Chapter Robust Performance and Introduction to the Structured Singular Value Function Introduction As discussed in Lecture 0, a process is better desc

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1 Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c

2 Chapter Robust Performance and Introduction to the Structured Singular Value Function Introduction As discussed in Lecture 0, a process is better described in terms of a set of plants centered around a nominal model The robust stabilization problem is concerned with nding non conservative conditions on the stable nominal closed loop system that guarantee the stability of all possible closed loop systems An equally important problem is the robust performance problem which is concerned with nding non conservative conditions on the nominal closed loop system that guarnatee that the performance is met for all possible closed loop systems Robust Disturbance Rejection We will focus our discussion on one prototype problem, namely, the robust disturbance rejection problem shown in Figure This motivates the following problem: Robust Disturbance Rejection Problem (RP) Find conditions on the nominal closed-loop system (P o K) such that K robustly stabilizes all P, where fp j P (I W )P o kk < g: k(i P K) ; W k for all P From Lecture 0, a performance objective in terms of the H -norm of some closed loop map between some exogenous input w, to a regulated variable z, is mathematically equivalent to a robust stabilization problem with a perturbation block mapping the regulated output z to the exogenous input w Obviously, the new perturbed system is stable if and only if kt zw k, which is the performance

3 - - - m to k () m e? - W - W m - -?? P 0 - ; m - - K Figure : Uncertain Plant with Disturbance objective Notice that if the performance objective consists of several closed loop maps, then several perturbation blocks can be introduced in exactly the same fashion Proceeding for RP, we can \wrap" a frequency-weighted perturbation from the output to the input of interest, which results in the model of Figure Next, we can re-arrange the system into the - W z - w w z W?? m - - m P o ; K Figure : Robust Performance Model M- feedback form (a nominal stable M in feedback with the perturbation ) as in Figure In this case, however, there are multiple inputs and outputs to consider We use the following procedure to generate M and : Dene w i z i to be the output and input, respectively, of the perturbation i For a total of m perturbations, compute the matrix transfer function M as the map from w : w z z w m z m In other words, all the blocks are removed, and the transfer functions \seen" by the blocks from each input w j to each output z i are calculated and used as the (i j) th element of M The perturbation matrix will have the structure m i k < : () For a SISO system, each i is a scalar, so that becomes a diagonal matrix with complex entries In the MIMO case, is block-diagonal

4 Example (Robust Disturbance Rejection) Applying the robust performance procedure to Figure yields: ;W (I P 0 K) ; P 0 K ;W (I P 0 K) ; P 0 K M : () W (I P 0 K) ; W (I P 0 K) ; The transfer functions on the diagonal are identical to those in the single-block robust stability and disturbance-rejection problems, respectively, while the o-diagonal terms account for the interaction between the two constraints Having found the appropriate M and, we have thereby reduced the robust performance problem to a stability problem for the system of Figure - l - M - m! Figure : M- Feedback Form A sucient condition for robust stability is given by the small gain theorem, namely, max [M(jw)] max [(jw)] < for all w: Since is norm bounded by one, this condition translates to kmk This condition, however, is far from necessary since has a block diagonal structure The Structured Singular Value For an unstructured perturbation, the supremum of the maximum singular value of M (ie kmk ) provides a clean and numerically tractable method for evaluating robust stability Recall that, for the standard M- loop, the system fails to be robustly stable if there exists an admissible such that (I ; M) is singular What distinguishes the current situation from the unstructured case is that we have placed constraints on the set Given this more limited set of admissible perturbations, we desire a measure of robust stability similar to kmk This can be derived from the structured singular value (M) Denition The structured singular value of a complex matrix M with respect to a class of perturbations is given by (M) : () inff max () j det(i ; M) 0g If det(i ; M) 0 for all, then (M) 0

