Stability Margin Based Design of Multivariable Controllers

Transcription

1 Stability Margin Based Design of Multivariable Controllers Iván D. Díaz-Rodríguez Sangjin Han Shankar P. Bhattacharyya Dept. of Electrical and Computer Engineering Texas A&M University College Station, Texas, USA ivan diaz 1st IEEE Conference on Control Technology and Applications, August 27-3, 217. The Mauna Lani Bay Hotel and Bungalows Kohala Coast, Hawai i. CCTA 217 Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

2 Overview 1 Introduction Research content Motivation 2 Methodology Transformation of the multivariable plant into a diagonal transfer function matrix Transformation of the diagonal controller matrix into the corresponding MIMO controller Design of the controller to obtain predesigned gain margin, phase margin, and gain crossover frequency for the Smith-McMillan plants Gain and phase margin design for MIMO plants 3 PI Controller Design 4 Example: PI controller design for a TITO system 5 Conclusions Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

3 Introduction - Research content Consider a multivariable continuous-time Linear Time Invariant (LTI) plant with an n n transfer function matrix P(s). An n n controller C(s) in a unity feedback configuration. The design problem is to find C(s) suchthatthecontrolsystemachieves predesigned: - Gain Margin (GM) - Phase Margin (PM) Methodology 1 Transform the multivariable system P(s) into a diagonal transfer function matrix. 2 Introduce a diagonal controller transfer function matrix. 3 Design each diagonal controller to obtain predesigned gain margin, phase margin, and gain crossover frequency. 4 Transform the diagonal controller matrix into the MIMO controller C(s) for the original plant P(s). 5 Verify that the gain and phase margins of this multivariable system is the minimum gain margin and the minimum phase margin of the Smith-McMillan plants. Illustrative example Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

4 Introduction - Motivation New approach to design a controller for a MIMO systems by taking advantage of the simplicity of SISO system design tools. PI controller is one of the most widely used controllers in the control industry. Design tool to analyze the achievable performance of the system. Multi objective design simultaneous achievement of the design specifications most often required in applications. Gain Margin Phase margin Gain crossover frequency Robustness measures of importance. Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

5 Transformation of the multivariable plant into a diagonal transfer function matrix Multivariable systems can be transformed into a diagonal transfer function matrix called the Smith-McMillan form. The transfer function matrix P(s) canbe expressed as P(s) = 1 N(s) (1) d(s) where d(s) is the least common multiple of the denominators of the elements of P(s) andn(s) is a polynomial matrix. The Smith form of N(s) is given by S(s) =Y (s)n(s)u(s) (2) where S(s) is a diagonal polynomial matrix and Y (s), U(s) areunimodular polynomial matrices. Let P d (s) = S(s) d(s). (3) Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

6 Transformation of the multivariable plant into a diagonal transfer function matrix Then P d (s) is the Smith-McMillan form of P(s) andp i (s) willbecalledthe Smith-McMillan plants for i =1, 2,...n. SinceP d (s) is of diagonal form we can introduce a diagonal controller matrix C d (s) where each controller C i (s) is designed for the corresponding P i (s) giving the multiple SISO loops as in Figure below. + 2 C C n 2 P P n Figure: Multiple SISO unity feedback block diagram Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

7 Transformation of the diagonal controller matrix into the corresponding MIMO controller After designing the controller C i (s) for each Smith-McMillan plant P i (s), we can transform the diagonal controller matrix C d (s) into the MIMO controller matrix C(s) via C(s) =U(s)C d (s)y (s) (4) where U(s) andy (s) are the unimodular matrices in (2). C(s) isthe MIMO controller for the MIMO plant P(s). Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

8 Design of the controller to obtain predesigned gain margin, phase margin, and gain crossover frequency for the Smith-McMillan plants The controller C i (s) canbedesignedtoachievespecificgainmargin,phasemargin,andgain crossover frequency for the Smith-McMillan plant P i (s). It is important to consider the relative degree of P i (s) whendesigningc i (s) inordertoensurethatc(s) isproper. Lemma Let r k be the relative degree of the controller C k (s) for the Smith-McMillan plant P k (s). Ifr k for k =1, 2,...,n satisfies n o min r k dik U dkj Y k=1,2,...,n, 8i, j =1, 2,...,n (5) where dik U and d kj Y are the degree of (i, k) th and (k, j) th polynomials of the unimodular matrices U(s) and Y (s), respectively,thenthemimocontrollerc(s) will be proper. Proof. See D. N. Mohsenizadeh, L. H. Keel, and S. P. Bhattacharyya, Multivariable controller synthesis using siso design methods, in215 54thIEEE Conference on Decision and Control (CDC). IEEE, 215, pp Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

