Control in The Presence of Uncertainty

Control in The Presence of Uncertainty Adaptive Estimation and Control Dispense del Corso di Controllo Robusto e Adattativo A.A. 2024/2025 Alessandro Astolfi Dipartimento di Ingegneria Civile e Ingegneria Informatica Università di Roma Tor Vergata Parameter Estimation In the first part of the course we have introduced some basic analysis and design tools and we have seen how it is possible to construct models of linear, and some nonlinear, dynamical systems which are parameterized in terms of parameters and available, measured, signals. In what follows we assume that the parameters are constant, that is their time derivative is identically equal to zero. Occasionally we may discuss the implication of the fact that parameters may vary with time. The availability of dynamic models allows exploiting simple time-domain and frequency-domain techniques to (asymptotically) estimate the parameters using dynamical systems: the so-called adaptive estimators. The use of dynamical systems as mechanism to perform the estimate lends itself to the on-line implementation of the proposed estimation schemes. AA (DICII) RAC - Adaptive Estimation and Control 1 / 102 Parameter Estimation The essential idea behind any parameter estimation method is the comparison between the observed system response y (t) = y (θ, t) and the response ŷ (θ̂, t) of a parameterized model, the structure of which is the same as the model that has generated y (t). The comparison is used to continuously update the parameter estimate θ̂ in such a way that y (t) and ŷ (θ̂, t) are close, for t sufficiently large. Note that, without additional assumptions, it is in general not possible to conclude that y (θ, t) = ŷ (θ̂, t), for all t ≥ 0, implies θ̂(t) = θ, or even that θ̂(t) is close to θ. The on-line estimation procedure thus involves three steps: model parameterization; generation of the update law for θ̂; design of the input signal to guarantee convergence of θ̂(t) to θ. Note that the final step can be accomplished only if one is allowed to run experiments on the plant. However, we will see that while parameter convergence is important in estimation problem it is actually not important in adaptive control, since one could achieve adaptive stabilization without recovering the actual value of the parameter θ. AA (DICII) RAC - Adaptive Estimation and Control 2 / 102 Parameter Estimation – One Unknown Parameter Consider the simplest possible linear, SISO, plant, namely the plant described by the input-output relation y (t) = θu(t), with input u ∈ L∞ , output y (t) (∈ L∞ ), and unknown scalar parameter θ. Assume that u and y are measured. The goal of a parameter estimation algorithm is to generate an estimate for θ. If the measured data were noise-free, one could simply write θ̂(t) = y (t) , u(t) provided u(t) ≠ 0, for all t. This naive approach is undesirable since the division may be ill-conditioned and the effect of measurement noise is amplified by the division. Note that one could consider alternative versions of this naive approach, such as θ̂(t) = W (s) y , W (s) u where W (s) is any L∞ stable transfer function (note that one could also use other operators, such as a moving average, or a windowed integrator or a time delay). Recall that the notation W (s) y is a shorthand for L−1 (W (s)L(y (t))). AA (DICII) RAC - Adaptive Estimation and Control 3 / 102 Parameter Estimation – One Unknown Parameter To generate a recursive, that is dynamic, scheme we construct a predicted output as ŷ (t) = θ̂(t)u(t) and the prediction, or estimation, error ϵ1 = y − ŷ = y − θ̂u = (θ − θ̂)u = −θ̃u, where θ̃ = θ̂ − θ is the parameter estimation error. Note that ϵ1 is measurable and θ̃˙ = θ̂˙ = w , where w is to be designed to provide the on-line parameter update law. The update law θ̂˙ = w is obtained by minimizing a cost criteria involving ϵ1 with respect to θ̂. The simplest possible cost is J(θ̂) = ϵ21 (y − θ̂u)2 = , 2 2 which can be continuously minimized using the gradient system θ̂˙ = −γ∇J(θ̂) = γ(y − θ̂u)u = γϵ1 u = −γ θ̃u 2 , where γ > 0 is the so-called adaptation gain. AA (DICII) RAC - Adaptive Estimation and Control 4 / 102 Parameter Estimation – One Unknown Parameter To assess the properties of the proposed adaptation mechanism consider the Lyapunov function θ̃2 V (θ̃) = 2γ and recall that θ̃˙ = −γu 2 θ̃, hence V̇ = −u 2 θ̃2 = −ϵ21 ≤ 0, which implies that the equilibrium θ̃ = 0 is stable and that ϵ1 ∈ L2 . In addition, since u ∈ L∞ , θ̃˙ ∈ L∞ ∩ L2 . This, however, does not guarantee convergence of θ̃˙ and ϵ1 to zero. ˙ If u̇ ∈ L∞ then ϵ̇1 ∈ L∞ and θ̃¨ ∈ L∞ , hence lim ϵ1 (t) = 0 and lim θ̃(t) = 0. t→∞ t→∞ Can we conclude convergence of θ̃(t) and, if so, what is its limit? AA (DICII) RAC - Adaptive Estimation and Control 5 / 102 Parameter Estimation – One Unknown Parameter Recall that, along the trajectories of the system, V (θ̃(t)) is non-increasing and it is bounded from below, since it is a positive definite function. Hence it has a limit, that is √ lim V (θ̃(t)) = V∞ ⇒ lim θ̂(t) = θ ± 2γV∞ . t→∞ t→∞ This implies that θ̂(t) converges to the true value θ if and only if V∞ = 0. This condition is not easy to verify, hence a different type of analysis is required. Recall that θ̃˙ = −γu 2 θ̃, hence θ̃(t) = e −γ t ∫ 0 u 2 (τ )dτ θ̃(0). As a result θ̃(t) converges to zero if and only if u ∈/ L2 . Note also that the estimate improves monotonically over time, that is ∣θ̃(t)∣ is monotonically non-increasing. To have exponential convergence of θ̃(t) to zero one needs the more restrictive condition 0 < α0 T0 ≤ ∫ t t+T0 u 2 (τ )dτ, for all t ≥ 0, and some α0 > 0 and T0 > 0. AA (DICII) RAC - Adaptive Estimation and Control 6 / 102 Parameter Estimation – The Scalar Case Consider the scalar system ẋ = −ax + bu, where the parameters a > 0 and b are constant and unknown, and u and x are available for measurement. The condition a > 0 implies that for any u ∈ L∞ one has x ∈ L∞ . Our objective is to generate update laws, to update the parameter estimates â and b̂, driven by the estimation error ϵ1 = x − x̂, where x̂ is an estimate of the state resulting from the use of â and b̂ (consistently with a certainty equivalence architecture). The state estimate x̂ is generated by a copy of the equation of the system, that is x̂˙ = −âx̂ + b̂u. (P) This equation is known as the parallel model. Alternatively, one could rewrite the plant as ẋ = −am x + (am − a)x + bu ⇔ x(s) = 1 [(am − a)x + bu] , s + am x̂(s) = 1 [(am − â)x + b̂u] . s + am yielding the so-called series parallel model: x̂˙ = −am x̂ +(am − â)x + b̂u AA (DICII) ⇔ RAC - Adaptive Estimation and Control (SP) 7 / 102 Parameter Estimation – The Scalar Case The estimation error ϵ1 = x − x̂ satisfies the differential equation ϵ̇1 = −aϵ1 + ãx̂ − b̃u for the parallel model (P), and the differential equation ϵ̇1 = −am ϵ1 + ãx − b̃u. for the series parallel model (SP). Consider now the storage function S(ϵ1 ) = ϵ21 and note that 2 Ṡ = −aϵ21 + ϵ1 ãx̂ − ϵ1 b̃u = −aϵ21 + [ ϵ1 x̂ −ϵ1 u ] [ ã ], b̃ for the parallel model (P), and Ṡ = −am ϵ21 + ϵ1 ãx − ϵ1 b̃u = −am ϵ21 + [ ϵ1 x −ϵ1 u ] [ ã ], b̃ for the series parallel model (SP). AA (DICII) RAC - Adaptive Estimation and Control 8 / 102 Parameter Estimation – The Scalar Case The time derivatives of the storage function reveal that the (P) and (SP) estimation error equations describe passive systems with input (ã, b̃) and outputs (ϵ1 x̂, −ϵ1 u) and (ϵ1 x, −ϵ1 u), respectively. Interconnecting each of these passive systems to the passive system ξ˙ = v , with ξ(t) ∈ IR 2 1 and v (t)IR 2 and storage function Sξ (ξ) = ξ ⊺ ξ, via the interconnection equations (note 2 the negative feedback, consistent with the Passivity Theorem) v =[ ϵ1 x̂ ], −ϵ1 u [ ã ] = −ξ, b̃ (P) v =[ ϵ1 x ], −ϵ1 u [ ã ] = −ξ, b̃ (SP) yields ϵ̇1 = −aϵ1 + ãx̂ − b̃u, ã˙ = −ϵ1 x̂, b̃˙ = ϵ1 u, (P) ϵ̇1 = −am ϵ1 + ãx − b̃u, ã˙ = −ϵ1 x, b̃˙ = ϵ1 u. (SP) and These equations are not implementable, because ã and b̃ are not known. AA (DICII) RAC - Adaptive Estimation and Control 9 / 102 Parameter Estimation – The Scalar Case ˙ They, however, allow implementing the update laws (recall that ã˙ = â˙ and b̃˙ = b̂) â˙ = −ϵ1 x̂, b̂˙ = ϵ1 u, (P) â˙ = −ϵ1 x, ˙ b̂ = ϵ1 u. (SP) For both systems the overall storage function is ST (ϵ1 , ã, b̃) = 1 2 (ϵ1 + ã2 + b̃ 2 ) 2 and it is such that ṠT = −aϵ21 ≤ 0 (P) and ṠT = −am ϵ21 ≤ 0 (SP) (recall that a > 0 and am > 0 and note that the (SP) system is passive by design). We conclude that ϵ1 ∈ L2 ∩ L∞ , ã ∈ L∞ , b̃ ∈ L∞ , and x̃ ∈ L∞ , provided u ∈ L∞ . These imply lim ϵ1 (t) = 0, t→∞ ˙ lim â(t) = 0, t→∞ ˙ lim b̂(t) = 0, t→∞ lim (ã2 (t) + b̃ 2 (t)) = 2 lim S(t), t→∞ t→∞ but do not imply convergence of ã and/or b̃, which requires additional assumptions on u. This procedure is called signal chasing and it is widely used in adaptive systems. AA (DICII) RAC - Adaptive Estimation and Control 10 / 102 Parameter Estimation – The Scalar Case To summarize, we have seen two schemes for estimating the parameters of the plant ẋ = −ax + bu, with a > 0 and u ∈ L∞ , namely (P) (SP) x̂˙ = −âx̂ + b̂u x̂˙ = −am x̂ + (am − â)x + b̂u â˙ = −(x − x̂)x̂ â˙ = −(x − x̂)x b̂˙ = (x − x̂)u b̂˙ = (x − x̂)u Note that the update laws can be scaled, by multiplying the right-hand sides by constants 1 ã2 b̃ 2 γ1 > 0 and γ2 > 0 and exploiting the storage function ST (ϵ1 , ã, b̃) = (ϵ21 + + ). 