# INTRODUCTION TO DIGITAL CONTROL

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1 ECE4540/5540: Digital Control Systems INTRODUCTION TO DIGITAL CONTROL.: Introduction In ECE450/ECE550 Feedback Control Systems, welearnedhow to make an analog controller D(s) to control a linear-time-invariant (LTI) plant G(s). We used feedback to: Reduce error due to disturbance, Reduce error due to model mismatch, Stabilize an unstable plant, Improve transient response. w(t) r(t) e(t) D(s) u(t) G(s) y(t) v(t) There are advantages to using a digital computer to replace D(s). Flexible: Modifications to design easy to implement. Accurate: Does not rely on imprecise R, C,... values. Advanced algorithms are possible (nonlinear, adaptive...). Logic statements may be included. There are problems with using a digital computer:

2 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 2 D(s) knows e(t) at each instant in time but digital computers cannot process input infinitely quickly. D(s) outputs u(t) at each instant of time but... e(t) is an infinite-precision value, but digital computers must work with finite-precision values. Digital computers are finite. The necessary compromises are to sample the input e(t) to get e[k] =e(kt), tocomputeanoutputu[k] that is held constant from u(kt) to u((k + )T ), andto quantize internalcomputervariables. e(t) r(t) A2D Q e[k] u[k] D(z) D2A zoh w(t) u(t) G(s) y(t) v(t) D(z) is the transfer function of the controller; discrete-time. We need to know how to design D(z), howquicklytosample(/t ), and how finely to quantize. Fortunately, many of the techniques are similar either in principle or practice to analog-control techniques. Proceeding, we look at: Review. Examine impact A2D/D2A with approximate control method. Study z-transform. More in-depth study of A2D/D2A using z-transform knowledge. Control design of D(z) using transform methods. Implementation using DSP. Adaptive form of digital control called Adaptive Inverse Control.

3 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 3.2: Review: System modeling Goals of feedback control Change dynamic response of a system to have desired properties. Ensure that closed-loop system is stable. Constrain system output to track a reference input with acceptable transient and steady-state responses. Reject disturbances. These goals are accomplished via frequency-domain (Laplace) analysis and design tools for continuous-time systems. Dynamic response We wish to control linear time-invariant (LTI) systems. These dynamics may be specified via linear, constant-coefficient ordinary differential equations (LCCODE). Examples include: Mechanical systems: Use Newton s laws. Electrical systems: Use Kirchoff s laws. Electro-mechanical systems (generator/motor). Thermodynamic and fluid-dynamic systems. EXAMPLE: Second-order system in standard form : ÿ(t) + 2ζω n ẏ(t) + ωn 2 y(t) = ω2 n u(t). u(t) istheinputand y(t) istheoutput.

4 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 4 ẏ(t) = dy(t) dt and ÿ(t) = d2 y(t) dt 2. The Laplace Transform is a tool to help analyze dynamic systems. Y (s) = H(s)U(s), where Y (s) is Laplace transform of output y(t); U(s) is Laplace transform of input u(t); H(s) is transfer function the Laplace tx of impulse response h(t). KEY IDENTITY: L{ẏ(t)} = sy(s) for system initially at rest. EXAMPLE: Second-order system: s 2 Y (s) + 2ζω n sy(s) + ω 2 n Y (s) = ω2 n U(s) Y (s) = ω 2 n U(s). s 2 + 2ζω n s + ωn 2 Transforms for systems with LCCODE representations can be written as Y (s) = H(s)U(s), where H(s) = b ms m + b m s m + +b s + b 0 a n s n + a n s n + +a s + a 0, and where n m for physical systems. These can be represented in MATLAB using vectors of numerator and denominator polynomials: num=[bm b b0]; den=[an a a0]; sys=tf(num,den); Can also represent these systems by factoring the polynomials into zero-pole-gain form: H(s) = K m i= (s z i) n i= (s p i).

5 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 5 u sys=zpk(z,p,k); % in MATLAB Input signals of interest include the following: u(t) = k δ(t)... U(s) = k impulse u(t) = k (t)... U(s) = k/s step u(t) = kt (t)... U(s) = k/s 2 ramp u(t) = kt 2 /2(t)... U(s) = k/s 3 parabola kω u(t) = k sin(ωt) (t)... U(s) = sinusoid s 2 + ω 2 MATLAB s impulse, step, and lsim commandscanbeusedto find output time histories. The Final Value Theorem states that if a system is stable and has a final, constant value, lim x(t) = lim sx(s). t s 0 This is useful when investigating steady-state errors. Block diagrams Useful when analyzing systems comprised of a number of sub-units. U(s) H(s) Y (s) Y (s) = H(s)U(s) U(s) H (s) H 2 (s) Y (s) Y (s) = [H (s)h 2 (s)] U(s)

