Digital implementation of discrete-time controllers

Transcription

1 Schweizerische Gesellschaft für Automatik Association Suisse pour l Automatique Associazione Svizzera di Controllo Automatico Swiss Society for Automatic Control Advanced Control Digital implementation of discrete-time controllers u x x q ỹ Scope Keywords Prerequisites Contact Learn the issues influencing the implementation of a given controller. Learn how to circument or solve problems arising when implementing a controller. Causes and effects of quantization, propagation of quantization errors, scaling, well-conditioned realizations, case studies Discrete-time models of LTI systems Silvano Balemi, silvano.balemi@supsi.ch Version.0 Date June 3, 20

2 2 Silvano Balemi June 3, 20

3 Contents Introduction Realization of code for controllers Controller given as a transfer function Controller given as a state-space realization Sources of quantization errors Arithmetic operations Parameter quantization Converter quantization Propagation of quantization errors The stochastic error bound Other effects of quantization Static errors Limit cycles Remedies against quantization effects Use of lower order subcomputations Dither signals Discretization through δ operator Noise shaping Scaling of variables Conclusions A. Quantization errors A.2 Sinusoidal encoder interpolation A.3 Positioning of a mass A.4 Control of a force-feedback system B. First order plant B.2 Limit cycles (Franklin, Powell, Workman, pb 0.0) B.3 Poles sensitivity (Franklin, Powell, Workman, pb 0.8) June 3, 20 Silvano Balemi 3

4 Introduction Control design does not end with the determination of the mathematical controller satisfying the application specifications and with a simulation proving the validity of the design. The implementation of a controller may require additional design steps which are necessary to make sure that the controller still behaves satisfactorily when implemented in hardware. In particular, numerical issues may require particular implementation forms of controller transfer functions or additional measures. In the following the basic procedure for implementing a controller from its transfer function is presented. Then issues affecting the implementation of a controller are addressed. Different causes of quantization will be presented. Then the propagation of the error to the quantities of interest will be analyzed. Finally other effect like limit cycles and static errors will be presented together with some remedies. 2 Realization of code for controllers 2. Controller given as a transfer function Consider a controller transfer function from the error to the actuation, of the form C(z) = U(z) E(z) = b n z n +b n z n +b n 2 z n b z +b 0 z n +a n z n +a n 2 z n a z +a 0 Then the following steps can be performed (z n +a n z n a 0 ) U(z) = (b n z n b z +b 0 ) E(z) (+a n z a 0 z n ) U(z) = (b n b z n+ +b 0 z n ) E(z) U(z)+a n z U(z)+... +a 0 z n U(z) = b n E(z)+... +b z n+ E(z)+b 0 z n E(z) Now consider the z-transform U(z), which corresponds to the sequence {u k } = {u 0,u,u 2,...,u k,u k,u k+,...}. Remember also that the z-transform z U(z) corresponds to the delayed sequence {u k } = {0,u 0,u,u 2,...,u k 2,u k,u k,...}. Then the equation above for z-transforms corresponds to the following equation for sequencies: {u k }+a n {u k }+... +a 0 {u k n } = b n {e k }+... +b {e k n+ }+b 0 z n {e k n } In this expression the sum of sequences indicates the sum of the corresponding element of each sequence. Comparison of the k-th elements of each sequence yields u k +a n u k a 0 u k n = b n e k b e k n+ +b 0 z n e k n 4 Silvano Balemi June 3, 20

5 A final rearrangement leads finally to u k = a n u k... a 0 u k n +b n e k b e k n+ +b 0 z n e k n This means that n variable for the past values of the actuation u and up to n+ variable for the current and past values of the error e must be stored. Example Consider the controller C(z) = z +3 z 2 +z + The equation for the update of the actuation value u is u k = u k u k 2 +e k +3 e k 2 A corresponding pseudo-code for the controller implementation would look like as follows ek_=0; ek_2=0; uk_=0; uk_2=0; while TRUE { yk=read_yk(); ek=yrefk-yk; uk=-uk_-uk_2+ek_-3*ek_2; write(uk); uk_2=uk_; uk_=uk; ek_2=ek_; ek_=ek; } 2.2 Controller given as a state-space realization Suppose the controller design method delivers instead a controller given in a statespace realization { xk+ = Φ x k +Γ u k y k = C x k +D u k where x k is a vector of n state variables. Then the code implementing the controller must define n variables for the state vector. The n state equations and the output equations can be written in a straight-forward manner (do this as an exercise). 3 Sources of quantization errors The operations needed to implement a controller in a digital controller are forcibly affected by quantization errors. The main sources of quantization error are arithmetic operations, the non-ideal representation of parameters and converter quantization. June 3, 20 Silvano Balemi 5