2 γ1 γ2 The two schemes differ in their performance in the presence of noise. If x is corrupted by noise, that is one measures x + ν then the scheme (SP) may generate biased estimates because of the presence of the x 2 term in the â˙ equation. AA (DICII) RAC - Adaptive Estimation and Control 11 / 102 Parameter Estimation – The Scalar Case Exercise (Assignment 1) Consider the system ẋ = −ax + bu. Let a = 1 and b = 2. Run simulations for the adaptive estimation algorithms based on the (P) and (SP) models. Evaluate the performance of the algorithms for various input signals, as a function of the adaptive gains, and of the presence, or otherwise, of noise on the measurements. 2πt ). This is the case, for Suppose that a is slowly time varying, that is a = 1 + 0.1 sin ( 24×3600 example, if a changes during the day as a function of the ambient temperature. Assess the performance of the designed adaptive estimation algorithms in this scenario. AA (DICII) RAC - Adaptive Estimation and Control 12 / 102 Parameter Estimation – The General Case Consider the linear system ẋ = Ap x + Bp u, with state x(t) ∈ IR and input u(t) ∈ IR m , both available for measurement, and Ap ∈ IR n×n and Bp ∈ IR n×m unknown. Suppose that Ap is asymptotically stable and u ∈ L∞ . n Define the parallel model (P) x̂˙ = Âp x̂ + B̂p u, with state x̂(t) ∈ IR n , where Âp (t) ∈ IR n×n and B̂p (t) ∈ IR n×m are the estimates of Ap and Bp at time t to be generated by update laws. Similarly we could consider the series parallel model (SP) x̂˙ = Am x̂ + (Âp − Am )x + B̂p u. In what follows we focus on the parallel model: similar considerations can be carried out for the series parallel model. AA (DICII) RAC - Adaptive Estimation and Control 13 / 102 Parameter Estimation – The General Case The estimation error equation for the parallel model is ϵ̇1 = Ap ϵ1 − Ãp x̂ − B̃p u, where ϵ1 = x − x̂, Ãp = Âp − Ap and B̃p = B̂p − Bp . Consider the storage function S(ϵ1 ) = ϵ⊺1 Pϵ1 , where P = P ⊺ > 0 is the solution of the Lyapunov equation A⊺p P + PAp = −I , and note that Ṡ = ϵ̇⊺1 Pϵ1 + ϵ⊺1 P ϵ̇1 = ϵ⊺1 (A⊺p P + PAp )ϵ1 − 2ϵ⊺1 P Ãp x̂ − 2ϵ⊺1 P B̃p u. Recall the properties of the trace operator (A and B matrices, v and w vectors) tr(AB) = tr(BA), As a result tr(A + B) = tr(A) + tr(B), tr(v w ⊺ ) = w ⊺ v . Ṡ = −ϵ⊺1 ϵ1 − 2 tr(Ã⊺p Pϵ1 x̂ ⊺ ) − 2 tr(B̃p⊺ Pϵ1 u ⊺ ), which suggests that the estimation error system is passive from the input (Ãp , B̃p ) to the output (−Pϵ1 x̂ ⊺ , −Pϵ1 u ⊺ ). AA (DICII) RAC - Adaptive Estimation and Control 14 / 102 Parameter Estimation – The General Case Consider now the system Ξ̇ = W , with Ξ(t) ∈ R n×q and W (t) ∈ IR n×q , with q = n or q = m, and the storage function SΞ = tr ( Ξ⊺ PΞ ). γ A direct computation yields ṠΞ = 2 tr ( Ξ⊺ P Ξ̇ ) γ which suggests that the system is passive from the input P Ξ̇ = PW to the output Ξ. Exploiting the two passivity properties we conclude that the update laws Â˙ p = γ1 ϵ1 x̂ ⊺ , B̂˙ p = γ2 ϵ1 u ⊺ , with γ1 > 0 and γ2 > 0 yield ϵ1 ∈ L2 ∩ L∞ , AA (DICII) Âp ∈ L∞ , B̂p ∈ L∞ , x̂ ∈ L∞ , RAC - Adaptive Estimation and Control lim ϵ1 = 0. t→∞ 15 / 102 Parameter Estimation – The General Case Exercise Show that the function tr ( Ξ⊺ MΞ ) γ with M = M ⊺ > 0 is positive definite in Ξ, that is the function is zero for Ξ = 0 and positive otherwise. Exercise Consider the series parallel model and derive the update laws for the estimates Âp and B̂p . Show that, since the matrix Am is known, one could use the storage function SΞ = tr ( Ξ⊺ Ξ ). γ Explain why this storage function cannot be used to derive an update law in the case in which the parallel model is used. AA (DICII) RAC - Adaptive Estimation and Control 16 / 102 Parameter Estimation – Sufficiently Rich Signals Consider again the scalar system ẋ = −ax + bu, with a and b constant and unknown. Assume that a > 0 and that u ∈ L∞ and let x(0) be the initial state. Consider the abstract problem of determining for which classes of inputs it is possible, regardless of any specific algorithm, to estimate a and b by processing input/output data, that is measurements of u and x. Clearly, if u = 0 one has x(t) = e −at x(0), which shows that the measured signals do not carry any information on b and the information on a disappears exponentially fast. In principle, one could obtain a from the relation a=− x(t) 1 ln > 0, T x(t − T ) for t ≥ T , where T > 0 is a user selected parameter. This relation becomes rapidly numerically ill-conditioned, it is extremely sensitive to measurement noise, and cannot be used if x(0) = 0. AA (DICII) RAC - Adaptive Estimation and Control 17 / 102 Parameter Estimation – Sufficiently Rich Signals Assume now that u = u0 , with u0 constant. Then x(t) = e −at (x(0) − b b u0 ) + u0 , a a which reveals that one could estimate b/a (the DC gain of the system). Again one could estimate a using delayed measurements of x(t) in what would be a non-robust scheme. Assume, finally, that u(t) = sin ω0 t, with ω0 ≠ 0. Then x(t) = A sin(ω0 t + ϕ) + exp, A= ∣b∣ , ∣jω0 + a∣ ϕ = (sign(b) − 1) π ω0 − tan−1 , 2 a where ”exp” indicates exponentially decaying terms, which reveals that one could determine a and b from measurements of the steady state response of the system. This example demonstrates that the on-line estimate of a and b is possible only if the output signal carries sufficient information on the parameters and, in turn, this information is the result of a particular selection of input signal. AA (DICII) RAC - Adaptive Estimation and Control 18 / 102 Parameter Estimation – Sufficiently Rich Signals Definition A signal u ∶ IR ≥0 → IR is sufficient rich of order n if it contains at least frequencies. n distinct 2 Exercise m Show that the signal u(t) = ∑ Ai sin ωi t, with Ai ≠ 0, for all i, and ωi ≠ ωk , for all i ≠ k, i=1 is sufficiently rich of order n, for any n ≤ 2m. A more general definition of richness that includes signals which are not linear combination of sinusoids can also be given. AA (DICII) RAC - Adaptive Estimation and Control 19 / 102 Parameter Estimation – Sufficiently Rich Signals Definition A signal u ∶ IR ≥0 → IR m is said to be stationary if the limit Ru (t) = lim ∫ T →∞ t0 +T t0 u(τ )u ⊺ (t + τ )dτ exists, uniformly in t0 . The matrix Ru (t) is called the auto-covariance of u. The matrix Ru (t) is positive semi-definite and it is Fourier transformable, that is Su (ω) = ∫ ∞ −∞ e −jωτ Ru (τ )dτ, where Su (ω) is the so-called spectral measure of u, is well defined. Definition A stationary signal u ∶ IR ≥0 → IR is sufficiently rich of order n if the support of the spectral measure Su (ω) of u contains at least n points. AA (DICII) RAC - Adaptive Estimation and Control 20 / 102 Parameter Estimation – Sufficiently Rich Signals Definition A signal u ∶ IR ≥0 → IR m is persistently exciting if there exist a constant t0 and positive constants T and α such that 1 T ∫t t+T u(τ )u ⊺ (τ )dτ ≥ αI > 0, for all t ≥ t0 . Consider now the equation ϕ = H(s)u, with H(s) a proper and stable transfer function and ϕ(t) ∈ IR n . Theorem Let u ∶ IR ≥0 → IR be a stationary signal. Assume that H(jω1 ), ⋯, H(jωn ) are linearly independent (on CI n ) for all ωi ∈ IR, with ωi ≠ ωk for i ≠ k. Then ϕ is persistently exciting if and only if u is sufficiently rich of order n. AA (DICII) RAC - Adaptive Estimation and Control 21 / 102 Parameter Estimation – Sufficiently Rich Signals Exercise Consider the linear, single-input, single-output, system described by the relation y (s) = kG (s)u(s), with s2 + 1 G (s) = , (s + 1)3 and k an unknown parameter. Show that the system can be parameterized as y = θ ϕ, where θ = k and ϕ = G (s)u. Show that if u is sufficiently rich of order 1 then ϕ is not persistently exciting, while it is persistently exciting if u is sufficiently rich of order n ≥ 2. AA (DICII) RAC - Adaptive Estimation and Control 22 / 102 Parameter Estimation – Sufficiently Rich Signals Theorem Consider the system ẋ = Ap x + Bp u, with x(t) ∈ IR and u(t) ∈ IR . Assume that the state x is measurable, that u ∈ L∞ , and that Ap and Bp are constant matrices to estimate. Assume, in addition, that Ap is asymptotically stable and that the pair (Ap , Bp ) is reachable. n m Consider the (P) and the (SP) parameter estimation algorithm. Let m = 1. If the input u is sufficiently rich of order n + 1 then the estimates Âp and B̂p converge exponentially