6 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 6 i k H (s) U(s) Y (s) Y (s) = [H (s) + H 2 (s)] U(s) H 2 (s) R(s) U (s) Y 2 (s) H (s) H 2 (s) U 2 (s) Y (s) Y (s) = H (s) + H 2 (s)h (s) R(s) Block-diagram algebra (or Mason s rule) may be used to reduce block diagrams to a single transfer function. U(s) H(s) Y (s) Y 2 (s) U(s) H(s) H(s) Y (s) Y 2 (s) U (s) U 2 (s) H(s) Y (s) U (s) U 2 (s) H(s) H(s) Y (s) R(s) H (s) Y (s) R(s) H 2 (s) H 2 (s) H (s) Y (s) H 2 (s) Unity Feedback

7 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 7.3: Review: Actual and desired dynamic response The poles of H(s) determine (qualitatively) the dynamic response of the system. The zeros of H(s) quantify the relationship. If the system has only real poles, each one has the form H(s) = s + σ. If σ>0, thesystemisstable,andh(t) = e σ t (t). Thetimeconstant is τ = /σ, and the response of the system to an impulse or step decays to steady-state in about 4 or 5 time constants. impulse([0 ],[ ]); step([0 ],[ ]); h(t) e σ t e y(t) K K ( e t/tau ) System response. K = DC gain Response to initial condition t = τ Time (sec τ) t = τ Time (sec τ) If a system has complex-conjugate poles, each may be written as: H(s) = ω 2 n. s 2 + 2ζω n s + ωn 2 We can extract two more parameters from this equation: σ = ζω n and ω d = ω n ζ 2.

8 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 8 σ plays the same role as above it specifies decay rate of the response. I(s) θ = sin (ζ ) ωd is the oscillation frequency of the output. Note: ω d = ω n unless ζ = 0. ζ is the damping ratio and it also plays a ω n σ ω d R(s) role in decay rate and overshoot. Impulse response h(t) = ωn e σ t sin(ω d t) (t). ( Step response y(t) = e σ t cos(ω d t) + σ ) sin(ω d t). ω d y(t) Impulse Responses of 2nd-Order Systems ζ = 0.8 ζ = ω n t y(t) Step Responses of 2nd-Order Systems ζ = Asummarychartofimpulseresponsesandstepresponsesversus pole locations is: I(s) ω n t I(s) R(s) R(s) Impulse responses vs. pole locations Step responses vs. pole locations

9 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 9 Time-domain specifications determine where poles SHOULD be placed in the s-plane (step-response). t p M p t r t s t Rise time tr =timetogofrom0% to 90% of final value. Settling time ts =timeuntilpermanently within % of final value. Overshoot Mp =maximumper- 0 CENT overshoot Some approximate relationships useful for initial design processes are: M p, % t r.8/ω n... ω n.8/t r t s 4.6/σ... σ 4.6/t s M p e πζ/ ζ 2... ζ fn(m p ) ζ

10 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 0.4: Review: Feedback control r(t) y(t) D(s) G(s) Y (s) R(s) = D(s)G(s) + D(s)G(s) = T (s). Stability depends on roots of denominator of T (s): + D(s)G(s) = 0. Routh test used to determine stability. Steady-state error found from (for unity-feedback case) E(s) R(s) = + D(s)G(s). ess = lim t e(t) = lim s 0 se(s) if the limit exists. System type = 0 iff e ss is finite for unit-step (t). System type = iff e ss is finite for unit-ramp r(t). System type = 2 iff e ss is finite for unit-parabola p(t).. For unity-feedback systems, K p = lim D(s)G(s), s 0 K v = lim sd(s)g(s). s 0 K a = lim s 2 D(s)G(s). s 0 position error constant velocity error constant acceleration error constant Steady-state errors versus system type for unity feedback: Step input Ramp input Parabola input Type 0 + K p Type 0 K v Type K a

11 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL Some types of controllers Proportional ctrlr: u(t) = Ke(t). D(s) = K. Integral ctrlr u(t) = K t e(t) dt. D(s) = K T I T I s Derivative ctrlr. u(t) = KT D ė(t) D(s) = KT D s Combinations: PI: D(s) = K ( + T I s ); PD: D(s) = K ( ( + T D s) ; PID: D(s) = K + ) T I s + T Ds. Lead: D(s) = K Ts + αts +, Lag: D(s) = K Ts + αts +, α < (approx PD) α > (approx PI; often, K = α) Lead/Lag: D(s) = K (T s + )(T 2 s + ) (α T s + )(α 2 T 2 s + ), α <, α 2 >. Root locus Arootlocusplotshows(parametrically)thepossiblelocations of the roots of the equation + K b(s) a(s) = 0. For a unity-gain feedback system, T (s) = D(s)G(s) + D(s)G(s). The poles of the closed-loop system T (s) depend on the open-loop transfer functions D(s)G(s). Suppose D(s) = KD 0 (s).