9 Gain and phase margin design for MIMO plants In order to design the controller C i (s) foreachsmith-mcmillanplantp i (s) onecanspecifygain and phase margins for the corresponding SISO loop. Let g i and i be the gain margin and phase margin for the i th SISO loop for i =1, 2,...,n. Define as 2 6 := (6) Consider G(s) = P(s)C(s) infig2.multivariablestabilitymarginscanbedefinedasfollows: For gain margins replace by K in (6) and find the smallest K, calledk,suchthattheloopin Fig. 2 becomes marginally unstable. A similar definition applies to phase margin where is replaced by e j. G(s) Figure: Unity feedback MIMO system with a perturbation matrix Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

10 Gain and phase margin design for MIMO plants We need a preliminary lemma to prove the main result. Lemma det(i + P(s)C(s)) = n i=1 (1 + P i (s)c i (s)). This lemma in fact proves that the stability of the multivariable system in Fig 3 is equivalent to that of the Smith-McMillan loops. Theorem Suppose C(s) in (4) is a proper controller such that C i (s) stabilizes the corresponding Smith-McMillan plant P i (s). ThenC(s) stabilizes P(s) with a gain margin G and phase margin : G = min {g i }, (7) i=1,2,...,n = min i=1,2,...,n { i }, (8) where g i and i are the gain and phase margins of the SISO loops C i (s)p i (s) for i =1, 2,...n. This theorem shows that the gain and phase margins for the multivariable system are the minimum of the gain margins and the minimum of the phase margins of the multiple SISO Smith-McMillan loops. Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

11 PI controller design Computation of the Stabilizing Set Consider a continuous-time LTI SISO system P(s) = N(s) D(s) (9) and a PI controller C(s) = K Ps + K I s. (1) The computation of the stabilizing set uses the generalized Hermite-Biehler theorem and is known as the signature method. See Chapter 2 in 1 for a complete treatment of the method. 1 Bhattacharyya, S. P., Datta, A., and Keel, L. H. Linear Control Theory: Structure, Robustness, and Optimization, CRCPress,29. Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

12 PI controller design Constant gain and constant phase loci for PI controllers Let P(s) andc(s) bethesisoplantandcontrollertransferfunctions.thefrequencyresponse of the plant and controller are P(j!), C(j!), respectively, where! 2 [, 1]. For the PI controller Then, we have for fixed!, Equations (12) and (13) can be written as C(j!) = jk P! + K I. (11) j! C(j!) 2 = KP 2 + K I 2! 2 =: m2 (12) KI \C(j!) =arctan =: (13) K P! where (K P ) 2 a 2 + (K I ) 2 b 2 =1 and K I = ck P (14) a 2 = m 2, b 2 = m 2! 2, c =! tan (15) Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

13 PI controller design Constant gain and constant phase loci for PI controllers Thus constant gain loci are ellipses and constant phase loci are straight lines in K P, K I space. The major and minor axes of the ellipse are given by a and b. Theslopeofthestraightline(14) is c. Suppose! g is the prescribed closed-loop gain crossover frequency. Then C(j! g 1 ) = P(j! g =: mg (16) ) If g is the desired phase margin in radians, \C(j! g )= + g \P(j! g )=: g (17) Combining (14), (16), and (17) we obtain the ellipse and straight line corresponding to m = m g and = g,givingthedesignpoint(kp, K I )deliveringphasemargin g at gain crossover frequency! g,iftheseintersectionpointslieinthestabilizingset.iftheellipseandstraightline only intersect outside the stabilizing set, the specifications (! g, g )areunattainableandhaveto be modified. The graphical procedure (intersection of ellipse and straight line in the stabilizing set) makes it a convenient tool for computer-aided design. Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