fast to the unknown parameters Ap and Bp . Let m > 1. If each input ui is sufficiently rich of order n + 1 and uncorrelated, that is each ui contains different angular frequencies, then the estimates Âp and B̂p converge exponentially fast to the unknown parameters Ap and Bp . AA (DICII) RAC - Adaptive Estimation and Control 23 / 102 Parameter Estimation – SPR Design The foregoing parameter estimation algorithms requires full state measurements. In practice this is unfeasible, hence one has to develop estimation methods which rely solely on input and output signals. To this end consider the linear parametric model z = W (s)θ⊺ ψ, derived on the basis of input-output signals, and assume that W (s) is stable and ψ ∈ L∞ . Since θ is a constant vector, for any transfer function L(s) one could write the alternative, equivalent from an input-output perspective, model z = W (s)L(s)θ⊺ ϕ, where ϕ = L−1 (s)ψ. Suppose L(s) is such that L−1 (s) is stable and proper and W (s)L(s) is strictly proper and SPR. Note that it is always possible to manipulate the model to satisfy the above conditions: this may require the redefinition of the filtered signals z and ϕ. AA (DICII) RAC - Adaptive Estimation and Control 24 / 102 Parameter Estimation – SPR Design Let θ̂(t) be the estimate of θ at time t. Then the estimate of z is given by ẑ = W (s)L(s)θ̂⊺ ϕ and the estimation error is defined as ϵ = z − ẑ = W (s)L(s)(−θ̃⊺ ϕ). Let ė = Ae + B(−θ̃⊺ ϕ), ϵ = Ce, be a state-space representation of the error equation, i.e. W (s)L(s) = C (sI − A)−1 B. By SPR-ness of the transfer function W (s)L(s) there exist matrices P = P ⊺ > 0 and Q = Q ⊺ > 0 such that A⊺ P + PA = −Q, PB = C ⊺ . Consider now the storage function S(e) = 1 ⊺ e Pe and note that 2 1 1 1 Ṡ = − e ⊺ Qe + e ⊺ PB(−θ̃⊺ ϕ) = − e ⊺ Qe + ϵ(−θ̃⊺ ϕ) = − e ⊺ Qe + (−ϵϕ⊺ )θ̃ 2 2 2 which shows that the error system is strictly passive from the input θ̃ to the output −ϵϕ. AA (DICII) RAC - Adaptive Estimation and Control 25 / 102 Parameter Estimation – SPR Design Consider now the interconnection of the error system with the passive system ξ˙ = Γw where Γ = Γ⊺ > 0, via the interconnection equations w = −ϵϕ⊺ , These yield θ̃˙ = −ξ˙ = −Γ(−ϵϕ⊺ ) = Γϕϵ θ̃ = −ξ. Ô⇒ θ̂˙ = Γϕϵ. One can therefore conclude that e ∈ L∞ , ϵ ∈ L∞ , θ̂ ∈ L∞ and e ∈ L2 . Theorem Suppose that ϕ̇ ∈ L∞ and that ϕ is persistently exciting. Then θ̂(t) converges to θ exponentially fast. AA (DICII) RAC - Adaptive Estimation and Control 26 / 102 Parameter Estimation – SPR Design Exercise Consider the system y (s) = b1 s + b0 u(s), (s + 1)(s + 2) where b1 and b0 have to be estimated. Show that the system can be described by the parametric model y= 1 θ⊺ ψ = W (s)θ⊺ ψ, (s + 1)(s + 2) where θ⊺ = [b1 b0 ] and ψ = [u̇ u]⊺ . Select L(s) = s + 2 and show that W (s)L(s) is SPR and that the parameterized model can be written as 1 ⊺ θ ϕ. y= s +1 Determine explicitly the term ϕ and show that is it is measurable. AA (DICII) RAC - Adaptive Estimation and Control 27 / 102 Parameter Estimation – Gradient Identifiers An alternative to the class of parameter identifiers which rely on the interconnection of passive systems can be obtained exploiting the gradient method. The idea of the gradient estimators is to develop an algebraic estimation error equation from which one could define a cost function to be continuously minimized. Consider again the parameterized model z = W (s)θ⊺ ψ and note that, since θ is constant, one could write the equivalent model z = θ⊺ ϕ, where ϕ = W (s)ψ. This justifies the definition of the estimate ẑ = θ̂⊺ ϕ, of the estimation error ϵ = z − ẑ = θ⊺ ϕ − θ̂⊺ ϕ = −θ̃⊺ ϕ, and of the instantaneous cost function J(θ̂) = ϵ2 (z − θ̂⊺ ϕ)2 = . 2 2 Note that other cost functions could be considered, for example a cost computed on the basis of a weighted integral of the square of the estimation error. AA (DICII) RAC - Adaptive Estimation and Control 28 / 102 Parameter Estimation – Gradient Identifiers Note that the cost function J(θ̂) is quadratic in the unknown θ̂, hence it is (globally) convex and the gradient method provides a globally converging (under proper assumptions) minimization algorithm. The algorithm is obtained by defining a trajectory that evolves on the basis of the differential equation θ̂˙ = −Γ∇J(θ̂) = Γ(z − θ̂⊺ ϕ)ϕ = Γϕϵ, where Γ = Γ⊺ > 0. The gradient method yields an adaptive law which has the same form as that obtained using the SPR design. This is the consequence of the fact that one could take L(s) = W −1 (s) in the SPR design. Theorem Consider the gradient algorithm and assume ϕ ∈ L∞ . Then ϵ ∈ L∞ ∩ L2 , θ̂ ∈ L∞ , θ̂˙ ∈ L∞ , and θ̂˙ ∈ L2 . In addition, if ϕ is persistently exciting then θ̂(t) converges to θ exponentially. AA (DICII) RAC - Adaptive Estimation and Control 29 / 102 Parameter Estimation – Gradient Identifiers Proof. We prove only the first claim. To begin with note that, since θ is constant, one has θ̃˙ = Γϕϵ. Consider now the Lyapunov function V (θ̃) = from which we prove the claim. θ̃⊺ Γ−1 θ̃ and note that V̇ = θ̃⊺ ϕϵ = −ϵ2 ≤ 0, 2 Interestingly, the convergence proof of the gradient method does not use J(θ̂) as Lyapunov function, which may therefore be increasing. Explain why this is the case! One major disadvantage of the gradient algorithm, hence also of the algorithm based on the SPR design, is that the parameter estimates are coupled, that is one can only prove that the function V (θ̃(t)) is non-increasing, whereas one may be interested in a parameter-wise property, that is the property that each ∣θ̃i (t)∣ be non-increasing. Before addressing this issue, we show that the gradient identifiers can incorporate prior knowledge on the parameters. AA (DICII) RAC - Adaptive Estimation and Control 30 / 102 Parameter Estimation – Gradient Identifiers with Projection Suppose, now, that some a priori information on the parameter is available and that this can be expressed as θ ∈ S, where S = {θ ∈ Θ ∣ g (θ) ≤ 0}, for some scalar function g . Suppose, in addition, that S is a convex set. Let S 0 be the interior of S and δS the boundary of S. The idea of the so-called gradient projection method is to modify the gradient-based adaptive scheme to guarantee that the parameter estimates, if initialized in S, that is θ̂(0) ∈ S, provides an estimate that remains in S for all t ≥ 0. To this end, let (note that ∇⊺g Γ ∇g is a scalar) ⎧ ⎪ Γϕϵ, ⎪ ⎪ ˙θ̂ = ⎪ ⎨ ⎪ −1 ⎪ ⊺ ⊺ ⎪ ⎪ ⎩ Γϕϵ − Γ ∇g [∇ g Γ ∇g ] ∇ g Γϕϵ, if θ̂(t) ∈ S 0 or θ̂(t) ∈ δS and (Γϕϵ)⊺ ∇g ≤ 0, otherwise. The projection term can be used with any parameter estimation algorithm. AA (DICII) RAC - Adaptive Estimation and Control 31 / 102 Parameter Estimation – Gradient Identifiers with Projection Theorem The gradient algorithm with projection retains all the properties established for the algorithm without projection and in addition it guarantees that θ̂(t) ∈ S for all t ≥ 0, provided θ̂(0) ∈ S. Proof. Note that θ̂(t) can escape the set S only through its boundary δS. Note now that on δS one has θ̂˙⊺ ∇g ≤ 0, that is the vector θ̂˙ points either inside S or along the tangent plane to δS at any point of δS. The conclusion therefore follows by convexity of S. Consider now the same Lyapunov function used for the method without projection and note that the projection term is such that (recall that θ̃˙ = θ̂˙ and that θ̃⊺ ∇g = (θ̂ − θ)⊺ ∇g ) − θ̃⊺ Γ−1 Γ∇g ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ≥ 0 by convexity of S × 1 ∇⊺ g Γϕϵ ∇⊺g Γ∇g ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ≥ 0 by definition ≤ 0, therefore the term introduced by the project can only make V̇ more negative. AA (DICII) RAC - Adaptive Estimation and Control 32 / 102 Parameter Estimation – DREM The method of Dynamic Regression Extension and Mixing (DREM) is a method which allows obtaining decoupled estimators for each component of the parameter vector θ. This is achieved by filtering and post-processing the measured signals. To begin with consider the parametric model z = θ⊺ ϕ, with θ ∈ IR q . The first step in DREM is to introduce a linear, single-input, q-output, bounded-input bounded-output stable filter H and to define the vector Z and the matrix Φ(t) ∈ IR q×q as Z = Hz, Φ = HϕT . These signals satisfy the q relations Z = Φθ. Recall now that for any square matrix M one has adj{M}M = detM I . Hence adj{Φ}Z = adj{Φ}Φ θ = detΦ θ yielding, by setting Z̃ = adj{Φ}Z, the q decoupled equations Z̃i = detΦ θi . AA (DICII) RAC - Adaptive Estimation and Control 33 / 102 Parameter Estimation – DREM Theorem Consider the scalar equations Z̃i = detΦ θi and the gradient estimator θ̂˙i = γi detΦ (Z̃i − detΦ θ̂i ) , with γi > 0. Then the parameter estimation error θ̃ is such that θ̃˙i = −γi (detΦ)2 θ̃i ⇒ ∣θ̃i (t1 )∣ ≤ ∣θ̃i (t0 )∣, t1 ≥ t0 . Moreover lim θ̃i (t) = 0 t→∞ AA (DICII) ⇔ detΦ ∈/ L2 . RAC - Adaptive Estimation and Control 34 / 102 Parameter Estimation – DREM The filter H can be constructed considering q filters of the form Hi (s) = bi , s + τi or Hi (s) = e −τi s , with bi ≠ 0, for all i ∈ [1, q], τi > 0, for all i ∈ [1, q], and τi ≠ τj for all [1, q] ∋ i ≠ j ∈ [1, q]. Exercise (Assignment 2) Consider the system ẋ = −ax + bu, with a > 0, b ≠ 0 and a and b unknown. Determine a parameterization of the considered system and design a DREM parameter identifier and a gradient identifier. Run simulations for the DREM and gradient estimators assuming a = 0.4, b = 0.4, and selecting, for the DREM estimator, H1 (s) = 1 , s +1 H2 (s) = 2 . s +2 Consider the cases u(t) = 10 and u(t) = 10 sin 5t2 . Compare the performance of the estimators in terms of transient response, speed of response and overall performance. Plot the phase portraits of the parameter estimation errors and comment on the results. AA (DICII) RAC - Adaptive Estimation and Control 35 / 102 Parameter Estimation – SPR Design for Bilinear Models Consider the bilinear parametric model y = k0 (θ⊺ ϕ + z0 ) and assume, for simplicity, that the sign of k0 is known and positive. The more general case in which the sign is unknown requires some modification and the introduction of what is called a Nussbaum-type function. Suppose in addition that all signals are bounded. One naive way to deal with the parameter identification problem for the bilinear model is to rewrite it as ϕ y = k0 θ⊺ ϕ + k0 z0 = [ k0 θ⊺ k0 ] [ ] z0 and regard this as a re-parameterized linear model with parameters θ1 = k0 θ and θ2 = k0 . One can therefore generate estimates θ̂1 and θ̂2 of θ1 and θ2 and then derive estimates for θ and k0 as θ̂1 θ̂ = , k̂0 = θ̂2 . θ̂2 This approach requires the knowledge of a lower bound on k0 and the use of a projection-based algorithm, hence may not provide the best solution. AA (DICII) RAC - Adaptive Estimation and Control 36 / 102 Parameter Estimation – SPR Design for Bilinear Models Consider, instead, the SPR design for the model z = W (s)L(s)k0 (θ⊺ ψ + z1 ) , where z1 = L−1 (s)z0 and ψ = L−1 (s)ϕ and L(s) is such that L−1 (s) is proper and stable and W (s)L(s) is strictly proper and SPR. The estimate ẑ of z and the estimation error are defined as ẑ = W (s)L(s)k̂0 (θ̂⊺ ψ + z1 ) , Note now that ϵ = z − ẑ. ϵ = W (s)L(s) (k0 θ⊺ ψ − k̂0 θ̂⊺ ψ − k̃0 z1 ) , which can be rewritten as ϵ = W (s)L(s) (−k0 θ̃⊺ ψ − k̃0 (θ̂⊺ ψ + z1 )) , where k̃0 = k̂0 − k0 and θ̃ = θ̂ − θ. AA (DICII) RAC - Adaptive Estimation and Control 37 / 102 Parameter Estimation – SPR Design for Bilinear Models Let ė = Ae + B (−k0 θ̃⊺ ψ − k̃0 (θ̂⊺ ψ + z1 )) , ϵ = Ce, be a state-space representation of the error equation, i.e. W (s)L(s) = C (sI − A)−1 B. By SPR-ness of the transfer function W (s)L(s) there exist, similarly to the case of the SPR design for the linear model, matrices P = P ⊺ > 0 and L = L⊺ > 0 such that A⊺ P + PA = −L and PB = C ⊺ . 1 One could therefore select the storage function S(e) = e ⊺ Pe and note that 2 1 1 Ṡ = − e ⊺ Le + e ⊺ PB(−k0 θ̃⊺ ψ − k̃0 (θ̂⊺ ψ + z1 )) = − e ⊺ Le + (−k0 ψ ⊺ ϵ)θ̃ + ((−θ̂⊺ ψ − z1 )ϵ)k̃0 , 2 2 which shows that the error system is strictly passive from the inputs θ̃ and k̃0 to the outputs −k0 ψϵ and (−θ̂⊺ ψ − z1 )ϵ. Observe that the first output is not measurable, since it contains the unmeasured parameter k0 . One, could however still use a passive interconnection using the trivial 1 observation that k0 = 1 and making sure that the passive system used to interconnect k0 1 the first output to the first input is scaled by . k0 AA (DICII) RAC - Adaptive Estimation and Control 38 / 102 Parameter Estimation – SPR Design for Bilinear Models To this end consider the systems 1 ξ˙ = Γw , k0 η̇ = γω, with Γ = Γ⊺ > 0 and γ > 0. These are passive from the input w to the output ξ and from the input ω to the output η. Note that the ξ system is not realizable since k10 is unknown. Consider now the interconnection equations w = −k0 ψϵ, θ̃ = −ξ, ω = −θ̂⊺ ψ − z1 , k̃0 = −η. Note that even if one output signal and the ξ system are not realizable, their interconnection is, leading to the parameter update laws θ̂˙ = Γψϵ, k̂˙ 0 = γϵ(θ̂ψ + z1 ). By the Passivity Theorem we conclude stability of the interconnected system, thus e ∈ L∞ , θ̃ ∈ L∞ , k̃0 ∈ L∞ and e ∈ L2 , θ̂˙ ∈ L2 , and k̂˙ 0 ∈ L2 . Finally, if ψ̇ ∈ L∞ , ψ is persistently exciting, and ω ∈ L2 then θ̂(t) converges to θ and k̂0 converges to a constant. AA (DICII) RAC - Adaptive Estimation and Control 39 / 102 Normalization We have, up to now, discussed several parameter estimation methods which require boundedness of the external signals, that is the the input and output signals. This is achieved assuming that the input is in L∞ and the system is stable. We now consider how such parameter estimation methods could be modified to be used with signals which are not necessarily bounded and with systems which may be unstable. Considering these scenario is crucial to be able to exploit parameter estimation algorithms in a closed-loop configuration to control potentially unstable systems, that is systems for which bounded properties of the signals have to be proved and cannot be assumed. We consider only two simple examples which contain all the essential ingredients of a process, called normalization, which allows obtaining bounded/converging estimates even if the measured signals are not bounded. The normalization can then be used in more involved schemes with minor modifications. AA (DICII) RAC - Adaptive Estimation and Control 40 / 102 Normalization – One Unknown Parameter Consider the linear, SISO, plant described by the relation y (t) = θu(t), with input u, output y and unknown θ. When all signals are bounded, the estimate of θ has been obtained considering an estimated output ŷ = θ̂u and the cost function J(θ̂) = (y − ŷ )2 (y − θ̂u)2 = . 2 2 If u and y are not in L∞ then the minimization problem min J(θ̂) does not make sense. θ̂ This issue can be solved defining normalized measurements ū = u , m ȳ = y , m where m > 0 is the so-called normalizing signal, defined as m2 = 1 + ns2 , with the signal ns u selected such that ∈ L∞ , for example ns = u, yielding m2 = 1 + u 2 . m AA (DICII) RAC - Adaptive Estimation and Control 41 / 102 Normalization – One Unknown Parameter Since ū ∈ L∞ and ȳ ∈ L∞ , we can now follow the same approach that we have used in the case in which all signal were bounded defining the estimated output ȳˆ = θ̂ū, and the estimation error ϵ̄1 = ȳ − ȳˆ = ȳ − θ̂ū. The adaptive law is now determined by solving the well-posed optimization problem min J(θ̂) = min θ̂ θ̂ (ȳ − θ̂ū)2 (y − θ̂u)2 , = min 2 2m2 θ̂ which yields the gradient-based update law ϵ1 θ̂˙ = γϵ̄1 ū = γ 2 u = γϵu, m where ϵ = ϵ1 u = −θ̃ 2 is referred to as the normalized estimation error. m2 m 2 ˙ hence letting V (θ̃) = θ̃ yields Note that θ̃˙ = θ̂, 2γ V̇ = −θ̃2 ū 2 = −ϵ2 m2 ≤ 0. AA (DICII) RAC - Adaptive Estimation and Control 42 / 102 Normalization – One Unknown Parameter We conclude that θ̃ ∈ L∞ , θ̂ ∈ L∞ , and ϵm ∈ L2 . Since ū ∈ L∞ , then ϵ ∈ L∞ and ϵm ∈ L∞ , which imply that θ̃˙ ∈ L2 ∩ L∞ and that . « ϵm ∈ L∞ . In conclusion lim ϵm = 0, t→∞ lim θ̃˙ = 0, t→∞ which means that even in the case of unbounded signals it is possible to develop an adaptive estimator that guarantees bounded parameter estimate and a speed of adaptation that is bounded in an L2 and L∞ sense. Selecting ns = 0, that is m = 1, yields the un-normalized update law, from which we can establish bounded parameter estimate, but we cannot guarantee boundedness of θ̃˙ in an Lp sense, unless u ∈ L∞ . AA (DICII) RAC - Adaptive Estimation and Control 43 / 102 Normalization – The Scalar Case To illustrate how the normalization process can be used to estimate the parameters of a dynamical system consider the system (for simplicity we have selected the ”b” matrix to be equal to 1) ẋ = −ax + u, which we reparameterize as x= 1 [(am − a)x + u] , s + am with am > 0. We do not assume that u ∈ L∞ , but only that it is piece-wise continuous, and a is arbitrary (that is, not positive). The objective is to estimate a using measurement of x and u. To this end, consider an estimate x̂ of x defined as 1 x̂ = [(am − â)x + u] , s + am and the estimation error ϵ implicitly defined as ϵ = x − x̂ − 1 1 (ãx − ns2 ϵ) ns2 ϵ = s + am s + am ⇔ ϵ̇ = −am ϵ + ãx − ns2 ϵ where ns is the normalizing signal to be defined. AA (DICII) RAC - Adaptive Estimation and Control 44 / 102 Normalization – The Scalar Case Consider now the storage function S(ϵ) = 12 ϵ2 and note that Ṡ = −am ϵ2 − ns2 ϵ2 + ϵx ã, which reveals a passivity property from the input ã to the output ϵx. Consider now the negative feedback interconnection of the estimation error system with the passive system ξ˙ = w , via the interconnection equations w = ϵx and ã = −ξ. By the Passivity Theorem we conclude that the overall system has a storage function 1 1 ST (ϵ, ã) = ϵ2 + ã2 , which is such that 2 2 ṠT = −am ϵ2 − ns2 ϵ2 ≤ 0. As a result ϵ ∈ L∞ , ã ∈ L∞ , ϵ ∈ L2 , and ns ϵ ∈ L2 . Let now m2 = 1 + ns2 and assume that x ∈ L∞ (for example one could select ns = x, yielding m2 = 1 + x 2 ). As a result one could m write x ã˙ = â˙ = −ϵx = −ϵm , m which implies that ã˙ ∈ L2 (and â˙ ∈ L2 ). AA (DICII) RAC - Adaptive Estimation and Control 45 / 102 Normalization The signal ns is used to normalize the effect of unbounded x, yielding a bounded speed of adaptation regardless of boundedness of x. The normalizing term −ns2 ϵ is similar to a nonlinear damping term, which is used in some nonlinear control design methods to strengthen the negativity of the time derivative of a Lyapunov function. Despite the normalization, it is not possible to establish the property that ã˙ ∈ L∞ , which is essential to prove convergence of ã(t). The normalization procedure can be used with any of the schemes that we have seen. For example, in the SPR design one has to define the normalized estimation error ϵ = z − ẑ − W (s)L(s)ns2 ϵ and in the gradient method one has to select the normalized estimation error/cost ϵ= z − ẑ , 1 + ns2 J(θ̂) = (1 + ns2 ) ϵ2 , 2 where, in both cases, m2 = 1 + ns2 is such that the ratio of the regressor with m is bounded. AA (DICII) RAC - Adaptive Estimation and Control 46 / 102 Adaptive Luenberger Observers Consider the SISO system ẋ = Ax + bu, y = cx, with state x(t) ∈ IR n , input u(t) ∈ IR and output y (t) ∈ IR. Assume, for simplicity, that u is a piecewise continuous and bounded function of time and A is asymptotically stable. Assume, in addition, that the plant is reachable and observable. Consider the problem of estimating, simultaneously, the parameters of the system and the states from measurements of the input and output signals. Observe that this problem is partially ill-posed, since we cannot estimate the matrices A, b and c, but only the parameters of the transfer function W (s) = c(sI − A)−1 b. The reason for this is that there are infinitely many, algebraically equivalent, state-space representations giving the same transfer function. This shortcoming, however, allows selecting a particular structure, minimally parameterized, which facilitates the solution of the simultaneous state and parameter estimation problem. AA (DICII) RAC - Adaptive Estimation and Control 47 / 102 Adaptive Luenberger Observers Assume that the triple (A, b, c) is in observer form, that is it is such that ⎡ ⎢ ⎢ ⎢ ⎢ A=⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −an−1 −an−2 ⋮ −a1 −a0 1 0 ⋮ 0 0 0 1 ⋯ 0 0 ⋯ ⋯ ⋱ ⋯ ⋯ 0 0 ⋮ 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ bn−1 ⎢ ⎢ b ⎢ n−2 b=⎢ ⎢ ⋮ ⎢ ⎢ b0 ⎣ ⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦ c =[ 1 0 ⋯ 0 0 ], and note that ẋ = Ax + bu = (S − ac)x + bu = Sx − ay + bu, where a = [ an−1 an−2 ⋯ a1 ⊺ a0 ] , and S is the upper shift matrix. Recall, finally, that the vectors a and b identify the transfer function of the system, that is c(sI − A)−1 b = AA (DICII) sn bn−1 s n−1 + bn−2 s n−2 + ⋯b0 . + an−1 s n−1 + an−2 s n−2 + ⋯a0 RAC - Adaptive Estimation and Control 48 / 102 Adaptive Luenberger Observers The adaptive Luenberger observer is given by the system x̂˙ = Âx̂ + b̂u + L(y − ŷ ), ⋆ ⋆ ŷ = c x̂, ⋆ with  = S − âc, L = a − â, a selected such that S − a c is Hurwitz, and â and b̂ estimates of a and b,respectively. A wide class of adaptive law can be used to generate â and b̂, for example starting from the parametric model z = θ⊺ ϕ, where z and ϕ are filtered, measured, signals obtained ⊺ from u and y , and θ = [ a⊺ b ⊺ ] . Theorem An adaptive observer formed by the combination of the proposed adaptive Luenberger observer with any parameter estimation law (which uses normalization if required) based on the plan parametric model z = θ⊺ ϕ is such that all signals are in L∞ ; the output estimation error ỹ = y − ŷ converges to zero as t → ∞; if u is sufficiently rich of order 2n then the state estimation error x − x̂ and the parameter estimation error θ − θ̂ converge exponentially to zero. AA (DICII) RAC - Adaptive Estimation and Control 49 / 102 Adaptive Observers – Frequency Estimation using the I&I Approach Consider the problem of estimating the amplitude, phase and frequency of the signal y (t) = E1 sin(ω1 t + ϕ1 ). As already discussed, the signal y can be regarded as the output of the system ẏ = x, ẋ = −θ1 y , with θ1 = ω12 , and E1 and ϕ1 related to the initial states y (0) and x(0). This problem can be solved using the parameter estimation methods and the adaptive Luenberger observer discussed in the previous lectures. We present, herein, an alternative approach based on what is called the I&I (Immersion and Invariance) approach, which regards parameter and state estimation problems in a unified way and as reduced order observer design problems. AA (DICII) RAC - Adaptive Estimation and Control 50 / 102 Adaptive Observers – Frequency Estimation using the I&I Approach To apply the I&I methodology define the state and parameter estimation errors zx = k1 y + x̂(k12 + θ1 ) − x, zθ = y x̂ + θ̂1 − θ1 , with k1 > 0. These error variables differ from the classical error variables x̂ − x and θ̂ − θ in that they contain additional injection terms. Consider now the dynamics of the estimation error variables, that is the equations ˙ 12 + θ1 ) + θ1 y , żx = k1 x + x̂(k żθ = x x̂ + y x̂˙ + θ̂˙1 . Using the relations x = −zx + k1 y + x̂(k12 + θ), θ1 = −zθ + y x̂ + θ̂1 , yields ˙ żx = −k1 zx + (k12 + θ1 )(y + k1 x̂ + x̂), żθ = −x̂zx − x̂ 2 zθ − ∆ + θ̂˙1 , where ∆ = ∆(y , x̂, θ̂1 ) is a function of known signals. AA (DICII) RAC - Adaptive Estimation and Control 51 / 102 Adaptive Observers – Frequency Estimation using the I&I Approach Selecting x̂˙ = −k1 x̂ − y , θ̂˙1 = ∆(y , x̂, θ̂1 ), yields the state and parameter estimation error system [ Let V (zx , zθ ) = −k1 żx ]=[ żθ −x̂ 0 zx ][ ]. zθ −x̂ 2 z2 zx2 + θ . Then V̇ = −zx2 − zθ zx x̂ − zθ2 x̂ 2 < 0, for all (zx , zθ x̂) ≠ (0, 0). 2k1 2 . As a result zx ∈ L∞ ∩ L2 , zθ ∈ L∞ , zθ x̂ ∈ L2 and, since x̂ ∈ L∞ , żx ∈ L∞ converges (exponentially) to zero and zθ x̂ converges to zero. « and zθ x̂ ∈ L∞ , zx We conclude that, provided E1 ≠ 0 and ω1 ≠ 0, θest = y x̂ + θ̂1 is an asymptotic estimate of θ1 = ω 2 and xest = k1 y + x̂(k12 + θest ) is an asymptotic estimate of x. AA (DICII) RAC - Adaptive Estimation and Control 52 / 102 Adaptive Observers – Frequency Estimation using the I&I Approach Exercise (Assignment 3) To study the performance of the I&I frequency estimator consider the two signals y1 (t) = sin 100t and y2 (t) = sin t. Implement the I&I frequency estimator and run simulations showing the effectiveness of the approach. Experiment with different values of the gain k1 and discuss what is a good selection for such a gain. Suppose that the measured signals are perturbed by a periodic disturbance, that is one measures y1 + d, with d = 0.001 sin 50t. Run simulations for this scenario and discuss the impact that the disturbance has on the convergence of the frequency estimates. AA (DICII) RAC - Adaptive Estimation and Control 53 / 102 Model Reference Adaptive Control Model Reference Adaptive Control (MRAC) is one of the main adaptive control approaches. The basis structure of a MRAC scheme is described in the figure below. The reference model is selected to generate the desired output trajectory that the plant output has to follow. The error signal e1 describes the mismatch between the desired behaviour and the actual behaviour. The closed-loop system is composed of a standard feedback loop containing the plant and a parameterized controller together with an adjustment/learning mechanism which generates on-line the controller parameter θc . AA (DICII) RAC - Adaptive Estimation and Control 54 / 102 Model Reference Adaptive Control MRAC could be direct or indirect and with, or without, normalization. In direct MRAC the parameter vector of the controller is updated directly by an adaptation mechanism, whereas in indirect MRAC the parameter vector of the controller is obtained via some algebraic equations that relate the controller parameters with the plant parameters, which are estimated on-line. In direct and indirect MRAC with/without normalization the controller is combined with a parameter estimation algorithms with/without normalization. This design procedure allows using a wide class of adaptive laws relying, for example, on the gradient descent or on the SPR design. In the case of un-normalized MRAC the controller is modified to give an error equation the form of which allows using passivity arguments. As a result, the design stage is more complex, but the study of the stability properties relies purely on passivity arguments. AA (DICII) RAC - Adaptive Estimation and Control 55 / 102 MRAC – A Scalar Example: Adaptive Regulation Consider the scalar system ẋ = ax + u, with a an unknown constant. The control objective is to determine a function u = u(x, t) such that the state of the closed-loop system converges asymptotically to zero for any initial condition x(0). To this end let am > 0 be a parameter selected by the designer and note that selecting u = −kx, with k = a + am yields the closed-loop system ẋ = −am x, that is the closed-loop system behaves like the model ẋm = −am xm . The control law u = −kx = −(a + am )x is often referred to as the known parameter controller and it is clearly non-implementable. As a result, one has to replace k with an estimate k̂, computed by viewing the problem as an on-line parameter estimation problem, yielding the control law u = −k̂x. Note that this control law is nonlinear in the state (x, k̂), despite the fact that the underlying system is linear. AA (DICII) RAC - Adaptive Estimation and Control 56 / 102 MRAC – A Scalar Example: Adaptive Regulation To obtain the update law for k̂ rewrite the system as ẋ = ax − kx + kx + u = (a − k)x + kx + u = −am x + kx + u or, equivalently, as 1 (kx + u). s + am These equations provide two equivalent parameterizations of the system in terms of the controller parameter k. x= Since x and u are measured and am is known one could consider a wide class of parameter identifiers. The one that generates the simplest possible update law, yielding a simple stability proof, is the one based on the use of the un-normalized SPR design. Since 1 is SPR, one could directly apply the SPR design with the state estimate s + am x̂ = 1 1 (k̂x + u) = (0), s + am s + am where the last equality has been obtained noting that u = −k̂x. Selecting x̂(0) = 0, one then has x̂(t) = 0, for all t ≥ 0. AA (DICII) RAC - Adaptive Estimation and Control 57 / 102 MRAC – A Scalar Example: Adaptive Regulation Defining the regulation error ϵ1 = x − x̂ = x and letting k̃ = k̂ − k yields ϵ̇1 = −am ϵ1 − k̃x, Selecting the storage function S = ϵ1 = 1 (−k̃x). s + am ϵ21 yields 2 Ṡ = −am ϵ21 − k̃ϵ1 x, which reveals a passivity property from the input k̃ to the output −ϵ1 x = −x 2 . Consider now the interconnection with the passive system ξ˙ = γv with output ξ and γ > 0, via the interconnection equations v = −ϵ1 x, k̃ = −ξ, yielding the update law k̃˙ = γϵ1 x = γx 2 . The interconnected system can be studied with the storage function ST (ϵ1 , ξ) = ϵ21 ξ 2 + , 2 2γ yielding ṠT = −am ϵ21 ≤ 0. AA (DICII) RAC - Adaptive Estimation and Control 58 / 102 MRAC – A Scalar Example: Adaptive Regulation As a result ϵ1 ∈ L∞ ∩ L2 , k̂ ∈ L∞ and ϵ̇1 ∈ L∞ . This implies that lim ϵ1 (t) = lim x(t) = 0. t→∞ t→∞ ˙ Finally, boundedness of k̂ and convergence of x imply lim k̂(t) = 0 and lim u(t) = 0. t→∞ t→∞ Note that, even if we are able to establish adaptive regulation (to zero), we are not able to prove convergence of k̂ to k = a + am , that is the closed-loop system does not behave as the model ẋm = −am xm . Lack of parameter convergence to the actual value is not crucial in adaptive regulation: as this example demonstrates adaptive regulation can be achieved regardless of the convergence of the estimated parameter to its true value. For this scalar example one could integrate the system and obtain the relation √ lim k̂(t) = k̂∞ = a + γx 2 (0) + (k̂(0) − a)2 ≥ a, t→∞ showing that the k̂(t) converges to a stabilizing gain (for almost all initial conditions), the value of which depends upon the initial conditions and the adaptive gain γ. The selection of k̂(0) and γ affects the value of k̂∞ and the transient response of k̂ and of x. AA (DICII) RAC - Adaptive Estimation and Control 59 / 102 MRAC – A Scalar Example: Adaptive Tracking Consider now a slightly more general problem, that is the problem of designing an adaptive control law such that the state of the system ẋ = ax + bu ⇔ x= b u, s −a with a and b unknown constants, tracks the state of the (known) reference model ẋm = −am xm + bm r ⇔ xm = bm r, s + am with am > 0, for any bounded, piecewise continuous, reference signal r , and it is such that all closed-loop signals are bounded. To achieve the considered control objective one could select the non-implementable known parameter controller u = −kx + lr , with k and l such that xm (s) x(s) bl bm = = = , r (s) s − a + bk s + am r (s) am + a bm and l = , which are well defined provided b ≠ 0 (i.e. the system is b b controllable). that is k = AA (DICII) RAC - Adaptive Estimation and Control 60 / 102 MRAC – A Scalar Example: Adaptive Tracking The known parameter controller guarantees that, for any initial condition x(0) of the system and any initial condition xm (0) of the model lim ∣x(t) − xm (t)∣ = 0, exponentially. t→∞ In an adaptive scheme, the known parameter controller is replaced by the controller ˆ, u = −k̂x + lr with k̂ and lˆ to be computed on-line from the available signals. To design the update laws for k̂ and lˆ rewrite the system as ẋ = −am x + bm r + b(kx − lr + u) ⇔ xm ³¹¹ ¹ ¹ ¹ ¹ · ¹ ¹ ¹ ¹ ¹ µ bm b x= r+ (kx − lr + u), s + am s + am and note that the error variable e = x − xm is such that e= b (kx − lr + u). s + am The equation describing the error variable is in the form of a bilinear parametric model, hence we could use the corresponding parameter estimation method. AA (DICII) RAC - Adaptive Estimation and Control 61 / 102 MRAC – A Scalar Example: Adaptive Tracking Define the estimate ê of e as ê = b̂ ˆ + u) = 1 (0), (k̂x − lr s + am s + am ˆ . This equation where the last equality has been obtained recalling that u = −k̂x + lr implies that the estimation error ϵ1 = e − ê coincides with the tracking error e, that is there is no need to generate ê. As a result, defining k̃ = k̂ − k and l˜ = lˆ− l and recalling that ϵ1 = e yields ˜) ϵ̇1 = −am ϵ1 + b(−k̃x + lr ⇔ ϵ1 = e = b ˜ ). (−k̃x + lr s + am Similarly to the regulation problem, consider the storage function S = ˜ ) = −am ϵ21 + ∣b∣ [ −ϵ1 xsign(b) Ṡ = −am ϵ21 + ϵ1 b(−k̃x + lr ϵ21 2 and note that k̃ ϵ1 r sign(b) ] [ ˜ ] , l which reveals a passivity property from the inputs k̃ and l˜ to the outputs −ϵ1 xsign(b) and ϵ1 r sign(b). AA (DICII) RAC - Adaptive Estimation and Control 62 / 102 MRAC – A Scalar Example: Adaptive Tracking The passivity property suggests the selection of the update laws ˙ k̃˙ = γ1 ϵ1 xsign(b) = k̂, ˙ l̃˙ = −γ2 ϵ1 r sign(b) = l̂, with γ1 > 0 and γ2 > 0, which are implementable provided the sign of b is known. It follows that ϵ1 ∈ L∞ ∩ L2 , k̃ ∈ L∞ , and l˜ ∈ L∞ . Thus x ∈ L∞ , u ∈ L∞ , ϵ̇1 ∈ L∞ , which imply that lim ϵ1 (t) = 0. t→∞ We have therefore established that (provided the sign of b is known) the proposed adaptive control law is such that all closed-loop signals are bounded and the state x of the system tracks, asymptotically, the state xm of the model, for any bounded, piecewise continuous, signal r . This asymptotic tracking result does not imply that k̂ and lˆ converge to k and l, respectively, unless the signal r is sufficiently rich of order 2 (in which case one has exponential convergence of k̂ and lˆ to k and l, respectively). The assumption that the sign of b is known can be relaxed with a slightly more elaborate construction, which we do not discuss for brevity. AA (DICII) RAC - Adaptive Estimation and Control 63 / 102 MRAC – A Scalar Example: Time Varying Parameter To understand the role of the assumption that the unknown parameters are constant, consider the adaptive regulation problem for the system ẋ = a(t)x + u, where a(t) is an unknown (non-constant) parameter which is such that a(t) is bounded for all t ≥ 0. Suppose in addition that there exists an unknown la such that, for some known constant ∆a > 0, ∣a(t) − la ∣ ≤ ∆a . One could attempt to study this regulation problem using the same storage function used 2 2 in the constant parameter case, that is ST = x2 + ã2 , where ã = â − a, the time derivative of which along the trajectories of the system with state x and ã is ṠT = ax 2 + xu + ãã˙ = âx 2 − ãx 2 + xu + ãâ˙ − ãȧ. The selection u = −âx − x and â˙ = x 2 , yields ṠT = −x 2 − ãȧ ≤ −x 2 + ∣ã∣∣ȧ∣, which contains a term upon which we have no control. AA (DICII) RAC - Adaptive Estimation and Control 64 / 102 MRAC – A Scalar Example: Time Varying Parameter The main reason why this approach fails is that we use a in the storage function, via the term ã2 . This is used to guarantee boundedness of â, without guaranteeing convergence of â to a, thus replacing a with another constant, for example la , may achieve the same objective without yielding any undesired time derivative. Consider therefore the modified storage function S̃T = x 2 (â − la )2 + 2 2 yielding (note that a = la + (a–la )) S̃˙ T = âx 2 + xu + [(â–la )â˙ − (â–la )x 2 ] + (a–la )x 2 . This suggests the selection u = −x − âx − ∆a x, â˙ = x 2 , which gives S̃˙ T ≤ −x 2 , hence x ∈ L∞ ∩ L2 , â ∈ L∞ , ẋ ∈ L∞ , thus lim x(t) = 0. t→∞ Note that the control law does not use