Digital PID Controller DesignDigital PID Controller Design

1 Digital PID Controller Design 1. Digital PID Controller Design [n]. r[n] e[n] u[n] y[n]. C(z) P (z). Plant and Controller N(z). ( ) NC ((z)). G( ) =. G(z) ; C( ) =. C(z) : D(z) DC (z). The characteristic polynomial of the closed loop system (z) := DC (z)D(z) + NC (z)N (z). 2. Digital PID Controller Design TCHEBYSHEV REPRESENTATION AND. ROOT CLUSTERING. Tchebyshev representation of real polynomials Consider a real polynomial P (z) = an z n + an 1 z n 1 + + a1 z + a0. The image of P (z) evaluated on the circle C of radius , centered at the origin is: j .. P (z) : z = e ; 0 2 : jj . j . j . As the coe cients ai are real P e and P e are conjugate complex numbers, and so it su ces to determine the image of the upper half of the circle: j .. P (z) : z = e ; 0 : 3. Digital PID Controller Design . Since z k z= ej = k (cos k + j sin k );. k ). j . P e = (an n cos n + + a1 cos + a0 ) +j (an n sin n + + a1 sin ).

2 | {z } | {z }. ). R( ; ). R( ). I( ; ). I( . ) + j I( ;. = R( ; ): Consider ( ej )k = k cos k + j k sin k . ( . Write u = cos and de ne the generalized Tchebyshev polynomials as follows: ck (u; ) = k ck (u); sk (u; ) = k sk (u); k = 0; 1; 2 . and note that 1 d [ck (u; )]. sk (u; ) = ; k = 1; 2; . k du . ck+1 (u; ) = uck (u; ) 1 u2 sk (u; ); k = 1; 2; . 4. Digital PID Controller Design The ggeneralized Tchebyshev y polynomials p y are for k = 1;; 5: k ck (u; ) sk (u; ). 1 u . 2 2 (2u2 1) 2 2 u 2 . 3 3 ( 4u3 + 3u) 3 (4u2 1). 4 4 (8u4 8u2 + 1) 4 ( 8u3 + 4u). 5 5 ( 16u5 + 20u3 5u) 5 (16u4 12u2 + 1).. p With this notation, P ej = R(u; ) + j 1 u2 T (u; ) =: Pc (u; ). where R(u; ) = an cn (u; ) + an 1 cn 1 (u; ) + + a1 c1 (u; ) + a0. T (u; ) = an sn (u; ) + an 1 sn 1 (u; ) + + a1 s1 (u; ): R(u; ) and T (u; ) are polynomials in u and . p The complex plane image p g of P ((z)) as z traverses the upper pp half of the circle C can be obtained by evaluating Pc (u; ) as u runs from 1 to +1.)

3 5. Digital PID Controller Design LEMMA. For a xed > 0, (a) if P (z) has no roots on the circle of radius > 0, 0. (R (u; ) ; T (u; )) have no common roots for u 2 [ 1; 1] and R ( 1; ) 6= 0. (b) if P (z) has 2m roots at z = (z = + ), th R (u;. then ( )) and d T (u;. ( )) have h m roots t each h att u = +1 (u ( = 1). 1). (c) if P (z) has 2m 1 roots at z = (z = + ), then R (u; ) and T (u; ). have m and m 1 roots, respectively at u = +1 (u = 1). p (d) if P (z) has qi pairs of complex roots at z = ui j 1 u2i , for ui 6= 1, then R (u; ) and T (u; ) each have qi real roots at u = ui . When the circle of interest is the unit circle, that is = 1, we will write Pc (u; 1) = Pc (u) and also R(u 1) =: R(u);. R(u; R(u) T (u;. (u 1) =: T (u): (u). 6. Digital PID Controller Design Interlacing Conditions for Root Clustering and Schur Stability THEOREM. P (z) has all its zeros strictly within C if and only if (a) R(u; ) has n real distinct zeros ri , i = 1; 2; ; n in ( 1; 1).))

