Nonlinear Systems Examples Sheet: Solutions

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1 Nonlinear Sstems Eamples Sheet: Solutions Mark Cannon, Michaelmas Term 7 Equilibrium points. (a). Solving ẋ =sin 4 3 =for gives =as an equilibrium point. This is the onl equilibrium because there is onl one point ( =)wheresin = since sin < < =) sin 4 < 3 for all apple, 6= sin apple =) sin 4 < 3 for all > Lapunov s direct method, invariant sets and linearization. To eplain the significance of constants a, b, c, we first give a derivation of the dnamics (this is not asked for in the question). momentum of the craft in z-coordinates (Fig. ) is given b 3 3 I! H = I!, I = 4 I 5,! = 4 5 I z The angular where I, I, I z are the moments of inertia about,, and z-aes (assumed to be aligned with the spacecraft s principle aes). Since there is no torque acting on the craft: d (I!) =I! +! I! = dt (where the! I! term is needed because z-coordinates are fied to and hence rotate with the spacecraft). So the full dnamics are given b.5.5 4! = a = b! = c! a =(I I z )/I, b =(I I z )/I, c =(I I )/I z and the constants a, b, c are all positive if I >I >I z. z.5 = = sin() 4 6 = 3 = sin 4 () 8 Figure : solution of 3 =sin 4 for question (b). In terms of state variables (, )=(, ẋ): ẋ =ẋ = ẋ =ẍ = ( ) 5 +sin( /) At an equilibrium point ẋ =ẋ =. But ẋ =implies =, so ẋ = =) sin( /) = =) =or Therefore equilibrium points are (, )=(, ẋ) =(, ) and (, ).! Figure : Rotating spacecraft. (a). Equilibrium points:! = () =or =, i.e. at least two of!, and must be zero for! = = =. Therefore

2 3 4 ever point in state space ling on the! -ais, the -ais, or the -ais is an equilibrium point. (b). To show stabilit of the equilibrium at! =,trv = p! +q! +r! z as a Lapunov function. Clearl V is positive definite if p, q, r are all positive. Also V =(p!! + q! + r! z ) =(pa Hence choosing p, q, r so that qb + rc)! p>, q>, r>, and pa qb + rc =, (which is alwas possible since q =(pa + rc)/b is positive for an chosen positive p, r), results in V =, impling that! =is a stable equilibrium point b Lapunov s direct method. (c). Di erentiating the function V = c! + b! z + ac! + ab! z + bc(!! ) (for constant! )withrespecttot along sstem trajectories ields V =c +b {z } = + ac! + ab! z + bc(!! ) (4ac +ab +bc!! ) {z } = i.e. V =. Also V =onl if! =(±!,, ), and V> whenever! 6= ±!, 6=or 6=, so that V is a locall positive definite function centered at the equilibrium (±!,, ). Therefore V = implies that ever point on the! -ais in state space is a stable equilibrium, and hence that rotation at an constant velocit about the -ais alone is stable. [Note that rotational motion about the z-ais is likewise stable since a, c and!, can be swapped in the dnamics and in the definition of V. However rotation about the -ais is unstable, as shown b the linearized sstem at! =(,!, ): 3 3! a! c! which has eigenvalues ± p ac! and, and is therefore unstable.] 3. (a). The positive definite function V = has derivative: V = ẋ = b() which is negative definite due to b() > whenever 6=. Therefore =is asmptoticall stable, and since V!as! it follows that =is globall asmptoticall stable b Lapunov s direct method. (b). At an equilibrium point ẋ =. Hence ẍ = c() =implies = since the condition c() > whenever 6= implies that c() can onl be equal to zero if =. Therefore the onl equilibrium point is the origin of state space: (, ẋ) =(, ). The function V (, ẋ) is positive definite and has derivative! V =ẋẍ + c()ẋ = ẋb(ẋ) apple and hence (, ẋ) =(, ) is stable b Lapunov s direct method. To appl the local invariant set theorem, we need to show that: (i) the level sets {(, ẋ) : V (, ẋ) apple V } are bounded for some V ; (ii) V apple ; (iii) the sstem dnamics are continuous and V is continuousl di erentiable in and ẋ. Here (i) is satisfied because V is increasing in both (since sign(c()) =sign()) andẋ; (ii) is demonstrated above; and (iii) holds since b(ẋ), = ẋ, = c() are all continuous functions of and ẋ. Let R = {(, ẋ) : V = } and let M be the largest invariant set contained in R, then R = {(, ẋ) : ẋ =}