la , pretty much like a standard adaptive law, and uses solely a bound on the maximum variation of a and no bound on ȧ. This example reveals that this method, known as the congelation of variables method, can be interpreted as the combination of an adaptive controller, to deal with the unknown constant la , and a robust controller, to counteract the perturbation a(t)–la . AA (DICII) RAC - Adaptive Estimation and Control 65 / 102 MRAC – A Scalar Example: Time Varying Parameter The congelation of variable design can be also given a passivity interpretation similarly to the standard design. Let u = −x − âx − ∆a x. Consider the closed-loop system ẋ = −x + (a − la − ∆a )x − (â − la )x, with â − la regarded as an input, and the storage function S(x) = x2 . Note that 2 ≤0 ³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ·¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹µ Ṡ = −x + (a − la − ∆a ) x 2 − (â − la )x 2 ≤ −x 2 − (â − la )x 2 , 2 which suggests that the system is passive from the input â − la to the output −x 2 and justifies the use of the parameter update law . ­ â − la = â˙ = x 2 . The use of the congealed variable la allows defining a new passive interconnection, from which an update law can be designed. This passivity property cannot be enforced with respect to the input variable â − a, because of the non-zero time derivative of a. AA (DICII) RAC - Adaptive Estimation and Control 66 / 102 MRAC – A Scalar Example: The I&I Approach An alternative to the classical MRAC design is offered by the so-called I&I adaptive control approach, in which the adaptive control problem is regarded as a partial state feedback control problem. To illustrate the I&I approach, consider the system ẋ = ax + u, with a an unknown constant. Consider now the extended system, with states x and a and control u, described by the equations ȧ = 0, ẋ = ax + u. The control objective is to design u, as functions of x only, such that the equilibrium (a, 0) is stable and x converges to zero with target dynamics ẋm = −am xm , with am > 0. The control objective can be achieved selecting the known parameter controller u = −am x − ax. AA (DICII) RAC - Adaptive Estimation and Control 67 / 102 MRAC – A Scalar Example: The I&I Approach In the spirit of designing a reduced order observer for the state a, define an extended system ȧ = 0, ẋ = ax + u, â˙ = w , with u = −am x − aest x and w such that a can be asymptotically reconstructed from x and â, that is there exists a function aest = aest (x, â) which provides some sort of estimate of a. Define, consistently with the standard reduced order observer design procedure, the error variable aest ³¹¹ ¹ ¹ ¹ ¹ ¹ ¹·¹ ¹ ¹ ¹ ¹ ¹ ¹ µ z =â + β(x) −a, with β to be selected, and note that ż = â˙ + ∂β ∂β ∂β ∂β (ax − am x − aest x) = â˙ + (−zx − am x) = (â˙ − am x) − ( x) z. ∂x ∂x ∂x ∂x AA (DICII) RAC - Adaptive Estimation and Control 68 / 102 MRAC – A Scalar Example: The I&I Approach ∂β x yields Selecting â˙ = am ∂x ∂β x) z. ∂x The function β has to be selected to render the equilibrium z = 0 stable. This is achieved ∂β selecting β such that x > 0 for all x ≠ 0, for example ∂x ż = − ( x2 1 , or log(1 + x 2 ). 2 2 Note that the first selection yields the classical update law, whereas the second one yields a normalized update law. β(x) = Using the first selection of β yields the closed-loop system, in the a, x and z coordinates, ȧ = 0, ẋ = −am x − xz, ż = −x 2 z. We conclude that z ∈ L∞ , xz ∈ L2 , x ∈ L∞ and lim x(t) = 0, and the x variables achieves t→∞ the target dynamics with an L2 perturbation. Note that the update law and the parameter estimates are, respectively, â˙ = am x 2 , AA (DICII) aest = â + RAC - Adaptive Estimation and Control x2 . 2 69 / 102 MRAC – A Scalar Example Exercise (Assignment 4) Consider the system ẋ = ax + u, with a constant and unknown. Use, in the simulations, a = 1 and am = 1. Design a (classical) MRAC adaptive controller and an I&I adaptive controller to achieve adaptive regulation (with at least two selections of the function β for the I&I design). Compare the performance of the resulting closed-loop systems in terms of speed of response, control amplitude, and in the presence of measurement noise, that is the t measured variable is x + d, with d(t) = 0.1 sin . 5 Finally, compare the performance of the resulting controllers in the case in which the parameter is given by a =1+ AA (DICII) 1 sin 10t, 10 a = 1 + 10 sin RAC - Adaptive Estimation and Control t . 10 70 / 102 MRAC – The General State Feedback Case Consider the linear system ẋ = Ax + Bu, with state x(t) ∈ IR n and input u(t) ∈ IR m , both available for measurement, and A ∈ IR n×n and B ∈ IR n×m unknown. Suppose that the system is reachable. The control objective is to determine u such that all closed-loop signals are bounded and the plant state follows the state xm of the reference model ẋm = Am xm + Bm r , with Am ∈ IR Hurwitz, Bm ∈ IR , and r (t) ∈ IR m a bounded and piecewise continuous reference signal. If the matrices A and B were known one could select n×n n×m u = −Kx + Lr , yielding the closed-loop system ẋ = (A − BK )x + BLr , with K and L such that A − BK = Am , BL = Bm . This implies that the input-state behaviour of the closed-loop system coincides with the input-state behaviour of the model. In general it may be impossible to find matrices K and L satisfying the above matching conditions and existence of such matrices may rely on a particular structure for A and/or B. AA (DICII) RAC - Adaptive Estimation and Control 71 / 102 MRAC – The General State Feedback Case Suppose that K and L satisfying the matching conditions exist. Then one could select u = −K̂ x + L̂r , with K̂ and L̂ estimates of K and L, respectively. Using these definitions the system can be rewritten as ẋ = Ax + B(−K̂ x + L̂r ) = (Am + BK )x + BLr + B(−K̂ x + L̂r ) − BLr = Am x + Bm r − B K̃ x + B L̃r , with K̃ = K̂ − K and L̃ = L̂ − L. Let e = x − xm and note that ė = Am e + B(−K̃ x + L̃r ), which depends on the unknown B. In the scalar case we have exploited the fact that the sign of B is, by assumption, known. This assumption can be extended to the general case assuming that L is symmetric and positive (similarly if it is negative) definite and rewriting the error system as ė = Am e + Bm L−1 (−K̃ x + L̃r ). AA (DICII) RAC - Adaptive Estimation and Control 72 / 102 MRAC – The General State Feedback Case Consider now the storage function S(e) = e ⊺ Pe, with P = P ⊺ > 0, such that A⊺m P + PAm = −Q, with Q = Q ⊺ > 0, and note that Ṡ = −e ⊺ Qe + 2e ⊺ PBm L−1 (−K̃ x + L̃r ) = −e ⊺ Qe − 2tr (K̃ ⊺ L−1 Bm⊺ Pex ⊺ ) + 2tr (L̃⊺ L−1 Bm⊺ er ⊺ ). To define the update laws consider the storage function Supdate = tr (K̃ ⊺ L−1 K̃ + L̃⊺ L−1 L̃) and note that ˙ Ṡupdate = 2tr(K̃ ⊺ L−1 K̃˙ + L̃⊺ L−1 L̃). Both systems, that is the error system and the system defining the update laws, are passive with respect to properly selected input and output signals. AA (DICII) RAC - Adaptive Estimation and Control 73 / 102 MRAC – The General State Feedback Case Interconnecting the systems in a negative feedback interconnection yields the update laws K̃˙ = K̂˙ = Bm⊺ Pex ⊺ , L̃˙ = L̂˙ = −Bm⊺ Per ⊺ , hence, by the Passivity Theorem, ṠT = Ṡ + Ṡupdate = −e ⊺ Qe ≤ 0. We conclude that e ∈ L∞ ∩ L2 , K̃ ∈ L∞ , L̃ ∈ L∞ , hence K̃˙ ∈ L∞ and L̃˙ ∈ L∞ . These imply that ė ∈ L∞ . As a result limt→∞ e(t) = 0. The assumption that L is positive definite has strong implications on the structure of B and on the selection of Bm . In fact, if L = L⊺ > 0 then BLB ⊺ = Bm B ⊺ = BBm⊺ ≥ 0, which imposes non-generic symmetry and positivity conditions on Bm B ⊺ . Note that BLB ⊺ cannot be strictly positive definite unless m = n and rankB = n. AA (DICII) RAC - Adaptive Estimation and Control 74 / 102 MRAC for SISO Systems We have introduced several examples to illustrate the design of MRAC schemes for systems in which the whole state vector is measurable. We now consider the more realistic and challenging problem in which only the output of the system is available for measurement. For simplicity we focus, however, on SISO systems. Consider the SISO system (the plant) ẋ = Ax + Bu, y = Cx, with x(t) ∈ IR n , u(t) ∈ IR, and y (t) ∈ IR, and A, B and C matrices of appropriate dimensions, and its associated transfer function G (s) = C (sI − A)−1 B = k Z (s) , R(s) with Z (s) and R(s) monic polynomials and k constant. AA (DICII) RAC - Adaptive Estimation and Control 75 / 102 MRAC for SISO Systems The reference model, selected by the designer, is described by the equations ẋm = Am xm + Bm r , ym = Cm xm , with xm (t) ∈ IR nm , ym (t) ∈ IR and r (t) ∈ IR a bounded and piecewise continuous function of time. The transfer function of the model is Wm (s) = Cm (sI − Am )−1 Bm = km Zm (s) , Rm (s) with Zm (s) and Rm (s) monic polynomials. To begin with we consider the problem of designing