4 ((b)) T ((u;; ) has n 1 real distinct zeros tj , j = 1;; 2;; ; n 1 in (( 1;; 1). ). (c) The zeros ri and tj interlace: 1 < r1 < t1 < r2 < t2 < < tn 1 < rn < +1: The three conditions given in the above Theorem may be referred to as interlacingg conditions on R(u;. ( ; ) and T ((u;; ). ). By setting = 1 in the above Theorem we obtain conditions for Schur stability in terms of interlacing of the zeros of R(u) and T (u). 7. Digital PID Controller Design Tchebyshev Representation of Rational Functions Let p Pi(z)jz= u+j p1 u2 =Ri(u; )+j 1 u2Ti(u; );i=1;2: .. p P1 (z)P2 (z 1 ) . Q(z)j Q( )jz= =. z= u+j . u+j 1 u2. 1 u P2 (z)P ( 11 ) z= u+j p1 u2. ( )P2 (z R(u; ). z }| 2 {. R1 (u; )R2 (u; ) + 1 u T1 (u; )T2 (u; ). =. R22 (u; ) + (1 u2 ) T22 (u; ). T (u; ). p z }| {. 2. 1 u ((T1 ((u;; ). )R2 ((u;; ) R1 ((u;; ). )T2 ((u;; )). +jj : R22 (u; ) + (1 u2 ) T22 (h; ). R(u R(u; ), ) T (u.)))))))

5 (u ) are rational functions of the real variable u which runs from -1 to +1. 8. Digital PID Controller Design ROOT COUNTING FORMULAS. LEMMA. Let the real polynomial P (z) have i roots in the interior of the circle C . and no roots on the circle. Then: 0 [ P ( )] = i LEMMA. L Q( ) = PP12 (z). Lett Q(z) (z). where h th reall polynomials the l i l P1 (z). ( ) and d P2 (z). ( ) have h i1 and d i2. roots, respectively in the interior of the circle C and no roots on the circle. Then 0 [ Q ( )] = (i1 i2 ) = +11 [ QC (u)]: 1 (u)]. 9. Digital PID Controller Design Let t1 ; ; tk denote the real distinct zeros of T (u; ) of odd multiplicity, for u 2 ( 1; 1), ordered as follows: 1 < t1 < t2 < < tk < + +1. SSuppose pp also that T ((u;; ) has p zeros at u = 1 and let f i (x0 ) denote the i-th derivative to f (x) evaluated at x = x0 . THEOREM. Let P (z) be a real polynomial with no roots on the circle C and suppose that T (u; ) has p zeros at u = 1.)

6 Then the number of roots i of P (z) in the interior of the circle C is given by Xk . 1 (p) . i = Sgn T ( 1; ) Sgn [R( 1; )] + 2 ( 1)j Sgn [R (tj ; )] + ( 1)k+1 Sgn [R(+1; )] : 2 j=1. 10. Digital PID Controller Design The result derived above can now be extended to the case of rational functions. Let Q(z) = PP12 (z). ( ). (z). where Pi (z); i = 1; 2 are real rational functions. Tchebyshev representation of Q(z) on the circle C . Let R(u; ); T (u; ) be de ned by: R(u; ) = R1 (u; )R2 (u; ) + (1 u2 )T1 (u; )T2 (u; ). T (u;. (u ) = T1 (u; (u )R2 (u;. (u ) R1 (u;. (u )T( u;. u ). Suppose that T (u; ) has p zeros at u = 1 and let t1 tk denote the real distinct zeros of T ((u;; ) of odd multiplicity p y ordered as 1 < t1 < t2 < < tk < +1. THEOREM. Let Q(z) = PP12 (z). (z). where Pi (z); i = 1; 2 are real polynomials with i1. and i2 zeros respectively inside the circle C and no zeros on it.)))))

7 Then Xk . 1 (p) j k+1. i1 i2 = Sgn T ( 1; ) Sgn [R( 1; )]+2 ( 1) Sgn [R (tj ; )]+( 1) Sgn [R(+1; )] : 2 j=1. 11. Digital PID Controller Design Digital PI. PI, PD AND PID CONTROLLERS. For PI controllers, . KP. z (KP + KI T ) z KI T +KP. C(z). ( ) = KP + KI T =. z 1 z 1. K1 (z K2 ) K1 K1 K2. = where KP = K1 K2 ; KI = : z 1 T. For F PD controllers, t ll . KD. KD. KP + T z T. K. KD z 1 KP + TD. C(z) = KP + =. T z z K1 (z K2 ). =: where KP = K1 K1 K2 ; KD = K1 K2 T: z The general formula of a discrete PID Controller , using backward di . di erences t preserve causality, to lit z KD z 1 K2 z 2 + K1 z + K0. C(z) = KP + KI T + =: where z 1 T z z(z 1). K0 + K1 + K2. KP = K1 2K0 ; KI = ; KD = K0 T: T. 12. Digital PID Controller Design COMPUTATION OF THE STABILIZING SET. Constant Gain Stabilization N(z). Plant G(z) =. D(z). The closed-loop p characteristic p polynomial y is (z) = D(z) + KN(z): Tchebyshev representations of D(z) and N(z).