6 3. Arithmetic operations The errors caused by arithmetic operations strongly depend on the representation of the numbers used in the operations. 3.. Fixed-point representation Quantization arises in multiplication with fixed-point numbers. In a fixed point representation, only a limited number of bits (typically 6 or 32 bits) is used for the representation of a number. In fixed-point data types, binary numbers are generally represented by an integer part (QI) and a fractional part (QF). Both contain a certain number of bits. This notation is called Q-format and is given in the form Q[QI] [QF] where Q designates the format notation, QI is the number of bits for the integer part and QF is the number of bits for the fractional part. The word length L of a Q-format number is given by L = QI +QF + (.) where the additional bit accounts for the sign bit of the number. For instance, a Q2.5 number indicates an 8-bit value with two integer bits and five fractional bits. Example 2 The Q0.3 format representations of the numbers and 0.5 are. Q0.3 and.00 Q0.3. Their multiplication in the Q0.3 format is. Q Q0.3 =.0 Q0.3 The result does not correspond to the expected value but the quantized value Thus, multiplications of numbers in fixed-point representation introduce quantization. On the contrary, no quantization occurs when summing two numbers(although underflow and overflow may occur) Floating-point representation Floating-point representations are based on two fixed-point numbers (a mantissa or fraction and an exponent). Typically, a 32 bit floating point number ( single precision float ) according to the IEEE 754 standard uses an 7+ bit number for the exponent and a 23+ bit number for the mantissa. A 64 bit floating point number ( double precision float ) according to the IEEE standard uses an 0+ bit number for the exponent and a 52+ bit number for the mantissa. Texas Instruments representation for signed fixed-point numbers 6 Silvano Balemi June 3, 20

7 As floating-point representations are based on fixed-point numbers, multiplication also causes quantizations of the mantissa and thus of the floating-point number. Moreoever, summation also causes quantization. This can be seen in the sequence M 2 E +M 2 2 E 2 = M 2 E + M 2 2 E E 2 2 E = (M +M 2) 2 E }{{} M 2 where M 2 is shifted to the right by E E 2 bits and thus potentially truncated. If errors in fixed-point representations are absolute, quantization errors in floatingpoint numbers are relative to the size of the number (quantization occurs only in the mantissa). The change from a fixed-point representation to a floating-point one (e.g. from a 6 bit fixed-point to a 32 bit floating point representation) reduces the impact of quantizations on computations, even though it does not eliminate it completely. Even worse, quantization problems become more tricky to identify. Consider for instance the operation a+b a in Matlab: the result is 0 if a is sufficiently large with respect to b. Quantization errors due to arithmetic operations can be minimized by a correct scaling of the variables and by increasing the number of bits if necessary. 3.2 Parameter quantization Quantization in parameters is also due to the non-ideal representation of numbers. Consider the fourth-order transfer function [] H(z) = z z z z z z z with two complex poles at ±j Quantization with 3 fractional bits leads to the quantized transfer function H(z) = which has two poles at.08! z 3 +.5z z +0.5 z z z z Sensitivity to parameter quantization The effect can be analyzed with the help of the sensitivity of the poles with respect to parameter changes. Consider the characteristic polynomial P(p j,α k ) = 0 wherethedependenceofthepolynomialonthespecificpolep j andontheparameter α k ismadeexplicit. Thecharacteristicpolynomialwithitsfirst-ordertermexpansion is P(p j +δp j,α k +δα k ) = 0 P(p j,α k ) + P }{{} z 0 δp j + P δα k = P δp z=pj α k z j + P δα k z=pj α k June 3, 20 Silvano Balemi 7

8 Then the following holds: δp j P/ α k P/ z δα k (.2) z=pj Example 3 For the characteristic polynomial n P(z, α k ) = (z p l ) l= of the form equation (.2) becomes P(z, α k ) = z n +α z n α n p n k j δp j = l j(p j p l ) δα k (.3) As for stable poles p j <, the expression (.3) is large for large k (see numerator of the fraction). Therefore, the largest sensitivity is obtained when k = n and the most critical parameter is α n. Moreover, the sensitivity becomes also large when two poles are close to each other (in which case the denominator becomes small). Therefore in a good controller design all closed-loop poles should be chosen to stay well apart. 3.3 Converter quantization A last source of quantization is given by the finite resolution of converters like ADconverters, DA-converters and PWM-modulators. They introduce a quantization which depends on the range and on the resolution in bits of the converter. Their effect is the same of that caused by the quantization of a variable. 4 Propagation of quantization errors Truncation(lower bound) and round-off(closer value) are two possible quantizations of a number. Both operations can be represented by the expression x = x q +ǫ where x q is the digital representation of x with error ǫ. However, the quantization in control systems should not be judged based on the error ǫ but on its effect on the quantities of interest. Consider the linear system Y(z) = H(z) U(z): the introduction of the quantization of a variable in the system perturbs the output as shown in Figure.. The perturbed output Ỹ(z) is Ỹ(z) = H(z) U(z) H q (z) E(z,x) 8 Silvano Balemi June 3, 20

