Stability of Parameter Adaptation Algorithms. Big picture

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1 ME5895, UConn, Fall 215 Prof. Xu Chen Big picture For ˆθ (k + 1) = ˆθ (k) + [correction term] we haven t talked about whether ˆθ(k) will converge to the true value θ if k. We haven t even talked about whether ˆθ (k) will stay bounded or not! Tools of stability evaluation: Lyapunov-based analysis, or hyperstability theory (topic of this lecture) PAA Stability-1

2 Outline 1. Big picture 2. Hyperstability theory Passivity Main results Positive real and strictly positive real Understanding the hyperstability theorem 3. Procedure of PAA stability analysis by hyperstability theory 4. Appendix Strictly positive real implies strict passivity PAA Stability-2 Hyperstability theory history Vasile M. Popov: born in 1928, Romania retired from University of Florida in 1993 developed hyperstability theory independently from Lyapnov theory PAA Stability-3

3 Hyperstability theory Consider a closed-loop system in Fig. 1 + w u LTI block Nonlinear block v Figure 1: Block diagram for hyperstability analysis The linear time invariant (LTI) block is realized by continuous-time case: discrete-time case: ẋ (t) = Ax (t) + Bu (t) v (t) = Cx (t) + Du (t) x (k + 1) = Ax (k) + Bu (k) v (k) = Cx (k) + Du (k) Hyperstability discusses conditions for nice behaviors in x. PAA Stability-4 Passive systems Definition (Passive system). The system v System w is called passive if t1 w T (t)v (t)dt γ 2, t 1 or w T (k)v (k) γ 2, k= where δ and γ depends on the initial conditions. intuition: t 1 w T (t)v (t)dt is the work/supply done to the system. By conservation of energy, t1 E (t 1 ) E () + w T (t)v (t)dt PAA Stability-5

4 Strictly passive systems If the equality is strict in the passivity definition, with t1 w T (t)v (t)dt γ 2 + δ t1 or in the discrete-time case v T (t)v (t)dt + ε t1 w T (t)w (t)dt, t 1 w T (k)v (k) γ 2 k= + δ v T (k)v (k) + ε w T (k)w (k), k= k= where δ, ε, but not both zero, the system is strictly passive. PAA Stability-6 Passivity of combined systems Fact (Passivity of connected systems). If two systems S 1 and S 2 are both passive, then the following parallel and feedback combination of S 1 and S 2 are also passive S v w S 2 v + S 1 w S 2 PAA Stability-7

5 Hyperstability theory Definition (Hyperstability). The feedback system in Fig. 1 is hyperstable if and only if there exist positive constants δ > and γ > such that x(t) < δ [ x() + γ], t > or x(k) < δ [ x() + γ], k > for all feedback blocks that satisfy the Popov inequality t1 w T (t)v (t)dt γ 2, t 1 or w T (k)v (k) γ 2, k= In other words, the LTI block is bounded in states for any initial conditions for any passive nonlinear blocks. PAA Stability-8 Hyperstability theory Definition (Asymptotic hyperstability). The feedback system below is asymptotically hyperstable if and only if it is hyperstable and for all bounded w satisfying the Popov inequality we have lim k x(k) = + u LTI block w Nonlinear block v PAA Stability-9

6 Hyperstability theory Theorem (Hyperstability). The feedback system in Fig. 1 is hyperstable if and only if the nonlinear block satisfies Popov inequality (i.e., it is passive) and the LTI transfer function is positive real. Theorem (Asymptotical hyperstability). The feedback system in Fig. 1 is asymptotically hyperstable if and only if the nonlinear block satisfies Popov inequality and the LTI transfer function is strictly positive real. intuition: a strictly passive system in feedback connection with a passive system gives an asymptotically stable closed loop. PAA Stability-1 Positive real and strictly positive real Positive real transfer function (continuous-time case): a SISO transfer function G (s) is called positive real (PR) if G (s) is real for real values of s Re{G (s)} > for Re{s} > The above is intuitive but not practical to evaluate. Equivalently, G (s) is PR if 1. G (s) does not possess any pole in Re{s} > (no unstable poles) 2. any pole on the imaginary axis jω does not repeat and the associated residue (i.e., the coefficient appearing in the partial fraction expansion) lim s jω (s jω )G (s) is non-negative 3. ω R where s = jω is not a pole of G (s), G (jω) + G ( jω) = 2Re{G (jω)} PAA Stability-11

