INPUT-OUTPUT APPROACH NUMERICAL EXAMPLES

EXERCISE Let us consider the linear dynamical system of order 2 with transfer function with Determine the gain 2 (H) of the input-output operator H associated with this system.

WEAKLY BOUNDED OPERATOR u H y causal Definition (weakly bounded operator): A causal operator is weakly bounded (or with finite gain) if Definition (gain of a weakly bounded operator): Let be a causal weakly bounded operator. The gain of H is given by

EXAMPLE: LINEAR ASYMPTOTICALLY STABLE DYNAMICAL SYSTEM Let In conclusion: H is bounded and weakly bounded with gain norm of F(s)

Let us consider the linear dynamical system of order 2 with transfer function with Determine the gain 2 (H) of the input-output operator H associated with this system. The systems is asymptotically stable. We then need to determine the norm of F(s). F(s) is a transfer function with gain F(0)=1 and either real or complex coniugate poles. In the latter case,we have a resonance pick if

u EXERCISE F(s) y (n=2) By using the small gain theorem, determine the values of k>0 such that the operator H with input u and output y is: 1) L -stable 2) L 2 -stable with finite gain. Provide also an estimate for the gain.

STABILITY OF INTERCONNECTED SYSTEMS: FEEDBACK H u 1 z 1 y H 1 1 y 2 H 2 z 2 u 2 Small gain theorem Let H be a well-posed causal operator obtained by connecting in feedback two causal and weakly bounded operators H 1 and H 2. If then, H is weakly bounded. Furthermore, Remark: it holds irrespectively of the signs at the summation nodes.

u H 1 F(s) y H 2 We next show that the two operators are weakly bounded and estimate their gain.

v w w k v j(v) v

v w It is an unbiased operator because:

v w It is an unbiased operator because: Let Zero bias gain: (bounded operator weakly bounded operator)

v w It is an unbiased operator because: Let Zero bias gain: if then

v w Gain of the operator: Is the gain equal to the zero bias gain? Not in general if, then Hence,

u F(s) y (order n=2) asymptotically stable linear system

ASYMPTOTICALLY STABLE LINEAR SYSTEM Let H is bounded since (zero bias gain) H is weakly bounded Given that the operator is affine, then:

u F(s) y (n=2) asymptotically stable linear system Let

u F(s) y (n=2) asymptotically stable linear system Let since

u H 1 F(s) y H 2 We next show that the two operators are weakly bounded and estimate their gain. Let

u H 1 F(s) y H 2 Let us estimate for which value of k the condition required by the small gain theorem is satisfied. Let

u H 1 F(s) y H 2 Let us estimate for which value of k the condition required by the small gain theorem is satisfied. Let For k< ½, the operator H is weakly bounded and, hence, L -stable with finite gain

u H 1 F(s) y H 2 Let us estimate the gain of the operator H via the small gain theorem. Let

u H 1 F(s) y H 2 We next show that the two operators are weakly bounded and estimate their gain. Let

v w w k v j(v) v

v w It is an unbiased operator because: Let Zero bias gain: (bounded operator weakly bounded operator)

v w It is an unbiased operator because: Let Zero bias gain: If then and hence

u F(s) y (n=2) asymptotically stable linear system

ASYMPTOTICALLY STABLE LINEAR SYSTEM Let H is bounded because (zero bias gain) H is weakly bounded Given that the operator is affine, then:

u F(s) y (n=2) asymptotically stable linear system Let perchè

u H 1 F(s) y H 2 We next show that the two operators are weakly bounded and estimate their gain. Let

u H 1 F(s) y H 2 Let us estimate for which value of k the condition required by the small gain theorem is satisfied. Let For k< ½, the operator H is weakly bounded and hence L 2 -stable with finite gain

u H 1 F(s) y H 2 Let us estimate the gain of the operator H via the small gain theorem. H Let

u EXERCISE F(s) y (n=2) By the circle criterion, determine the values of k>0 such that the operator H with input u and output y is L 2 -stable.

L 2 STABILITY IN SECTOR [k 1,k 2 ] S : y w e u y G(s) Theorem (Circle criterion for L 2 stability of a Lur e system) System S is L 2 -stable for any if the number of encirclements of G(s) Nyquist plot around O(k 1,k 2 ) is equal to the number of poles of G(s) with positive real part. Im Im Im Re -1 k 2-1 k 1 Re -1 k 1-1 k 2 Re 0 k 1 < k 2 k 1 < 0 < k 2 k 1 < k 2 0

L 2 STABILITY IN SECTOR [k 1,k 2 ] S : y w G(s) y Theorem (Circle criterion for L 2 stability of a Lur e system) System S is L 2 -stable for any if the number of encirclements of G(s) Nyquist plot around O(k 1,k 2 ) is equal to the number of poles of G(s) with positive real part. In our setting: e S : u F(s) y