9 u x x ỹ = u ǫ q ỹ x - x q + Figure.: LTI system with quantization of the internal variable x whereh q (z)isthetransferfunctionfromthequantizationerrortotheoutput. Then the error at the output becomes Y(z) = Y(z) Ỹ(z) = Hq (z) E(z,x) and in the time domain n δy n = h q k ǫ n k(x). k=0 4. The stochastic error bound There are several methods to estimate the value of the error at the output []. For most practical applications the stochastic error bound gives a useful estimate. Under the assumption that the quantization step q is small with respect to the variable range, then one can assume that the quantization error ǫ is uniformly distributed in the interval q 2 ǫ q 2 The variable ǫ has zero mean and variance σ 2 = q 2 /2. Because the uniform distribution is not easily tractable with analytical tools, the signal ǫ is approximated with a white noise w with the same mean value and variance. With this approximation it is possible to compute the variance of the error δy n (( k ) ( k )) E(δyn(k)) 2 = E h q (m) w(k m) h q (n) w(k n) m=0 n=0 ( k ) k = E h q (m) w(k m) h q (n) w(k n) m=0 n=0 k k = h q (m) h q (n) E (w(k n) w(k m)) m=0 n=0 But Then E(w(k m) w(k n)) = E(w 2 ) δ(m n) k E(δyn(k)) 2 = E(w 2 ) (h q (m)) 2 m=0 June 3, 20 Silvano Balemi 9

10 and for k approaching infinity σ 2 δy n = q2 2 (h q (m)) 2 (.4) m=0 In the presence of multiple independent quantization sources, the variance of the output δy is expressed by σ 2 = i σ 2 i where σ i denotes the variance of the output introduced by the quantization source i. Example 4 For the error system H q (z) = z α equation (.4) becomes {h k } = {,α,α 2,α 3,...} with impulse response sequence and thus σ 2 δy n = q2 2 ( +α 2 +(α 2 ) 2 +(α 3 ) 2 +(α 4 ) ) = q2 2 ( +α 2 +(α 2 ) 2 +(α 2 ) 3 +(α 2 ) ) = q2 2 σ δyn = q 2( α 2 ) α 2 is obtained. The computation of the noise variance at the output is very simple in Matlab: itisenoughtocomputetheimpulseresponseandtocomputethesumofthe square of its values with the command sigmady=q*norm(impulse(h))/sqrt(2). 5 Other effects of quantization Quantization introduces noise into the closed-loop system. Moreover, it can affect stability of the system because of the unwanted modification of the position of the poles. Other effects may occur. 5. Static errors In practical implementations static errors often occur, e.g. because of not perfectly known process gains. The standard solution is the introduction of an integral term. It is then a big surprise when the static error even increases instead of disappearing. This is due to the perturbation of the position of the poles because of the quantization of the controller parameters. An effective way to solve the problem is to separate the integral part and to place it separately in front of the rest of the controller (in a sort of serial decomposition as seen above). 0 Silvano Balemi June 3, 20

11 5.2 Limit cycles Quantization can cause limit cycles. Consider the contraction x k+ = α x k with α < presented in []. The value of x k converges exponentially to 0. Consider instead the same function with quantization as in x k+ = Q(α x k ) with α < where the quantization is given by a round-off of step q. Then lim k x k 0 as shown in Figure.2. x slope /α x x 2 x 3,x 4,..... q k q. k q q/2 α x 2. α k q α x Figure.2: Roundoff and limit cycle. Starting with x we obtain in sequence x 2 and x 3 which is a fixed point of the quantized contraction. α x Alimitcycleoccurswhenwegetstuckonthesamequantizationlevel. Figure.2 shows that this may happen when α x = α k q k q q/2 k q q 2 α (.5) Consider now the second order system shown in Figure.3. A limit cycle occurs when a pole reaches the unit circle. In fact, this happens when condition (.5) with α = a 2 is satisfied. A quantization step q = 0. should cause an oscillation of amplitude q = /0. = 0.5. The simulation shown in the top graph of 2 α Figure.7 confirms this value for the amplitude of the oscillation. June 3, 20 Silvano Balemi