3 5 6 and since ẍ =is necessar in order that the state remains in R, we have M = R \ {(, ẋ) : ẍ =} = {(, ẋ) : c() =} = {(, )}. From the local invariant set theorem, (, ẋ) therefore converges asmptoticall to R from all initial conditions within an bounded level set of V, impling that (, ) is asmptoticall stable. To show global asmptotic stabilit we need V to be radiall unbounded (in order to appl the global invariant set theorem) or equivalentl the level sets of V must cover the entire state space as V!. This condition requires Z c(s) ds!as!. 4. (a). The equilibrium points can be found b solving ẋ =ẋ =for and : ẋ = =) = ẋ =ẋ = =) ( ) = =) =,,. Hence the equilibrium points are (, )={(, ), (, ), (, )}. (b). The sstem and function V have the following properties. (i). V, ẋ and ẋ are continuous functions of and. (ii). The level sets: {(, ): V apple V } are finite and V is radiall unbounded since V!as!and/or!. (iii). Along sstem trajectories, V has derivative V (, )= ẋ + ( )ẋ = ( ) ( ) + ( ) = ( ) apple. Using the global invariant set theorem, (i)-(iii) impl that ever state trajector tends to an invariant set on which V =. (The same conclusion can be reached using the local invariant set theorem, since the level sets of V can be made arbitraril large b choosing V su cientl large.) From (iii), V (, )=is satisfied on the lines =and =. The invariant sets within these lines are defined b ẋ =(on =) and ẋ =(on =). But = ẋ = =) =,,, = ẋ = =) = and ever state trajector therefore tends asmptoticall to one of the three equilibrium points identified in (a). (c). Writing the sstem dnamics in the form ẋ = f(), = where the Jacobian matri of f is () =, ( ) (3 ) ( ) the linearization of the sstem at = =is given b ẋ = A, () =. A has eigenvalues T / ± p 5/, and it follows that the origin is an unstable equilibrium of the nonlinear sstem, b Lapunov s linearization method. (d). V has local minimum points at (, )=(, ) and (, ) (since V rv = = 3 = > at (, )=(, ) and (, )). Hence V + 4 is locall positive definite at (, )=(, ) and (, ), and from Lapunov s direct method these equilibrium points are therefore stable because V apple.

4 7 8 Other approaches for (d): The equilibrium at (, ) can be shown to be stable using the linearization method, since the linearization at this point is stable. However the linearization about (, ) has eigenvalues ±i p, and therefore does not allow an conclusions to be made about the stabilit of this equilibrium for the nonlinear sstem. 5. (a). Using matrices A, B, K and the given matri P we get ( marks): Q = (A BK) T P P (A BK) = Linear and passive sstems An level set of V contained entirel within this strip is invariant and hence is a region of attraction for =. The level sets are ellipsoidal, centred on the origin, and decrease in size as is reduced. Hence must be invariant for small enough. where eig(p )= : eig(q) = : 3 += =) = 3 ± p = =) =, 3 The equilibrium =is locall asmptoticall stable since: the linearized closed loop sstem about =is ẋ =(A BK) 6. Let = A + µi, then A T P + PA+µP = Q implies T P + P = A T P + PA+µP = Q, so P, Q > impl that Re{eig( )} <, so that Re{eig(A + µi)} <, and therefore Re{eig(A)} < µ (since A = V V =) = V ( µi)v ). (A BK) T P + P (A BK) = Q for positive definite P, Q implies ẋ =(A BK) is stable, i.e. Re[eig(A BK)] < so the nonlinear closed loop sstem is locall a.s. (b). From V = T P and ẋ =(A BK) (K) we get 7. (a). Di erentiating V with respect to t gives: V = e L( ) R L ( ) =ẋ e R L ( ) V = T (A BK) T P + P (A BK) (K) T (P + P ) = T Q (K) T P apple T Q + K T P But T P T Q = T (P Q) = apple, so V apple T Q + K T Q. (c). V apple T Q( K ), so V is negative definite in the region where K <, which is the strip between the dashed lines in the figure below. and since V, this implies that the dnamic sstem with e as input and ẋ as output is passive (in fact it is dissipative). (b). Let 3 and 4 be respectivel the charge on the capacitor and flu in the inductor in the right-hand branch of the circuit, and define V ( 3, 4 )= Z 4 L() d + Di erentiating w.r.t. t gives V = ẋ 3 e Z 3 C() d. R 4/L ( 4 ). Therefore, defining V = V + V and using the fact that ẋ +ẋ 3 = i (since the

5 9 currents in the two branches of the circuit must sum to i), we get V = Z V = ie L() d + R L ( ) Z 4 and V since V,V. L() d + R L ( 4 ) 4. Opening the switch forces i =, so V = R L ( ) Z C() d + R L ( 4 ) 4 Z 3 C() d and since the level sets {(,, 3, 4 ):V apple V } are bounded (when V is su cientl small), it follows from the local invariant set theorem that the sstem is (locall) asmptoticall stable. Specificall, =(,, 3, 4 ) must converge to the largest invariant set within the set of states such that V =, i.e. = 4 = and ẋ = ẋ 4 =, impling that converges asmptoticall to a stead state such that /C( )= 3 /C( 3 )=and, 4 =. This asmptotic stabilit propert is global if V,V are radiall unbounded. Note also that the same analsis can be applied to an number of LCR branches connected in parallel. (b). Closed-loop stabilit does not appl to nonlinearities bounded b the union of the two sectors defined in part (a), i.e. [ 3, ], since this includes nonlinearities not belonging to either of the sectors [ 3, ] and [, ]. In particular, the disc centred on the real ais and intersecting the real ais at and 3 does not entirel contain the bo in which G(j!) is known to lie, so it cannot be concluded from the circle criterion that the closed loop sstem will be stable. b Im{G(j!)} D(a, b) Figure 3: Bounds on the Nquist plot of G(j!). a Re{G(j!)} 8. (a). The rectangular region containing G(j!) lies within D(a, b) if a = 3 and b =, since D(a, b) is then just touching its corners (Fig. 3). The open-loop sstem is stable, and the circle criterion therefore implies that the closed-loop sstem with u = () will be asmptoticall stable if lies in the sector [ 3, ]. Clearl this is not the onl sector bound for for which the closedloop sstem is guaranteed to be stable b the circle criterion. fact a famil of discs D(a, b) containing G(j!) is generated as a is increased from we need to set a =and b = [, ]. /3, and to allow for the largest possible value of b In, corresponding to sector bounds