8 J p D e = RD (u) + j 1 u2 TD (u). jj p N e = RN (u) + j 1 u2 TN (u);. (u). j . p Nr (z). Note also that N e = RD (u) j 1 u2 TD (u) and N (z 1 ) = zl where Nr (z) is the reverse polynomial and l is the degree of N(z). 13. Digital PID Controller Design (z)N (z 1 ) = D(z)N (z 1 ) + KN(z)N (z 1 ).. (z)Nr (z) p p z l = RD (u) + j 1 u2 TD (u) RN (u) j 1 u2 TN (u). z=ej . 2 . +K RN (u) + 1 u2 TN2 (u). 2 . = RD (u)RN (u) + 1 u2 TD (u)TN (u) + K RN (u) + 1 u2 TN2 (u). | {z }. R(K;u). ( ). p +j 1 u2 [TD (u)RN (u) RD (u)TN (u)]. | {z }. T (u). p R(K u)) + j 1 u2 T (u): = R(K; ( ). The imaginary part of the above expression has been rendered independent of K as a result of multiplying (z) by N(z 11 ). ). parameter separation Ready to apply the root counting formulas 14. Digital PID Controller Design Constant Gain Stabilization Algorithm Let ti ; i = 1; 2 ; k denote the real zeros of odd multiplicity of the xed T (u), for u in ( 1; +1) and set t0 = 1; tk+1 = +1.

9 Write Sgn [R(K; tj )] = xj ; j = 0; 1; ; k + 1. Let i , iNr denote the number of zeros of (z) and Nr (z). inside the unit circle. For simplicity assume that N (z) has no unit circle zeros and therefore neither does Nr (z). 1 . i + iNr l = Sgn T (p) ( 1). 2. Xk . Sgn [R(K; 1)] + 2 ( 1)j Sgn [R (K; tj )] + ( 1)k+1 Sgn [R(K; +1)] : j=1. 15. Digital PID Controller Design E. Example l z 4 + 1:93z 3 + 2:2692z 2 + 0:1443z 0:7047. G(z) = 5 : z 0:2z 4 3:005z 3 3:9608z 2 0:0985z + 1:2311. Then RD (u) = 16u5 1:6u4 + 32:02u3 6:3216u2 13:9165u + 4:9919. TD (u) = 16u4 + 1:6u 1 6u3 24:02u 24 02u3 + 7:1216u 7 1216u + 3:9065. 3 9065. RN (u) = 8u4 7:72u3 3:4616u2 + 5:6457u 1:9739. TN (u) = 8u3 + 7:72u2 0:5384u 1:7857. and T (u) = TD (u)RN (u) RD (u)TN (u). = 11:2752u4 + 7:5669u3 + 16:7782u2 14:1655u + 1:203: The roots of T (u) of odd multiplicity and lying in ( 1; 1). are 0:0963.

10 0 0963 and 0:8358. 0 8358. 16. Digital PID Controller Design 11 2752u5 + 12:1307u R(K u) = 11:2752u R(K; 12 1307u4 40:6359u 40 6359u3 7:1779u 7 1779u2 + 40:8322u 40 8322u 16:8293 19:6615u 5:4727. 4 3 2.. +K 11:2752u + 9:7262u + 15:0696u 20:3653u + 7:085 : Since i = 5 for stability, and iNr = 2 and l = 4, we must have: . (p) . Sgn T ( ) ( 1) Sgn[R(K; 1)] 2 Sgn[R(K; 0:0963)]+2 Sgn[R(K; 0:8358)] Sgn[R(K; 1)] = 6. (p).. Since Sgn T ( 1) = +1, +1 we have the only feasible string given by: Sgn[R(K,-1)]. [ ( )] Sgn[R(K, [ ( )]. )] Sgn[R(K, [ ( )]. )] Sgn[R(K, [ ( 1)]. )]. 1 -1 1 -1. 17. Digital PID Controller Design This translates into the following set of inequalities: R(K;. ( ; 1)) = 23:348 + 21:5185K > 0 ) K > 1:085. R(K; 0:0963) = 12:998 + 5:2709K < 0 ) K < 2:466. R(K; 0:8358) = 0:9232 + 0:7673K > 0 ) K > 1:2032. R(K 1) = 0:4050. R(K; 0 4050 + 0:2403K. 0 2403K < 0 ) K < 1:6854: 1 6854.