12 z z a a2 Figure.3: Second order system with quantized coefficients. a =.78, a 2 = Remedies against quantization effects 6. Use of lower order subcomputations In order to mimize the effect of parameter quantization one should preferably use computations which are numerically robust. This includes the correct choice of the sampling frequency, an appropriate choice of the closed-loop poles and appropriate sequences of the operations. Veryshortsamplingtimest s causealldiscrete-timepolestobeclosetothepoint in the z plane (remember that for a given continuous-time pole s the discrete-time poleisz = e s ts ), whichleadstonumericalproblems(seeabove). Thushighsampling frequencies should be avoided. The poles should be designed to be well distributed in the z plane and not too close to each other in order to guarantee a low sensitivity to parameter variations (see equation (.3)). ALso, the implementation of a given controller is numerically more sound when its implementation is separated into lower order parts as shown in Figure.4(parallel decomposition) or in Figure.5 (serial decomposition). st or 2 nd order st or 2 nd order. st or 2 nd order Figure.4: Decomposition of a transfer function into first- or second-order parallel terms (second-order terms for complex pole pairs). 2 Silvano Balemi June 3, 20

13 st or 2 nd order st or 2 nd st or 2 nd order order Figure.5: Decomposition of a transfer function into first- or second-order serial terms (second-order terms for complex pole pairs) The first and second-order blocks can be of the form H (z) = +β z α z and H 2 (z) = +β z +β 2 z 2 α z α 2 z 2 as shown in Figure.6. Other realizations are also possible [2]. u(k) y(k) u(k) y(k) alpha z beta alpha z beta z alpha2 beta2 Figure.6: First and second order transfer function implementations. Note that the parallel decomposition corresponds to the so-called modal decomposition(use the Matlab command[am,bm,cm,dm]=canon(a,b,c,d, modal )). Be aware however, that a general modal decomposition cannot consist of first- or secondorder blocks alone in presence of multiple eigenvalues (the matrix Am then contains so-called Jordan blocks). Additional measures can be introduced. For instance, in order to avoid potential static errors due to numerical inaccuracies caused by quantizations, an integral part may be implemented explicitely in a separate block. Also, the variable resolution of integral terms can be increased (larger variable format). 6.2 Dither signals Consider again the situation shown in Figure.2 with the signal stuck at x 3. A high frequency signal (dither) of amplitude 3 q (i.e. 3 times the quantization step) applied before the quantizer causes the value to be quantized to change slightly: the quantized value will eventually descend to the lower step. Thus the value of the output drifts towards zero. Several signals can be used in these cases: a simple square signal with period equal to twice the sampling time (i.e. the fastest possible change) is usually a good selection. A dither applied before the quantization of a 2 y in the second order system of Figure.3 effectively removes the limit cycle as shown in the bottom graph of Figure.7. June 3, 20 Silvano Balemi 3

14 6 q=0 and q=0. showing Limit cycle (no dither) q=0 and q=0. showing reduced limit cycle from dither=4q Figure.7: Closed-loop system with limit cycle because of round-off and addition of dither, picture from [] 6.3 Discretization through δ operator The δ T operator (see [3], [4] and [5]) is very helpful in improving the numerical behavior (stability and precision) of controllers. Similarly to the known shift operator q defined by qx k := x k+ the δ operator is defined as follows δ T x k := x k+ x k T It can be easily seen that the application of the operator converges to the continuoustime derivative for T 0, i.e. lim δ T x k = d T 0 dt x(t) t=k T The frequency domain corresponding to the δ T operator, is obtained with the help of the δ transform. The corresponding complex variable γ can be expressed in function of the z variable as follows: γ = z T (.6) This corresponds to a change in coordinates where the point is moved to the origin in 0. Changes which appeared to be small relatively to a pole close or at 4 Silvano Balemi June 3, 20

15 when viewed from the origin 0 become large. This and similar transformations reduce quantization effects (e.g. problems with the integral terms) in particular for high sampling frequencies (i.e. for T such that s T << ). The three planes (s-plane, z-plane and γ-plane) are shown on figure.8. Note that the γ-plane converges to the s-plane and that the poles in γ converge to the continuous-time poles for T 0. s z γ 2 T T Figure.8: Stability region in the complex planes for the Laplace, Z and δ T tranforms Implementation with state-space realization Equivalently to the standard state-space representation { qxk = Φ x k +Γ u k y k = C x k +D u k the state-space representation with the δ T operator is { δt x k = Φ δt x k +Γ δt u k y k = C x k +D u k where Φ δt = Φ I T and Γ δt = Γ T (note that lim T 0Φ δt = A). The next state expressed in function of the previous state with the help of the state-space representation is given by { xk+ = x k +T Φ δt x k +T Γ δt u k = x k +(Φ I) x k +Γ u k y k = C x k +D u k = C x k +D u k Note that this corresponds to a change of a computation with the new matrix Φ I having eigenvalues at λ for each eigenvalue λ of Φ Implementation with transfer function With this method z is replaced by T γ + and thus a transfer function in γ is obtained. As the implementation (like for a z transform) the filter is implemented June 3, 20 Silvano Balemi 5