7 Positive real and strictly positive real Strictly positive real transfer function (continuous-time case): a SISO transfer function G (s) is strictly positive real (SPR) if 1. G (s) does not possess any pole in Re{s} 1. ω R, G (jω) + G ( jω) = 2Re{G (jω)} > Figure: example Nyquist plot of a SPR transfer function PAA Stability-12 Positive real and strictly positive real discrete-time case A SISO discrete-time transfer function G (z) is positive real (PR) if: 1. it does not possess any pole outside of the unit circle 2. any pole on the unit circle does not repeat and the associated redsidue is non-negative 3. ω π where z = e jω is not a pole of G (z), G(e jω ) + G(e jω ) = 2Re { G(e jω ) } G (z) is strictly positive real (SPR) if: 1. G(z) does not possess any pole outside of or on the unit circle on z-plane 2. ω < π, G(e jω ) + G(e jω ) = 2Re { G(e jω ) } > PAA Stability-13

8 Examples of PR and SPR transfer functions G (z) = c is SPR if c > G (z) = z a 1, a < 1 is asymptotically stable but not PR: 2Re { G ( e jω)} 1 = e jω a + 1 e jω a cosω a = a 2 2a cosω G (z) = z a z, a < 1 is asymptotically stable and SPR PAA Stability-14 Strictly positive real implies strict passivity It turns out [see Appendix]: Lemma: the LTI system G (s) = C (si A) 1 B + D (in minimal realization) is ẋ = Ax + Bu y = Cx + Du passive if G (s) is positive real strictly passive if G (s) is strictly positive real Analogous results hold for discrete-time systems. PAA Stability-15

9 Outline 1. Big picture 2. Hyperstability theory Passivity Main results Positive real and strictly positive real Understanding the hyperstability theorem 3. Procedure of PAA stability analysis by hyperstability theory 4. Appendix Strictly positive real implies strict passivity PAA Stability-16 Understanding the hyperstability theorem Example: consider a mass-spring-damper system mẍ + bẋ + kx = u with a general nonlinear feedback control law 1 G u x (s) = ms 2 + bs + k s G u v (s) = ms 2 + bs + k + u G u v (s) w Nonlinear block v t 1 u (t)v (t)dt is the total energy supplied to the system PAA Stability-17

10 Understanding the hyperstability theorem if the nonlinear block satisfies the Popov inequality t1 w (t)v (t)dt γ, 2 t 1 then from u (t) = w (t), the energy supplied to the system is bounded by t1 u (t)v (t)dt γ 2 energy conservation (assuming v () = v and x () = x ): 1 2 mv kx mv kx 2 = define state vector x = [x 1,x 2 ] T, x 1 = t1 u (t)v (t)dt γ 2 k 2 x, x 2 = m 2 v, then x (t) 2 2 x () γ 2 ( x () 2 + γ ) 2 which is a special case in the hyperstability definition PAA Stability-18 Understanding the hyperstability theorem w + u LTI block Nonlinear block intuition from the example: The nonlinear block satisfying Popov inequality assures bounded supply to the LTI system. Based on energy conservation, the energy of the LTI system is bounded. If the energy function is positive definite with respect to the states, then the states will be bounded. more intuition: If the LTI system is strictly passive, it consumes energy. The bounded supply will eventually be all consumed, hence the convergence to zero for the states. v PAA Stability-19

11 A remark about hyperstability An example of a system that is asymptotically hyperstable and stable: v + 1 s+1 w static gain k (> ) Stable systems may however be not hyperstable: for instance v + 1 s 1 w static gain k (> 1) is stable but not hyperstable ( 1 s 1 is unstable and hence not SPR) PAA Stability-2 Outline 1. Big picture 2. Hyperstability theory Passivity Main results Positive real and strictly positive real Understanding the hyperstability theorem 3. Procedure of PAA stability analysis by hyperstability theory 4. Appendix Strictly positive real implies strict passivity PAA Stability-21