12 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 2 closed-loop poles at + K (D 0 (s)g(s)) = 0 which is the root-locus form. Drawing the root locus allows us to select K for good pole locations. Intuition into the root-locus helps us design D 0 (s) with lead/ lag/ PI/ PID...controllers. Root-locus drawing rules The steps in drawing a 80 root locus follow from the basic phase definition. This is the locus of + K b(s) a(s) = 0, K 0 They are ( phase of b(s) ) a(s) = 80 STEP : On the s-plane, mark poles (roots of a(s)) byan and zeros (roots of b(s)) withan. There will be a branch of the locus departing from every pole and a branch arriving at every zero. STEP 2: Draw the locus on the real axis to the left of an odd number of real poles plus zeros. STEP 3: Draw the asymptotes, centered at α and leaving at angles φ, where n m = number of asymptotes. n = order of a(s) m = order of b(s) pi z i α = n m = a + b n m φ l = 80 + (l )360, l =, 2,...n m n m For n m > 0, therewillbeabranchofthelocusapproachingeachasymptote and departing to infinity. For n m < 0, therewillbeabranchofthelocusarrivingfrominfinityalong each asymptote. STEP 4: Compare locus departure angles from the poles and arrival angles at the zeros where qφ dep = ψ i φ i 80 l360 qψ arr = φ i ψ i l360 where q is the order of the pole or zero and l takes on q integer values so that the angles are between ±80. ψ i is the angle of the line going from the i th zero to the pole or zero whose

13 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 3 angle of departure or arrival is being computed. Similarly, φ i is the angle of the line from the i th pole. STEP 5: If further refinement is required at the stability boundary, assume s o = jω o and compute the point(s) where the locus crosses the imaginary axis for positive K. STEP 6: For the case of multiple roots, two loci come together at 80 and break away at ±90. Three loci segments approach each other at angles of 20 and depart at angles rotated by 60. STEP 7 Complete the locus, using the facts developed in the previous steps and making reference to the illustrative loci for guidance. The loci branches start at poles and end at zeros or infinity. STEP 8 Select the desired point on the locus that meets the specifications (s o ), then use the magnitude condition to find that the value of K associated with that point is K = b (s o ) /a (s o ). When K is negative, the definition of the root locus in terms of the phase relationship changes. We need to plot a 0 locus instead. 0 locus definition: The root locus of b(s)/a(s) is the set of points in the s-plane where the phase of b(s)/a(s) is 0. For this case, the steps above are modified as follows: STEP 2: Draw the locus on the real axis to the left of an even number of real poles plus zeros. STEP 3: The asymptotes depart at φ l = (l )360, l =, 2,...n m. n m STEP 4: The locus departure and arrival angles are modified to Note that the 80 term has been removed. qφ dep = ψ i φ i l360 qψ arr = φ i ψ i + l360. STEP 8 Select the desired point on the locus that meets the specifications (s o ), then use the magnitude condition to find that the value of K associated with that point is K = b (s o ) /a (s o ).

14 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 4.5: Review: Frequency response The frequency response of a system tells us directly the relative magnitude and phase of a system s output sinusoid if the system input is a sinusoid. [What about output frequency?] If the plant s transfer function is G (s), theopen-loopfrequency response is G ( jω) = G(s) s= jω. We can plot G( jω) as a function of ω in three ways: Bode Plot. Nyquist Plot. Nichols Plot (not covered in ECE450/550). Bode plots A Bodeplot isreallytwoplots:20 log0 G ( jω) versus ω and G( jω) versus ω. The Bode-magnitude plot is plotted on a log-log scale.the Bode-phase plot is plotted linear-log scale. The transfer function G(s) is broken up into its component parts, and each part is plotted separately. The factorization must be of the form KG(s) = K o (s) n (sτ + )(sτ 2 + ) (sτ a + )(sτ b + ), which separates real zeros and poles, complex zeros and poles, zeros or poles at the origin, delay, and gains. The following dictionary is used when plotting a transfer function which has been factored into Bode form.