16 with blocks exploiting the inverse complex variable γ, which in turn is realized with the expression γ z = T (.7) z derived from.6. The corresponding block scheme in shown in figure.9. γ T z Figure.9: Implementation of the basic building block for the δ T transformation. Example 5 Given is the transfer function C(z) = z z a Then using equation (.6) the new transfer function C(γ) = (γ T +) (γ T +) a = γ + T γ + a T is obtained. The filter C(γ) can be implemented with the new γ block as shown in Figure.0. Its pole is at γ = a T. a z a T γ T a T z Figure.0: Simple transfer function in z (left) and in γ (middle and right with the substitution γ = T z z ). T T Note that the δ operator can be applied also for any T. In fact, in the state-space realization the value of T is not present; In the transfer function implementation it only corresponds to a scaling of the variables without much influence on the numerical accuracy. 6 Silvano Balemi June 3, 20

17 6.4 Noise shaping Another method for addressing quantization problems is by collecting the quantization error and by feeding it, after filtering, back to the quantizer input. This is similar to a reduction of the quantization step (in particular with the shaping filter equal to ). Advantages are also the shift of the quantization noise spectrum to higher frequencies, with a reduction of the effect of the noise on the closed loop (usually having low-pass characteristic). Filter + - Figure.: Noise shaping with collection of the quantization error and feedback before the quantization 7 Scaling of variables Fixed-point numbers have a limited range. Thus algorithms implemented with fixedpoint numbers have to consider that all variables have to stay within the available range. On the other hand, the range must be exploited as much as possible in order to minimize the quantization errors. Thus a scaling of the variables has to be performed. Scaling assumes that all variables are bounded. If this is not the case, some saturations must be introduced, which do not harm the behavior of the system. Then the possible range of a variable must be increased or decreased in order to fit well into the available range. Many techniques exist [6]. In the simplest case of a linear algorithm, a node can be scaled by a factor K (i.e. its range amplified by a factor K) by multiplying all incoming values by K dividing the outgooing value by K See Figure.2 for an example. 8 Conclusions A careful implementation of digital controllers must consider quantization issues. Scaling is also essential. Much help can be provided by simulations. However, simulations should not be used to justify the own work but only to validate one s June 3, 20 Silvano Balemi 7

18 a a u u 2 e f c x b x 2 d u u 2 e c K f K x b/k x 2 K d Figure.2: Scaling of node x 2 by a factor K. choices. The author hopes that this document has provided some insight into the issue and provided some useful tools. 8 Silvano Balemi June 3, 20

19 Bibliography [] G. F. Franklin, J. D. Powell, and M. L. Workman. Digital Control of Dynamic Systems. Addison Wesley, 3rd edition, 997. [2] A.V. Oppenheim and R.W. Schafer. Discrete-time signal processing. Prentice- Hall, 2 edition, 998. [3] R. Agarwal and S. Burrus. New recursive digital filter structures having very low sensitivity and roundoff noise. IEEE Trans. Circuits Syst., CAS-22(2):92 927, December 975. [4] S. Nishimura, K. Hirano, and R. N. Pal. A new class of very low sensitivity and low roundoff noise recursive digital filter structures. IEEE Trans. Circuits Syst., CAS-28(2):52 58, December 98. [5] Richard H. Middleton and Graham C. Goodwin. Improved finite word length characteristics in digital control using delta operators. IEEE Trans. on Automatic Control, AC-3:05 02, November 986. [6] M. Steinbuch, G. Schoostra, and H.-T.Goh. Closed-loop scaling in fixed-point digital control. IEEE Trans. on Control Systems Tech., 2:32 37, December 994. [7] R. Bishop, editor. The control handbook. The electrical engineering handbook series. CRC Press/IEEE Press, [8] R.C. Agarwal and C.S. Burrus. New recursive digital filter structures having very low sensitivity and roundoff noise. IEEE Trans. on Circuits and Systems, CAS-22:92 927, December 975. [9] H. Hanslemann. Implementation of digital controllers, a survey. Automatica, 23():7 32, 987. [0] G.C. Goodwin. Efficient data representations for signal processing and control: Making most of a little. Proceddings of the 25th Chinese Control Conference, pages 9 37, August [] H.-M. Cheng and G.T.-C. Chiu. Coupling between sample rate and required wordlength for finite precision controller implementation with delta transform. June 3, 20 Silvano Balemi 9