12 PAA stability analysis by hperstability theory step 1: translate the adaptation algorithm to a feedback combination of a LTI block and a nonlinear block, as shown in Fig. 1 step 2: verify that the feedback block satisfies the Popov inequality step 3: check that the LTI block is strictly positive real step 4: show that the output of the feedback block is bounded. Then from the definition of asymptotic hyperstability, we conclude that the state x converges to zero PAA Stability-22 Example: hyperstability of RLS with constant adaptation gain Recall PAA with recursive least squares: a priori parameter update ˆθ (k + 1) = ˆθ (k) + a posteriori parameter update F (k)φ (k) 1 + φ T (k)f (k)φ (k) εo (k + 1) ˆθ (k + 1) = ˆθ (k) + F (k)φ (k)ε (k + 1) We use the a posteriori form to prove that the RLS with F (k) = F is always asymptotically hyperstable. PAA Stability-23

13 Example cont d step 1: transformation to a feedback structure ˆθ (k + 1) = ˆθ (k) + F φ (k)ε (k + 1) parameter estimation error (vector) θ (k) = ˆθ (k) θ: θ (k + 1) = θ (k) + F φ (k)ε (k + 1) a posteriori prediction error ε (k + 1) = y (k + 1) ˆθ T (k + 1)φ (k): ε (k + 1) = θ T φ (k) ˆθ T (k + 1)φ (k) = θ T (k + 1)φ (k) PAA Stability-24 Example cont d step 1: transformation to a feedback structure PAA equations: equivalent block diagram: θ (k + 1) = θ (k) + F φ (k)ε (k + 1) ε (k + 1) = θ T (k + 1)φ (k) + 1 ε(k+1) θ T (k+1)φ(k) θ(k+1) + + F z 1 φ(k) φ(k) PAA Stability-25

14 Example cont d step 2: Popov inequality for the feedback nonlinear block, need to prove θ T (k + 1)φ (k)ε (k + 1) γ, 2 k= θ (k + 1) θ (k) = F φ (k)ε (k + 1) gives = θ T (k + 1)φ (k)ε (k + 1) k= ( θ T (k + 1)F 1 θ (k + 1) θ T (k + 1)F 1 θ (k)) k= PAA Stability-26 Example cont d step 2: Popov inequality adding and substracting terms gives θ T (k + 1)φ (k)ε (k + 1) k= = ( θ T (k + 1)F 1 θ (k + 1) θ T (k + 1)F 1 θ (k)) k= = k= ( θ T (k + 1)F 1 θ (k + 1) ± 1 2 θ T (k)f 1 θ (k) ) θ T (k + 1)F 1 θ (k) PAA Stability-27

15 Example cont d step 2: Popov inequality Combining terms yields + ( θ T (k + 1)F 1 θ (k + 1) ± 1 ) k= 2 θ T (k)f 1 θ (k) θ T (k + 1)F 1 θ (k) 1 = ( θ T (k + 1)F 1 θ (k + 1) θ T (k)f 1 θ (k)) k= 2 k= 1 2 ( θ T (k + 1)F 1 θ (k + 1) 2 θ T (k + 1)F 1 θ (k) + θ T (k)f 1 θ (k)) } {{ } [ ] [ ] is equivalent to ( ) T ( ) F 1/2 θ (k + 1) F 1/2 θ (k) F 1/2 θ (k + 1) F 1/2 θ (k) PAA Stability-28 Example cont d step 2: Popov inequality the underlined term is also lower bounded: hence 1 ( θ T (k + 1)F 1 θ (k + 1) θ T (k)f 1 θ (k)) k= 2 = 1 2 θ T ( + 1)F 1 θ ( + 1) 1 2 θ T ()F 1 θ () 1 2 θ T ()F 1 θ () θ T (k + 1)φ (k)ε (k + 1) 1 k= 2 θ T ()F 1 θ () PAA Stability-29