5 Theorem The M- System is stable for all with kk < if and only if sup (M) :! Proof: Immediate, from the denition Clearly, if, then the norm of the smallest allowable destabilizing perturbation must by denition be greater than Properties of the Structured Singular Value It is important to note that is a function that depends on the perturbation class (sometimes, this function is denoted by to indicate this dependence) The following are useful properties of such a fucntion (M) 0 If fi j C g, then (M) (M), the spectral radius of M (which is equal to the magnitude of the eigenvalue of M with maximum magnitude) If f j is an arbitrary complex matrixg then max (M), from which sup! kmk Property shows that the spectral radius function is a particular function with respect to a perturbation class consisting of matrices of the form of scaled identity Property shows that the maximum singular value function is a particular function with respect to a perturbation class consisting of arbitrary norm bounded perturbations (no structural constraints) If fdiag( : : : n ) j i complexg, then (M) (D ; MD) for any D diag(d : : : d n ) jd i j > 0 The set of such scales is denoted D This can be seen by noting that det(i ; AB) det(i ; BA), so that det(i ; D ; MD) det(i ; MDD ; ) det(i ; M) The last equality arises since the diagonal matrices and D commute If diag( : : : n ) i complex, then (M) (M) max (M) This property follows from the following observation: If, then It is clear that the class of perturbations consisting of scaled identity matrices is a subset of which is a subset of the class of all unstructured perturbations From and we have that (M) (D ; MD) inf DD max (D ; MD) Computation of In general, there is no closed-form method for computing Upper and lower bounds may be computed and rened, however In these notes we will only be concerned with computing the upper bound If diag( : : : n ), then the upper bound on is something that is easy to calculate Furthermore, property above suggests that by inmizing max (D ; MD) over all possible diagonal scaling matrices, we obtain a better approximation of This turns out to be a convex optimization problem at each

6 () frequency, so that by inmizing over D at each frequency, the tightest upper bound over the set of D may be found for We may then ask when (if ever) this bound is tight In other words, when is it truly a least upper bound The answer is that for three or fewer 's, the bound is tight The proof of this is involved, and is beyond the scope of this class Unfortunately, for four or more perturbations, the bound is not tight, and there is no known method for computing exactly for more than three perturbations Robust Disturbance Rejection (SISO) As shown earlier, the disturbance rejection requirement could be converted to a robust stability problem with two blocks of uncertainty, as in Figure, where and are SISO stable systems Hence is the set of diagonal complex matrices (which result from evaluating at each frequency) Now, since this is a two-block problem, it should be possible to nd by inmizing max (D ; MD) We have D diag(d d ), so that (M) inf d d >0 8 >< >: W ; P 0 K d ; W K > P 0 K d P 0 K max () d W P 0 W d P 0 K P 0 K {z A() with the \pure" robust stability requirement occup ying the upper left diagonal, and the nominal performance requirement on the lower right Setting d d and xing!, and taking the denition of A() from (), we have Now, for nominal performance, we require that For robust stability, we nee 9 } > (M) inf f (A ()A())g: () max jj>0 W P 0 K : W P 0 K : P 0 K For robust performance, the necessary and sucient condition is A bit of algebra yields from which we have max (A A) jj () (8) (M) : (9) W K W P 0 K P 0 K W KP 0 P 0 K jj W P 0 P 0 K W P 0 K W inf max (A A) : P 0 K P 0 K (0) ()

7 W (j ω ) - ω o L(j ω )W (j ω ) L(j ω) Figure : Robust Performance/Nyquist Criterion This minimum occurs at jj jw P 0 j jw Kj () which is not equal to in general, so that sup! kmk In other words, is a less conservative measure than kk in this case Once again, there is a graphical interpretation of the SISO robust disturbance rejection problem, in terms of the Nyquist criterion From (), we have (M) () W P 0 K P 0 K W : () P 0 K Letting L represent the nominal loop gain P 0 K, this can be rewritten as: jw Lj jw j j Lj: () Graphically, we can represent this at each frequency! as a circle centered at ; of radius jw j, and a second circle centered at L of radius jw Lj Robust performance will be achieved as long as the two circles never intersect Loop-shaping Revisited Loop-shaping is a well-established method of control design that concentrates on the frequency-domain characteristics of the open-loop transfer function L P 0 K Based primarily on design experience, there are certain characteristics of the loop transfer function that translate into desirable control performance Other open-loop characteristics are known by experience to result in undesirable or unpredictable behavior This method diers from -synthesis and H methods, which concentrate on optimizing the characteristics of the closed-loop transfer function Since, presumably, a controller with good behavior designed by loop-shaping should be similar in some way to a controller designed by more recent methods, it is of interest to look for parallels in the heuristic rules of loop-shaping and the more methodical methods of -synthesis and H