20 Proceedings of the 2007 American Control Conference, pages , July Silvano Balemi June 3, 20

21 Exercises A. Quantization errors Given is the closed-loop system with the discrete transfer function G cl (z) = z3 +2 z 2 2 z 3. Determine the resolution needed for an A/D converter with range -0V a +0V if the stochastic estimate of the quantization error at the output of the system must be less than 0mV (Note that the transfer function from the noise introduced by the A/D converter to the output is the closed-loop transfer function). A.2 Sinusoidal encoder interpolation Given is a displacement sensor with sinusoidal signal interface. Such a sensor delivers two signals sin(α) and cos(α), where α is proportional to the measured displacement x. The signal period is l period = 2µm. Some companies offer integrated circuits estimating the angle α from the explicit computation or with the help of a table the inverse tangent tan ( sinα ) and by cosα keeping track of the number of periods. One other possibility is to retrieve the value of x from the signals sin(α) and cos(α) is to convert these signals to digital variables and to process them with the closed-loop estimator shown in Figure.3: The advantage of this method is the cos alpha alpha sin(alpha) cos(alpha) E alpha_e alpha_e encoder filter sin Figure.3: Closed-loop-estimation of the angle α = K x possibility to obtain an estimate of the speed directly from the loop, and an increased robustness in the presence of offsets, incorrect amplifications or distortions of the input signals. The signal at the input of the filter shown in Figure.3 is given by E = sin(α) cos(ˆα) cos(α) sin(ˆα) = sin(α ˆα) June 3, 20 Silvano Balemi 2

22 which can be approximated, for small values of α ˆα, to: E = sin(α ˆα) α ˆα showing that the signal E is approximately the error between the value of α and the estimate ˆα. Then,forα ˆαtheloopofFigure.3canbereplacedbytheloopofFigure.4. α + - FILTER ˆα Figure.4: Simplified system for the position estimation If this loop is stable, ˆα converges to the value of α. From ˆα it is then possible to obtain the position estimate ˆx of x with ˆx = ˆα K. Find the second order discrete-time open-loop filter such that the closed-loop filter poles are both at the frequency of 2kHz. The sampling frequency is 0kHz. The second order filter should have two poles at (double integrator term for zero error in case of a ramp input for x). 2. Find the resolution of the A/D converters such that the influence of the quantization error on the measurements of sin(α) and cos(α) corresponds to a static estimate error x err < 0nm. A.3 Positioning of a mass Given is a small mass m = 50g. The mass must be precisely positioned within 200ms with a discrete-time controller.. What is a reasonable sampling time? 2. Suppose the stroke is 0mm. What is the resolution of the A/D converter needed to position the mass statically within 0.µm of the target position? (assume for this point to use the controller the controller C(z) = 3020z 3000 for z+ the sampling time T = ms). 3. Suppose the maximum actuation force is ±0.5N. What is the resolution of the D/A converter for the actuation force needed to position statically the mass within 0.µm? (assume for this point to use the controller the controller C(z) = 3020z 3000 z+ for the sampling time T = ms). 22 Silvano Balemi June 3, 20

23 A.4 Control of a force-feedback system The steering system of Figure.5 consists of two identical platforms each composed of a brushless motor, an encoder, a gear, a torque sensor and a steering wheel. The objective of the control is to keep the two steering wheels at the same angular position and to let the users feel the torque produced by the other user. It is also possible to implement different torque and/or position ratios for the simulation of a bilateral servo system with increased positioning accuracy or force gain. Figure.5: Steering wheel system with force feedback. What closed-loop bandwidth of the system do you choose? 2. List the issues influencing the choice of the sampling frequency and propose a sampling frequency for the controller. 3. Would you use a prefilter for the measurements? What would be the cut-off frequency? June 3, 20 Silvano Balemi 23

24 Solved exercises B. First order plant Given is a first order system G(s) = sτ + with τ = ms. The sampling frequency is 0kHz. The objective is to control the process and to speed up the transients by 2 times.. Compute the simplest discrete-time controller. 2. Suppose the controller is C(z) = az b = 2z. What is the sensitivity of the z z closed-loop poles in function of the parameters a and b of the controller? 3. Which of the controller parameters a and b is the most critical one for stability reasons? 4. Give the anti-windup realization of the controller C(z) = 2z which exploits z FIR filters. Draw the block diagram with each single controller state. 5. Write a sample C code lines for the implementation of the controller C(z) = 2z z. 6. Suppose the measurement of the plant output is corrupted by much noise spread over all frequencies. Which anti-aliasing filter would you choose? 7. Please give the form of the controller you would use in the presence of the antialiasing filter and show the steps to obtain the controller parameter values. Solution:. The discrete-time transfer function is G ZOH (z) = z ( ) Z s z s (sτ +) = z ( Z s z s τ ) sτ + = z ( z z z z z e T τ = z z e T τ = e T τ z e T τ ) 24 Silvano Balemi June 3, 20