14 PI controller design Computation of the achievable performance Gain-Phase margin design curves The Gain-Phase margin design curves represent the actual performance in terms of gain margin (GM), phase margin (PM), and gain crossover frequency (! g )foraplantp(s) achievablewitha PI controller. The procedure to construct these design curves is the following: 1 Set a test range of g 2 [ min g, max g ]and! g 2 [! g min,! g max ]. 2 For fixed values of g and! g,plotthecorrespondingellipseandstraightlineusing(14), (16), and (17). 3 If an intersection point of the ellipse and the straight line lies outside of the stabilizing set, then this point is rejected and we go to step 2). 4 If an intersection point of the ellipse and straight line is contained in the stabilizing set, it represents the design point with the PI controller gains (KP, K I )thatsatisfiesthe! g. g and 5 Given the selected PI controller gains (KP, K i I ), the upper and lower GM of the system i are given by GM upper = K lb P K P and GM lower = K P ub KP where KP ub and K lb P are the K P values at the upper and lower boundary respectively of the stabilizing set following the straight line intersecting the ellipse. 6 Go to step 2 and repeat for all values of g and! g in the ranges. Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25 (18)

15 PI controller design Selecting an achievable GM, PM, and! g and retrieving the PI Controller gains The designer can select a desired point from the achievable performance Gain-Phase margin set and retrieve the controller gains corresponding to that simultaneous specification of desired GM, PM, and! g. The controller gain retrieval process is the following. (1) Select desired GM, PM, and! g from the achievable gain-phase margin set. (2) For the specified point, construct the ellipse and straight line by using the selected PM and! g in the constant gain and constant phase loci. (3) Take the intersection of the ellipse and straight line contained in the stabilizing set. This will provide the gains (K P, K I ). (4) The controller that satisfies the prescribed margin specifications is C(s) = K P s+k I s. Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

16 Example: PI controller design for a TITO system Let us consider a Two-Input Two-Output system as + C(s) P(s) Figure: MIMO unity feedback block diagram with P(s) = " 4 (s+1)(s+2) 2 s+1 1 s+1 6s+7 2(s 2 +3s+2) #. (19) The objective is to design the controller C(s) such that it satisfies predesigned gain margin, phase margin, and gain crossover frequency specifications. Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

17 Example: Transformation of the multivariable system into multiple SISO systems. The least common multiple of the denominators of P(s) is d(s) =(s +1)(s +2) (2) Then, we can rewrite P(s) as P(s) = apple 1 4 1(s +2) (s +1)(s +2) 2(s +2) (3s +3.5) {z } N(s) (21) The Smith form of N(s) isexpressedas S(s) = apple apple 1 4 (s +2) 2 {z } Y (s) apple 1 = s 2 2s 3 4 1(s +2) 2(s +2) (3s +3.5) {z } N(s) apple 1 1 (s +2) 4 1 {z } U(s) (22) Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

18 Example: Transformation of the multivariable system into multiple SISO systems. Dividing every element of S(s) by d(s) we get the Smith-McMillan form P d (s) = apple 1 (s+1)(s+2) s 3 s+2 (23) Considering Fig 1 with a diagonal controller C d (s), we have multiple SISO loops where we can apply the SISO controller design method discussed in the previous section. Let us consider C d (s) as # C d (s) = " KP1 s+k I1 s K P2 s+k I2 s(s+2) 2 Note that there are two additional poles included in C 2 (s). The relative degree must be r 2 = 2 for the controller C(s) to be proper. These poles can also be considered as additional design variables. (24) Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

19 Example: Computation of the PI stabilizing sets for the multiple SISO loops For C 1 (s)p 1 (s) therangeofk P1 for stability was determined to be K P1 2 ( 2, 1). For C 2 (s)p 2 (s) therangeofk P2 for stability was determined to be K P2 2 ( 9.272, ). By sweeping over K P1 and K P2 within the intervals we can generate the set of stabilizing (K P1, K I1 ) and (K P2, K I2 )sets. 4 Unbounded K P K P K P K I K I2 Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