L 2 STABILITY IN SECTOR [k 1,k 2 ] Im O(0,k) 0 k 1 < k 2 Re S : u F(s) y

Im O(0,k) Re

EXERCISE u F(s) y k y j(y) y (n=2) Value of k>0 such that the operatore H with input u and output y is L 2 -stable: small gain theorem k<1/2 circle criterion k<4

EXERCISE u F(s) y k y j(y) y (n=2) Value of k>0 such that the operatore H with input u and output y is L 2 -stable: small gain theorem k<1/2 circle criterion k<4 conservative result obtained by the small gain theorem. But the circle criterion derives from the small gain theorem

LUR E SYSTEM: SMALL GAIN THEOREM u 1 y 1 z 1 F(s) H S: H 1 H 2 y 2 z 2 u 2 System S (operator H): is L 2 -stable [for any function ] in sector [-k, k] if system with F(s) as transfer function is asymptotically stable and In this case, same condition but with in sector [0,k]

EXERCISE The figure below reports Nyquist plot of the transfer function G(s) of a linear system of order 3:

u EXERCISE G(s) y Determine k 1 and k 2 such that the Lur e system is L 2 -stable in the sector [k 1,k 2 ] by using the circle criterion

u G(s) y Determine k 1 and k 2 such that the Lur e system is L 2 -stable in the sector [k 1,k 2 ] by using the circle criterion Sol: G(s) has a real positive pole Nyquist plot of G(s) should encircle once the circle O(k 1,k 2 ) anticlockwise.

If we draw the circle centered in -3 and with radius 0.7, then, the circle criterion is satisfies. We then get -1/k 1 =-3-0.7 k 1 = 0.27-1/k 2 =-3+0.7 k 2 = 0.43

EX. 2 OF THE EXAM DATED SEPTEMBER 14, 2016 Let us consider the nonlinear system in the following figure where is a sector nonlinearity in [-2k, k], with k>0, and G s = 2 (1 0.01s) is the transfer function of a linear system of 1+s 2 1+0.001s order n=3. 1. Define the L 2 stability notion for the operator H with input y and output y. A causal operator is - stable if In our setup the Lebesgue space of interest is L 2

H u 1 z 1 y H 1 1 y 2 H 2 z 2 u 2 Small gain theorem Let H be a well-posed causal operator obtained by connecting in feedback two causal and weakly bounded operators H 1 and H 2. If then, H is weakly bounded, that is: Furthermore,

u 1 y 1 z 1 H S: H 1 H 2 y 2 G(s) z 2 u 2 Small gain theorem System S (operator H) is L 2 -stable (for any sector nonlinearity in [-k, k]) if

S : y w F(s) z Theorem (Circle criterion for L 2 stability of a Lur e system) System S is L 2 -stable for any if the number of encirclements of F(s) Nyquist plot around O(k 1,k 2 ) is equal to the number of poles of F(s) with positive real part. Im Im Im Re -1 k 2-1 k 1 Re -1 k 1-1 k 2 Re 0 k 1 < k 2 k 1 < 0 < k 2 k 1 < k 2 0

S : y w F(s) z Theorem (Circle criterion for L 2 stability of a Lur e system) System S is L 2 -stable for any if the number of encirclements of F(s)=-G(s) Nyquist plot around O(k 1,k 2 ) is equal to the number of poles of G(s) with positive real part. Im Im Im Re -1 k 2-1 k 1 Re -1 k 1-1 k 2 Re 0 k 1 < k 2 k 1 < 0 < k 2 k 1 < k 2 0

EX. 2 OF THE EXAM DATED SEPTEMBER 14, 2016 Let us consider the nonlinear system in the following figure where is a sector nonlinearity in [-2k, k], with k>0, and G s = 2 (1 0.01s) is the transfer function of a linear system of 1+s 2 1+0.001s order n=3. 3. Determine the maximum value for k>0 such that the operator H with input y and output y is L 2 stable via (a) the small gain theorem and (b) the circle criterion for the L 2 stability of H, specifying in a clear a precise way the adopted procedure and motivating the obtained result.

EX. 2 OF THE EXAM DATED SEPTEMBER 14, 2016 Let us consider the nonlinear system in the following figure where is a sector nonlinearity in [-2k, k], with k>0, and G s = 2 (1 0.01s) is the transfer function of a linear system of 1+s 2 1+0.001s order n=3. Solution (small gain theorem): Embed the sector nonlinearity in he symmetric sector [-2k, 2k] and impose that 2k G max < 1. Since G max = 2, then k < 1/4.

EX. 2 OF THE EXAM DATED SEPTEMBER 14, 2016 Let us consider the nonlinear system in the following figure where is a sector nonlinearity in [-2k, k], with k>0, and G s = 2 (1 0.01s) is the transfer function of a linear system of 1+s 2 1+0.001s order n=3. Solution (circle criterion): Find the largest k such that the Nyquist plot of G(s) is within O(-2k,k) k<1/2 Im 1/ k 1 / 2k Re