25 The simplest controller is a proportional one. The closed-loop system becomes G cl (z) = = K ( e T τ) z e T τ +K ( e T τ) K ( e T τ) z +K (K +) e T τ The desired characteristic polynomial is z e 2 T τ thus which gives K = e T τ. K (K +) e T τ = e 2 T τ 2. The closed-loop system is G cl (z) = = (a z b)( e T τ) (a z b)( e T τ)+(z )(z e T τ) z a ( e T τ) b ( e T τ) z 2 +z (a ( e T τ) e T τ) b ( e T τ)+e T τ with ± i With P(z) = z 2 + z (a ( e T τ) e T τ) b ( e T τ)+e T τ being the characteristic polynomial of the closed-loop system, i.e. the denominator of the transfer function above, the sensitivities are P(z) a P(z) z P(z) a P(z) z z ( e T τ) = = i 2z +(a ( e T τ) e T τ) z=p z=p z ( e T τ) = = i 2z +(a ( e T τ) e T τ) z=p2 z=p2 P(z) b ( e T τ) = P(z) = 0.740i 2z +(a ( e T τ) e T τ) z=p z z=p P(z) b ( e T τ) = P(z) = 0.740i 2z +(a ( e T τ) e T τ) z=p2 z z=p2 June 3, 20 Silvano Balemi 25

26 3. Thus both poles are most sensitive with respect to the parameter b. As they are equally close to the unit circle there is no particular difference for both poles. The correspondings sensitivity is complex, that means that the pole is being shifted not only in direction of the real axis but also of the imaginary axis. 4. The controller can be rewritten as C(z) = 2z z = 2 z z = which yields the controller of Figure.6 N(z) {}}{ 2 z z }{{} D(z) r 2 y e 2 z z u Figure.6: Controller with anti-windup measure 5. A sample code can be ek_=0; while(true) { wait4interrupt(); yk=read_yk(); ek=(yrefk-yk); uk=(2*ek-ek_+u_k_); if (uk>ukmax) then uk=ukmax; if (uk<ukmin) then uk=ukmin; write(uk); uk_=uk; ek_=ek; } 6. As the noise is spread over all frequencies, the low-pass filter should be chosen with cut-off frequency as low as possible, i.e. F(s) = τ s Silvano Balemi June 3, 20

27 Attention: this is a continuous-time filter! 7. First we must find the model of the continuous-time part, composed of plant and anti-aliasing filter. The extended plant is G ZOH (z) = z ( ) Z s z s (sτ +) (s τ +) 4 = z ( Z s z s 4 3 s+ + ) 3 s+ 4 τ τ = z ( z z z 4 3 z + ) z e T τ 3 z z e 4T τ = 4 3 z + z e T τ 3 z z e 4T τ = 3 z (e 4T τ 4e T τ +3)+(3e 5T τ 4e 4T τ +e T τ) z 2 z (e 4T τ +e T τ)+e 5T τ z b +b 0 = (.8) z 2 +z a +a 0 The chosen controller is a first order one of the form C(z) = which gives the closed-loop system G cl (z) = = a z +b z +γ (a z +b)(z b +b 0 ) (a z +b)(z b +b 0 )+(z +γ)(z 2 +z a +a 0 ) z 2 a b +z (b b +a b 0 )+b b 0 z 3 +z 2 (a b +γ +a )+z (b b +a b 0 +a 0 +γ a )+b b 0 +γ a 0 For a closed-loop bandwidth of at least 2 times the open-loop bandwidth of the system, in the case of 3 identical closed-loop poles, a good approximation is to choose p c i = 3 2 for the continuous-time poles and τ pd i = e 6 T τ for the discrete-time poles. The corresponding characteristic polynomial is chp(z) = z 3 3 z 2 e 6 T τ +3 z e 2 T τ e 8 T τ and the controller parameters a, b and γ can be found by solving the linear system b 0 a 3 e 6 T τ a b 0 b a b = 3 e 2 T τ a 0 0 b 0 a 0 γ e 8 T τ wheretheparametersa 0,a,b 0 andb comefrom(.8). Theresultingcontroller becomes 6.7 z C(z) = 0.76 z June 3, 20 Silvano Balemi 27

