Ti, (26 where T i > T d. In this for the effect of the controller paraeters k, T i, T d on the Bode plot shown in Figure 8 can be seen directly: Due to the integrator ter s the Bode plot enters at 20dB/dec and 90deg. At the frequency ω = T i rad/s the slope of the agnitude plot changes to 0dB/dec, while the phase plot changes to 0deg. At the frequency ω = T d rad/s the slope of the agnitude plot changes to +20dB/dec while the phase plot changes to +90deg. The proportional ter k shifts the agnitude plot by 20 log k db and has no effect on the phase plot. 9

21 4.7 Lead and lag filters 4 FREQUENCY DOMAIN ANALYSIS AND DESIGN Figure 8: Bode plot of a PID controller with k =, T d = 0.0, T i = 2 PID control is used to: Add an integral action to eliinate steady state errors Add phase to increase the phase argin ϕ (for ω > T d rad/s. Drawbacks of PID control are: Reduction of the phase for low frequencies ( for ω < T i rad/s Aplification of high frequency noise ( for ω > T d rad/s Several design strategies have been proposed to tune a PID controller (e.g. Ziegler-Nichols. One stepby-step procedure to get an initial estiate of the PID controller gains is as follows:. Choose a frequency range [ω in, ω ax ] around the gain cross-over frequency ω gc : ω in < ω gc, ω ax > ω gc. 2. Choose the integrator zero at ω = T i well below the lower bound of the range of interest: T i ω in. 3. Choose the differentiator zero at ω = T d such that sufficient phase is added around the gain crossover frequency ω gc : T d ω in provides a good starting point. 4. Calculate the proportional gain k such that the gain cross-over frequency ω gc is at the desired frequency: ( 20 log k = 20 log P (iω c 20 log }{{} (it d ω c + iω c +. iω c T i agnitude plant in db }{{} agnitude PID controller fro step If needed, calculate K p, K i, K d fro k, T i, T d using (25 and (26 to tune the step response fro step Lead and lag filters Lead filter The transfer function of a lead filter is given by C lead (s = K T s + αt s +, (27 20

22 4.7 Lead and lag filters 4 FREQUENCY DOMAIN ANALYSIS AND DESIGN where α <. The lead filter is used to add phase, which coes at a cost of aplifiying high frequency noise. The Bode plot for a lead filter is shown in Figure 9. The break points in the Bode plot are at frequencies ω = T and ω = αt. The axiu phase added by the lead filter is φ ax = 90 2 arctan α which is added at the frequency ω ax = T α. The gain added at ω ax is given by 20 log C(iω = 20 log K + 0 log α. Furtherore, if the axiu phase which needs to be added is known, the lead filter paraeter α is given by α = sin φ ax + sin φ ax. Figure 9: Bode plot of a lead filter with k =, T = 0, α = 0. Lag filter The transfer function for a lag filter is given by C lead (s = K T s + αt s +, (28 where α > 0. Contrary to the lead filter, the lag filter is used to increase the low frequency gain, at the cost of reduction of the phase. The Bode plot for a lag filter is shown in Figure 0. The break points of a lag filter are exactly the sae as for the lead filter, naely ω = αt and ω = T. To reduce the effect of the lag filter on the phase argin ϕ, take T ω c. Table 8 gives soe iportant characteristics of the lead and lag filter. 2

23 4.7 Lead and lag filters 4 FREQUENCY DOMAIN ANALYSIS AND DESIGN Method Purpose Lead filter Add phase near the cross-over frequency Iprove phase argin φ and transient response Lag filter Increase gain at low frequencies Decrease steady state errors, while aintaining φ and transient respons When steady state errors are specified Applications When fast transient response is desired Results Increases syste bandwidth Decreases syste bandwidth Advantages Yields desired response Reduces the steady state error Speeds dynaic response Disadvantages Susceptible to easureent noise (high frequencies Slows down the transient response Table 8: Characteristics of lead and lag filters [] Figure 0: Bode plot of a lag filter with k =, T = 0, α = 0. 22

24 REFERENCES REFERENCES References [] O.H. Bosgra, H. Kwakernaak, and G. Meinsa. Design Methods for Control Systes Notes for a course of the Dutch Institute of Systes and Control, Winter ter [2] K.J. Åströ and R.M. Murray. Feedback Systes: An Introduction for Scientists and Engineers. Princeton University Press, 20. [3] J.M.A. Scherpen. Lecture notes for the course Regeltechniek - TBKRT05E (200/20. August