15 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 5 Dictionary of Bode-plot relationships (tempor Gain of K ;magnitude&phase db 0. 0 K > 0. 0 K < Delay; magnitude & phase db Zero at origin; magnitude & phase db Pole at origin; magnitude & phase db

16 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 6 Zero on real axis; magnitude & mp-phase & nmp-phase db ω n ωn 0ω n 0.ω n ωn 0ω n 0.ω n ωn 0ω n Pole on real axis; magnitude & mp-phase & nmp-phase 0.ω n ω n 0ω n 0.ω n ω n 0ω n db 0.ω n ωn 0ω n Complex zeros; magnitude & mp-phase & nmp-phase ω n ω n 0ω n 0.ω n ω n 0ω 0 n 0.ω n ω n 0ω Complex poles; magnitude & mp-phase & nmp-phase n ω n ω n 0ω n ω n ω n 0ω n 0.ω n ω n 0ω n 360 Bode-plot techniques It is useful to be able to plot the frequency response of a system by hand in order to Design simple systems without the aid of a computer, Check computer-based results, and

17 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 7 Understand the effect of compensation changes in design iterations. H.W. Bode developed plotting techniques in the 930s that enabled quick hand potting of the frequency response. His rules are: STEP : Manipulate the transfer function into the Bode form KG( jω) = K o ( jω) n ( jωτ + )( jωτ 2 + ) ( jωτ a + )( jωτ b + ) STEP 2: Determine the value of n for the K o ( jω) n term. Plot the low-frequency magnitude asymptote through the point K o at ω = rad/sec with the slope of n (or n 20 db per decade). STEP 3: Determine the break points whereω = /τ i.completethecompositemagnitude asymptotes by extending the low frequency asymptote until the first frequency break point, then stepping the slope by ± or±2, dependingonwhetherthebreakpointisfromafirstorsecond order term in the numerator or denominator, and continuing through all break points in ascending order. STEP 4: Sketch in the approximate magnitude curve by increasing from the asymptote by a factor of.4 (+3dB) at first order numerator breaks and decreasing it by a factor of ( 3dB) at first order denominator breaks. At second order break points, sketch in the resonant peak (or valley) using the relation that G( jω) = /(2ζ) at the break. STEP 5: Plot the low frequency asymptote of the phase curve, φ = n 90. STEP 6: As a guide, sketch in the approximate phase curve by changing the phase gradually over two decades by ±90 or ±80 at each break point in ascending order. For first order terms in the numerator, the gradual change of phase is +90 ;inthedenominator,thechangeis±80. STEP 7: Locate the asymptotes for each individual phase curve so that their phase change corresponds to the steps in the phase from the approximate curve indicated by Step 6. Sketch in each individual phase. STEP 8 Graphically add each phase curve. Use dividers if an accuracy of about ±5 is desired. If lessor accuracy is acceptable, the composite curve can be done by eye, keeping in mind that the curve will start at the lowest frequency asymptote and end onthehighestfrequency asymptote, and will approach the intermediate asymptotes to an extent that is determined by the proximity of the break points to each other. Nyquist plots Nyquist plot is a mapping of loop transfer function D(s)G(s) along the Nyquist path to a polar plot.

18 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 8 Think of Nyquist path as four parts: I: Origin. Sometimes a special case. II: + jω axis. FREQUENCY response of O.L. system! Just plot it as a polar plot. III: For many physical systems, zero. Some counter-examples. IV: Complex conjugate of II. The test: I(s) III II R(s) I IV N =#CWencirclementsof /K point when F(s) = D(s)G(s). P =#ofopen-loopunstablepoles. Z =#ofclosed-loopunstablepoles. Z = N + P The system is stable iff Z = 0. For poles on the jω-axis, use modified Nyquist path. Bode-plot performance At low frequency, approximate Bode plot with KG( jω) = K0 ( jω) n. n =systemtype...slope = 20n db/decade. K 0 = K p for type 0, = K v for type,... Gain margin (for systems that become unstable with increasing gain) =factorbywhichgainislessthan0dbwhenphase= 80. Phase margin (for same systems) = amount phase is greater than 80 when gain =.

19 ECE4540/5540, INTRODUCTION TO DIGITAL CONTROL 9 PM related to damping. ζ PM/00 when PM < 70. k B PM therefore related to Mp. Damping ratio ζ Damping ratio versus PM Phase margin M p, overshoot Overshoot fraction versus PM Phase margin Gain-phase relationship: For minimum-phase systems, G( jω) n 90 where n =slopeoflog-logplotofgain(n = if slope = 20 db/decade, n = 2 if slope = 40 db/decade...) Therefore want slope at gain crossover 20 db/decade for decent phase margin. Where from here? We have reviewed continuous-time control concepts. Can any of these be applied directly to the discrete-time control problem? We look next at two different ways to emulate continuous-time controllers using discrete-time methods, allowing a quick redesign and control implementation. However, we also find that these methods are not optimized and that we can do better by designing digital controllers from scratch.

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