16 Example cont d step 3: SPR condition w(k) + 1 ε(k+1) Nonlinear Block the identity block G ( z 1) = 1 is always SPR from steps 1-3, we conclude the adaptation system is asymptotically hyperstable this means ε (k + 1) will be bounded, and if w (k) is further shown to be bounded, ε (k + 1) converge to zero PAA Stability-3 Example cont d step 4: boundedness of the signal w(k) + 1 ε (k + 1) Nonlinear Block ε (k + 1) = w (k), so w (k) is bounded if ε (k + 1) is bounded thus hyperstability theorem gives that ε (k + 1) converges to zero PAA Stability-31

17 Example cont d intuition w(k) + 1 ε (k + 1) Nonlinear Block For this simple case, we can intuitively see why ε (k + 1) : Popov inequality gives k= ε (k + 1)w (k) γ2 ; as w (k) = ε (k + 1), so ε 2 (k + 1) γ 2 k= Let. ε (k + 1) must converge to to ensure the boundedness. PAA Stability-32 One remark Recall ε (k + 1) = ε o (k + 1) 1 + φ T (k)f φ (k) ε (k + 1) does not necessarily mean ε o (k + 1) need to show φ (k) is bounded: for instance, the plant needs to be input-output stable for y (k) to be bounded see details in Landau et al, Adaptive Control, 2nd Ed, Springer There are different PAAs with different stability and convergence requirements PAA Stability-33

18 Summary 1. Big picture 2. Hyperstability theory Passivity Main results Positive real and strictly positive real Understanding the hyperstability theorem 3. Procedure of PAA stability analysis by hyperstability theory 4. Appendix Strictly positive real implies strict passivity PAA Stability-34 Exercise In the following block diagrams, u and y are respectively the input and output of the overall system; h ( ) is a sector nonlinearity satisfying 2 x < h (x) < 5 x Check whether they satisfy the Popov inequality. u y u h ( ) y h ( ) u + y + u 1 as + 1, a > y PAA Stability-35

19 *Kalman Yakubovich Popov Lemma Kalman Yakubovich Popov (KYP) Lemma connects frequency-domain SPR conditions and time-domain system matrices: Lemma: Consider G (s) = C (si A) 1 B + D where (A,B) is controllable and (A,C) is observable. G (s) is strictly positive real if and only if there exist matrices P = P T, L, and W, and a positive constant ε such that PA + A T P = L T L εp PB = C T L T W W T W = D + D T Proof: see H. Khalil, Nonlinear Systems, Prentice Hall PAA Stability-36 *Kalman Yakubovich Popov Lemma Discrete-time version of KYP lemma: replace s with z and replace the matrix equalities with A T PA P = L T L εp B T PA C = K T L D + D T B T PB = K T K PAA Stability-37

20 *Strictly positive real implies strict passivity From KYP lemma, the following result can be shown: Lemma: the LTI system G (s) = C (si A) 1 B + D (in minimal realization) is ẋ = Ax + Bu y = Cx + Du passive if G (s) is positive real strictly passive if G (s) is strictly positive real Analogous results hold for discrete-time systems. PAA Stability-38 *Strictly positive real implies strict passivity Proof: Consider V = 1 2 x T Px: V (x (T )) V (x ()) = V dt = [ 1 ( ) ] 2 x T A T P + PA x + u T B T Px dt Let u and y be the input and the output of G (s). KYP lemma gives V (x (T )) V (x ()) = [ 1 ( ) ] 2 x T L T L + εp x + u T B T Px dt u T ydt = = = u T (Cx + Du)dt = [ ( u T B T P + W T L [ ( ) u T B T P + W T L ) x + 1 ( ) 2 ut D + D T u [ ( ) u T B T P + W T L x + 1 ] 2 ut W T Wu dt ] x + u T Du dt ] dt PAA Stability-39

21 *Strictly positive real implies strict passivity hence = = 1 2 u T ydt V (x (T )) + V (x ()) [ ( ) u T B T P + W T L x ut W T Wu + 1 ( ) 2 x T L T L + εp x u T B T Px (Lx + Wu) T (Lx + Wu)dt εx T Px 1 2 εx T Px > ] dt PAA Stability-4