8 L Identifying the sensitivity and complementary sensitivity functions from (), we can write the RP requirement as jw T j jw Sj : () Model uncertainty typically increases with frequency, so it is important that the complementary sensitivity function decreases with increasing frequency For disturbance rejection, which is typically most critical over a low frequency range, we require that S remain small The weighting functions W and W are designed to reect this, and so might take on the form of Figure Normally, at low W W Figure : Typical Weighting Functions frequency, L >> and at high frequency, L << Now, L T 0 S 0 () L L so that at low frequency, T 0 and S 0 L Thus we can approximate the RP requirement at the low end as: jw j W jw j (8) ) jlj ; jw j At high frequency, the approximation is T 0 L and S 0, which leads to: jw Lj jw j ) jlj ; jw j : (9) jw j These constraints are summarized in Figure, which also notes another design rule, which is that the 0 db crossing should occur at a slope no more negative than -0 db per decade If W and W do not overlap signicantly in frequency, then the upper and lower bounds reduce to jw j and jw j, respectively Example (Loop Shaping) Assume P 0 is minimum phase stable with relative degree Designing a controller by shaping the loop gain L P 0 K is not aected by P 0 just the relative degree is needed

9 W - W 0 db L(j ) -0 db/decade max - W W Figure : Typical Loop-shaping Problem Suppose the multiplicative uncertainty is described by s W 0(0:0s ) ie, the multiplicative perturbations of the plant are upper bounded by W at each frequency The objective is to track sinusoidal signals at the reference input in the frequency range [0 ] rads We would like to make the tracking error small however, we do not know yet by how much Let W have the following frequency response jw j a 0! 0 otherwise Note that this may not correspond to a stable W (s) however, this does not aect the resulting loop shape We are going to exhibit the design by trial and error Let At high frequency,! 0, b L(s) : cs ; jw j L! 0: jw j jw j If we pick c, then the largest value for b such that the above is satised is b 0 Hence 0 L(s) : s At low frequency,!, jw j a jlj : ; jw j ; jw j

10 which implies that a : Checking the RP condition Since jlj is decreasing and jw j is increasing in the range [0 ], the largest a can be solved for: a jl(j)j ; jw (j)j jw Sj jw T j 0:9 8! which implies RP is achieved and the tracking error is smaller than : in the range [0 ] If a better performance is desired, a possibly more complicated L needs to be used The discussion in this chapter has focused on perturbations that are arbitrary dynamic systems This alowed us to think of any class of structured perurbations as sets of arbitrary (structured) matrices at each frequency point These matrices correspond to evaluating the dynamic system at a given frequency In practical applications, some perturbations may be static and not dynamic These arise in problems with real parameter uncertainties We can still proceed as before and transform such problems to the general M- diagram In this case, will have a combination of both static and dynamic perturbations for such a class can be dened as before, and it will provide a necessary and sucient condition for robust stability The main issue here is computing a good upper bound for Of course, we can always embed this class of perturbations in a larger class containing dynamic perturbations and use D-scaling to obtain an upper bound This, however, gives conservative conditions Computing non-conservative upper bounds of for such perturbations remains an active area of research Rank-One Although we do not have methods for computing exactly, there is one particular situation where this is possible This situation occurs if M has rank, ie where a b C is given by However, M ab n Then it follows that with respect to containing complex diagonal perturbations inf f max () j det(i ; M) 0g: (M) det(i ; M) det(i ; ab ) det(i ; b a) 0 det I ; [ n ] ; [ n ] b a b a b na n b a b a b na n C A