20 Example: Construction of the Gain-Phase margin design curves for the multiple SISO loops For the system C 1 (s)p 1 (s), the evaluated range of! g is [.1, 2.1] and the range for PM is from o to 9 o.forthesystemc 2 (s)p 2 (s), the evaluated range of! g is [.1, 2.1] and the range for PM is from o to 12 o. GM PM (degrees) =.1 =.3 =.5 =.7 =.9 = 1.1 = 1.3 = 1.5 = 1.7 = 1.9 = 2.1 GM 1 1 g =.1:.2:2.5 PM: 71 GM: 5.94 PM: 6 GM: PM: 52 GM: PM: 56 GM: 2 PM: 83 GM: PM (degrees) =.1 =.3 =.5 =.7 =.9 = 1.1 = 1.3 = 1.5 = 1.7 = 1.9 = 2.1 Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

21 Example: Selection of simultaneous design specifications and retrieval of the PI controller gains For illustration purposes, we are going to select GM = 1, PM =6 o,and! g =.5 rad/sec for C 1 (s)p 1 (s) andgm =3.518, PM =6 o,and! g =.5 rad/secfor C 2 (s)p 2 (s). Retrieving the controller gains for the loops we have KP 1 =.424 and KI 1 =1.133 for C 1 (s) andkp 2 = 1.59 and KI 2 = 1.34 for C 2 (s). 4 PM = 6 o, g =.5 rad/s, unbounded K P 1 4 PM = 6 o, =.5 rad/s g 3 ub K P1 = 2 2 K P1 1 Ki1: Kp1:.424 K P Ki2: Kp2: K I2 ub = ub K P2 = K I K I2 Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

22 Example: Transformation of the diagonal controller C d (s) into the MIMO controller C(s) The final design step is to transform the diagonal controller C d (s) into the MIMO controller C(s) in (4) by substituting the unimodular matrices Y (s) andu(s) in (22). Then, using the assigned values of (KP 1, KI 1 )and(kp 2, KI 2 )wehave C(s) = ".371s+.618 s 1.61s+1.34 s(s+2).53s.67 s(s+2) 2.12s 2.68 s(s+2) 2 # (25) Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

23 Example: Verification We can verify the final results by computing the gain margin and phase margin of the multivariable system using the controller C(s). In Fig 2, the characteristic equation of the multivariable system is given by det[i + G(s)] (26) where G(s) = P(s)C(s) and is defined as in(6).let = k. Then det[i + G(s)] = s 7 +9s 6 +(32.637k)s 5 +( k)s 4 +( k k)s 3 +( k k)s 2 +(3.7833k k)s k 2. (27) We find that the range of k for the closed-loop system in Fig 2 to be stable is the roots of (27) with k =3.518 are 8 >< >: < k < (28) j j j j We see that two of the roots just cross the imaginary axis. Thus the gain margin is k =3.518 for the MIMO system. This is the same value that we selected in the desired specifications for C 2 (s)p 2 (s). Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25 (29)

24 Example: Verification Now, let = e j.equation(26)is det[i + G(s)] = s 7 +9s 6 +(32.637e j )s 5 +(2.33e j +56.)s 4 +(19.29e j.4484e j2 +48.)s 3 +(32.7e j.4218e j2 +16.)s 2 +(e j e j )s e j2 (3) The range for to keep the multivariable system stable is < <6 o. (31) The roots of (3) for =6 o are 8 >< >: j j j j j j j (32) Likewise, two of the roots just cross the imaginary axis. So, the phase margin is =6 o for the MIMO system. This is the same value that we selected in the desired specifications for the SISO system C 2 (s)p 2 (s). Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25

25 Conclusions We have presented an advanced tuning approach for MIMO systems to achieve classical stability margins. It was shown how SISO design methods, using PI controllers, can be used to design a MIMO controller to satisfy predesigned gain and phase margins for the MIMO system. Exploration of the system s capabilities by constructing the achievable performance presented as Gain-Phase margin design curves. It can be applied to an arbitrary plant where at least there exist a PI controller that stabilizes the system. Computational tool for the PI designer. The results presented should be extendible beyond PI controllers, and this is being researched. Díaz, Han, Bhattacharyya (TAMU) Stability Margin Based Design of Multivariable Controllers CCTA / 25