28 B.2 Limit cycles (Franklin, Powell, Workman, pb 0.0) Consider the system in Figure.7 with quantization step q = 0.. z x2 a z x a2 Scope Discrete Pulse Generator A/2 Constant Figure.7: Second order system with quantized coefficients and dither. Use Matlab/Simulink to simulate the response to the initial conditions x = 2 and x 2 = 0 with zero input. Use a 2 = 0.8,0.9,0.95 and 0.98 (fixing a = 2 a 2 ). Compare the amplitudes and frequencies of the limit cycles (if any) with the values predicted. 2. Add dither at the Nyquist frequency to the quantizer of a 2 y with amplitude A and find the dither amplitude that minimizes the peak output in the steady state. Solution:. The amplitude A L of the limit cycle is given by: A L = q 2 a 2 while the cycle frequency is given by the approximate position of the poles. Suppose in fact that p,2 = rcosϕ+jrsinϕ. The characteristic polynomial is z 2 2rcosϕ+r 2 = z 2 +a z +a 2 giving the pole argument ( ϕ = cos a ) 2r Considering that the limit cycles correspond to a pole on the unit circle, the ratio between the limit cycle period and the sampling period can be estimated as R = 2π ϕ = 2π ( ) cos a 2 28 Silvano Balemi June 3, 20

29 Figure.8 shows the limit cycles for various parameter values. Table. shows the predicted with the simulated data. 2 A=0*q a2= a2= a2= a2= Figure.8: Response of second order system with quantized coefficients: Note the limit cycles. 2. The limit cycle amplitudes for the various parameter values and for different dither amplitudes are show on Figure.9. B.3 Poles sensitivity (Franklin, Powell, Workman, pb 0.8) Consider implementation of the discrete compensator D(z) = z 2 (z 0.5)(z 0.55)(z 0.6)(z 0.65). If implemented as a cascade of first-order filters with coefficients stored to finite accuracy, what is the smallest gain perturbation that could cause instability? Which parameter would be most sensitive? 2. Draw an implementation of D(z) in controller normal form with parameters α, α 2, α 3 and α 4 and the characteristic polynomial z 4 +z 3 α +z 2 α 2 +zα 3 + June 3, 20 Silvano Balemi 29

30 a =.264 a =.34 a =.378 a =.4 a 2 = 0.8 a 2 = 0.9 a 2 = 0.95 a 2 = 0.98 Estimated amplitude Simulated amplitude Estimated cycle ratio Simulated cycle ratio Table.: Comparison between simulated and estimated values in limit cycles α 4. Compute the first-order sensitivities of these parameters according to the formula seen in the course. Which parameter is more sensitive? Compare the sensitivities of root locations for the cascade and the control forms for these nominal root positions. 3. Using the root locus, find the maximum deviations possible for the parameters of the controller normal form and compare with the results of part 2. Solution:. The sensitivities are shown in table.2 α =-0.5 α 2 =-0.55 α 3 =-0.6 α 4 =-0.65 pole pole pole pole Table.2: Sensitivities of the diagonal normal form 2. The parameters are the coefficients of the characteristic polynomial (z 0.5)(z 0.55)(z 0.6)(z 0.65) = z 4 2.3z z z The sensitivities are shown in table.3. Note that the highest sensitivity is that related to the parameter However, the most critical pole is not the one closest to the unit circle but the pole The equivalent root-locus problems for the 4 different parameters are parameter α G(z) = parameter α 2 G(z) = z 3 z 4 +α 2 z 2 +α 3 z +α 4 z 2 z 4 +α z 3 +α 3 z +α 4 30 Silvano Balemi June 3, 20

31 a2=0.8 band 0.5 band a2= A a2= A band a2= A band A Figure.9: Amplitude of limit cycles in function of the amplitude of a dither introduced before the quantizer of a 2. parameter α 3 z G(z) = z 4 +α z 3 +α 2 z 2 +α 4 parameter α 4 G(z) = z 4 +α z 3 +α 2 z 2 +α 3 z The root-locus plots are shown in Figure.20. June 3, 20 Silvano Balemi 3

32 α =-2.3 α 2 =.9775 α 3 = α 4 =0.073 pole pole pole pole Table.3: Sensitivities of the controller normal form 0.2 Root Locus 0.2 Root Locus Imag Axis 0 Imag Axis Real Axis Real Axis 0.2 Root Locus 0.2 Root Locus Imag Axis 0 Imag Axis Real Axis Real Axis Figure.20: Root-loci of closed-loop poles with varying parameters (α, α 2, α 3, α 4 in the sequence). The variation is ±0.00 around the nominal value. Note that changes are more severe for parameter α Silvano Balemi June 3, 20