11 m (M) N (M) jb m - - D m : and max () max i j i j Hence, inf ] ::: n 8 >< >: max j i j i [ n b a b a b na n : 9 > > Optimizing the RHS, it follows that (verify) P n $ (M) ia i j i X n i jb ia i j Notice that the SISO robust disturbance rejection problem is a rank-one problem This follows since Then M (M) which is the condition we derived before Coprime Factor Perturbations Consider the class of SISO systems N(s) D(s) ;W K P 0 [ ]: P 0 K P 0 K W W P 0 K P 0 K W P 0 K N 0 W D D 0 W k i k < where the nominal plant is N 0 D 0 with the property that both N 0 and D 0 are stable with no common zeros in the RHP Assume that K stabilizes N 0 D 0 This block diagram is shown in Figure - W z - w w z W ;? u? - - y - N 0 ; 0 K Figure : Coprime Factor Perturbation Model

12 M ; : The closed loop block diagram can be mapped to the M- diagram where ; W K W ; K D 0 N 0 K D 0 N 0 K Hence, M has rank and (M) W D 0 N 0 K ; W K D 0 N 0 K W D 0 N 0 K W K D 0 N 0 K W D 0 N 0 K [ ]: W D 0 N 0 K Robust Hurwitz Stability of Polynomials with Complex Perturbations Another application of the structured singular value with rank one matrices is the robust stabil- ity of a family of polynomials with complex perturbations of the coecients In this case let T n; n; : : : 0 and consider the polynomial family P (s ) s n (a n; n; n; )s n; : : : (a ) where a i, i, and i C and j i j We want to obtain a condition that is both necessary and sucient for the Hurwitz stability of the entire family of polynomials P (s ) We can write the polynomials in this family as P (s ) P (s 0) (0) P~(s ) ; n n; n; s a n; n; n; s : s : : : a 0 : : 0 0 () which can also be rewritten as n; 0 0 : : : 0 n; s n; 0 n; 0 : : : 0 n; s n; P (s ) P (s 0) : : : : 0 s 0 0 : : : We assume that the center polynomial P (s 0) is Hurwitz stable This implies that the stability of the entire family P (s ) is equivalent to the condition that : : : P (j! 0) n; 0 0 : : : 0 n; n; 0 n; 0 : : : 0 n; n; : : :

13 @ for all! R and j i j This is equivalent to the condition that 0 B n; n; n; n; for all! R and with kk Now using the concept of the structured singular value we arrive at the following condition which is both necessary and sucient for the Hurwitz stability of the entire family (M) < C A det I : : : 0 P (j! 0) 0 for all! R, where M P (j! 0) n; (j! ) n; n; (j! ) n; 0 : : : : Clearly this is a rank one matrix and by our previous discussion the structured singular value can be computed analytically resulting in the following test for all! R jp (j! 0)j X n i j n;i jj!j n;i <

14 Exercises Exercise In decentralized control, the plant is assumed to be diagonal and controllers are designed independently for each diagonal element If however, the real process is not completely decoupled, the interactions between these separate subsystems can drive the system to instability Consider the plant P P P (s) : P P Assume that P and P are stable and relatively small in comparison to the diagonal elements, and only a bound on their frequency response is available Suppose a controller K diag(k K ) is designed to stabilize the system P 0 diag(p P ) Set-up the problem as a stability robustness problem, ie, put the problem in the M ; form Derive a non-conservative condition (necessary and sucient) that guarantees the stability robustness of the above system Assume the o-diagonal elements are perturbed independently Reduce the result to the simplest form (an answer like (M) < is not acceptable this problem has an exact solution which is computable) How does your answer change if the o-diagonal elements are perturbed simultaneously with the same Exercise Consider the rank problem Suppose, contains only real perturbations Compute the exact expression of (M) Exercise Consider the set of plants characterized by the following sets of numerators and denominators of the transfer function: N(s) N 0 (s) N (s) D(s) D 0 (s) D (s) Where both N 0 and D 0 are polynomials in s, R n, and N D are polynomial row vectors The set of all plants is then given by: Let K be a con troller that stabilizes such that the system is stable N(s) f j R n j i j g D(s) N 0 Compute the exact stability margin ie, compute the largest D 0

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Chapter Stability Robustness Introduction Last chapter showed how the Nyquist stability criterion provides conditions for the stability robustness of

Chapter Stability Robustness Introduction Last chapter showed how the Nyquist stability criterion provides conditions for the stability robustness of Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter Stability