a controller such that the output of the system, in closed-loop with such a controller, tracks the output of the reference model for any given reference r , pretending that all parameters are known. AA (DICII) RAC - Adaptive Estimation and Control 76 / 102 MRAC for SISO Systems Plant assumptions Z (s) is a monic Hurwitz polynomial of degree m. An upper bound N of the degree n of R(s) is known. The relative degree of the system, that is rd = n − m, is known. The sign of the high frequency gain k is known (assume it is positive). Reference model assumptions Zm (s) and Rm (s) are monic Hurwitz polynomials of degree mm and nm , respectively and nm ≤ N. The relative degree of the model, that is rdm = nm − mm , is such that rdm = rd. The plant is minimum phase and can be unstable; pole-zero cancellations are allowed. Minimum phaseness is required because the controller cancels the zeroes of the plant. The assumptions on the relative degree and on the high frequency gain can be relaxed. AA (DICII) RAC - Adaptive Estimation and Control 77 / 102 MRAC for SISO Systems A trivial design (recall that all parameters are known) is obtained selecting u = C (s)r = km Zm (s) R(s) r, k Z (s) Rm (s) which yields the closed-loop system y = Wm (s)r . This design is viable only if R(s) is Hurwitz. It is however undesired because of its open-loop structure. Consider, instead, the feedback control law u = θ1⊺ αN−2 (s) αN−2 (s) u + θ2⊺ y + θ3 y + cr , Λ(s) Λ(s) where θ1 , θ2 , θ3 and c are constants, Λ(s) = Λ0 (s)Zm (s) is Hurwitz, monic, and of degree N − 1 (hence Λ0 (s) is monic, Hurwitz, and of degree N − 1 − mm ), and αN−2 (s) = [ s N−2 s N−3 ⋯ ⊺ 1 ] if N ≥ 2, otherwise αN−2 (s) = 0. AA (DICII) RAC - Adaptive Estimation and Control 78 / 102 MRAC for SISO Systems The controller parameter θ = [ θ1⊺ θ2⊺ θ3 ⊺ c ] = [ θ̄⊺ ⊺ c ] is to be selected such that the closed-loop transfer function equals Wm (s) and the closed-loop system is stable. The closed-loop transfer function is y (s) c k Z (s) Λ2 (s) = Gcl (s) = . ⊺ r (s) Λ(s)[(Λ(s) − θ1 αN−2 (s))R(s) − kZ (s)(θ2⊺ αN−2 (s) + θ3 Λ(s))] The control objective is therefore achieved if Λ(s)[(Λ(s) − Zm (s) c k Z (s) Λ2 (s) = km . − kZ (s)(θ2⊺ αN−2 (s) + θ3 Λ(s))] Rm (s) θ1⊺ αN−2 (s))R(s) The degree of the denominator of Gcl (s) is n + 2N − 2 and the degree of the denominator of Wm (s) is nm ≤ N, hence n + 2N − 2 − nm pole-zero cancellations have to occur. Such cancellations are stable since Z (s) and Λ(s) are Hurwitz. AA (DICII) RAC - Adaptive Estimation and Control 79 / 102 MRAC for SISO Systems Let c = km and recall that Λ(s) = Λ0 (s)Zm (s). Then the matching condition becomes k (Λ(s) − θ1⊺ αN−2 (s))R(s) − kZ (s)(θ2⊺ αN−2 (s) + θ3 Λ(s)) = Z (s)Λ0 (s)Rm (s), or θ1⊺ αN−2 (s)R(s) + k(θ2⊺ αN−2 (s) + θ3 Λ(s))Z (s) = Λ(s)R(s) − Z (s)Λ0 (s)Rm (s). Equating the coefficients of equal powers of s yields a set of linear equations of the form S θ̄ = p, where S ∈ IR (N+n−1)×(2N−1) depends on the coefficients of R(s), kZ (s) and Λ(s), and p ∈ IR N+n−1 depends on the coefficients of Λ(s)R(s) − Z (s)Λ0 (s)Rm (s). The existence of a solution θ̄ depends on the properties of the matrix S: one could have infinitely many solutions, one solution, or no solution. Note, finally, that the controller parameters are nonlinear functions of the coefficients of Z (s) and R(s). In the special case in which N = n and rd = 1 then θ̄ is a linear function of the coefficients of Z (s) and R(s). AA (DICII) RAC - Adaptive Estimation and Control 80 / 102 MRAC for SISO Systems Lemma Consider the equation S θ̄ = p. The equation has always at least one solution. In addition, if R(s) and Z (s) are coprime and N = n then the solution is unique. The developed known parameter controller assigns the closed-loop poles at the roots of the polynomial Z (s)Λ0 (s)Rm (s) and changes the high frequency gain from k to km . The transfer function matching is achieved by cancelling the zeros of the plant and replacing these by those of the reference model, that is by designing Λ(s) = Λ0 (s)Zm (s). This cancellation is feasible by the assumptions on the polynomials Z (s) and Λ0 (s)Zm (s). In our analysis the initial conditions have been neglected: for zero initial conditions of the plant and the model we have that y (t) = ym (t), for all t ≥ 0 and all bounded and piecewise continuous reference input r (t). For nonzero initial conditions, because all cancellations occur in CI− , one has that lim (y (t) − ym (t)) = 0, t→∞ exponentially. AA (DICII) RAC - Adaptive Estimation and Control 81 / 102 MRAC for SISO Systems A state-space realization of the controller is given by ω̇1 = F ω1 + gu, where ω1 (t) ∈ IR N−1 , ω2 (t) ∈ IR N−1 , θ = [ θ1⊺ ⎡ −λN−2 ⎢ ⎢ 1 ⎢ ⎢ 0 F =⎢ ⎢ ⎢ ⋮ ⎢ ⎢ 0 ⎣ u = θ⊺ ω, ω̇2 = F ω2 + gy , −λN−3 0 1 ⋮ 0 ⋯ ⋯ ⋯ ⋱ ⋯ −λ1 0 0 ⋮ 1 θ2⊺ θ3 −λ0 0 0 ⋮ 0 ⊺ c ] , ω = [ ω1⊺ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ g =⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ω2⊺ 1 0 0 ⋮ 0 y ⊺ r ] , ⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦ and Λ(s) = s N−1 + λN−2 s N−2 + ⋯ + λ1 s + λ0 = det(sI − F ). Note that F and g are such that (sI − F )−1 g = AA (DICII) αN−2 (s) . Λ(s) RAC - Adaptive Estimation and Control 82 / 102 MRAC for SISO Systems Exercise Consider a linear SISO system with transfer function G (s) = −2 s +5 s 2 − 2s + 1 and the reference model 3 r. s +3 Note that N = n = 2 and rd = rdm = 1. Let Λ(s) = s + 1 = Λ0 (s) and ym = u = θ1 1 1 u + θ2 y + θ3 y + cr . s +1 s +1 Show that c = −3/2 and that the equation S θ̄ = p is ⎡ 1 ⎢ ⎢ ⎢ 1 ⎢ ⎢ 1 ⎣ 0 1 −10 −2 6 −10 ⎤ ⎡ θ1 ⎤ ⎡ −10 ⎤ ⎥⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎥ ⎢ θ2 ⎥ = ⎢ 12 ⎥ , ⎥⎢ ⎥ ⎢ ⎥ ⎥ ⎢ θ3 ⎥ ⎢ −14 ⎥ ⎦⎣ ⎦ ⎣ ⎦ hence write a state-space realization of the controller achieving the model matching requirement. AA (DICII) RAC - Adaptive Estimation and Control 83 / 102 MRAC for SISO Systems Exercise Show that an alternative equivalent state-space realization of the controller is ω̇1 = F ⊺ ω1 + θ1 u, ω̇2 = F ⊺ ω2 + θ2 y , u = g ⊺ ω1 + g ⊺ ω2 + θ3 y + cr . Exercise Show that the controller can be implemented as in the diagram below, where C (s) = c AA (DICII) Λ(s) , Λ(s) − θ1⊺ αN−2 (s) F (s) = − RAC - Adaptive Estimation and Control 1 θ2⊺ αN−2 (s) + θ3 Λ(s) . c Λ(s) 84 / 102 MRAC for SISO Systems To obtain an adaptive version of the considered controller we follow an approach proposed by Feuer and Morse and focus mainly on the case rd = 1, since the complexity of the design increases with the relative degree. In addition, we assume that the reference model is SPR. Recalling the structure of the known parameter controller, it is reasonable to parameterize the adaptive controller as ω̇1 = F ω1 + gu, ω̇2 = F ω2 + gy , u = θ̂⊺ ω, where θ̂ is an estimate of θ to be computed online with an adaptive estimation algorithm. To design the estimation algorithm consider a state-space realization of the closed-loop system, that is Acl Bcl ³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹·¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ ³¹¹ ¹ ¹ · ¹ ¹ ¹ ¹µ ⎡ A ⎡ B ⎤ 0 0 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ξ˙ =⎢ 0 F 0 ⎥ ξ+ ⎢ g ⎥ (θ̂⊺ ω), ⎢ ⎥ ⎢ ⎥ ⎢ gC 0 F ⎥ ⎢ 0 ⎥ ⎣ ⎦ ⎣ ⎦ with ξ = [ x ⊺ ω1⊺ AA (DICII) Ccl ³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹·¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ y =[ C 0 0 ] ξ, ⊺ ω2⊺ ] . RAC - Adaptive Estimation and Control 85 / 102 MRAC for SISO Systems This can be re-written as y = Ccl ξ, = ξ˙ = Acl ξ + Bcl θ⊺ ω +Bcl (u − θ⊺ ω), ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹·¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ ξ˙ =A⋆cl ξ + c Bcl r +Bcl (u − θ⊺ ω), with A⋆cl such that y = Ccl ξ, c Ccl (sI − A⋆cl )−1 Bcl = Wm (s). With some manipulation we can therefore obtain an error equation of the form e1 = Wm (s)ρ(u − θ⊺ ω), with ρ = 1/c and e1 = y − ym , that is a bilinear parametric model, yielding the parameter update law θ̂˙ = −Γe1 ω, with Γ = Γ⊺ > 0. AA (DICII) RAC - Adaptive Estimation and Control 86 / 102 MRAC for SISO Systems Theorem Consider the MRAC scheme under the stated assumptions. Then the following holds. All signals are bounded and the tracking error e1 = y − ym converges asymptotically to zero for any bounded and piecewise continuous reference r . If r is sufficiently rich of order 2N, r˙ ∈ L∞ and Z (s) and R(s) are coprime then the parameter error θ̃ and the tracking error e1 converge to zero exponentially. AA (DICII) RAC - Adaptive Estimation and Control 87 / 102 MRAC for SISO Systems Exercise (Assignment 5) Consider the second order system with transfer function G (s) = k s + b0 , s 2 + a1 s + a0 where k, b0 > 0, a1 and a0 are unknown constants, and the reference model ym = 1 r. s +1 Let Λ(s) = s + 2. Verify that all assumptions for the design of a MRAC are met. Perform simulations of the system in closed-loop with the MRAC assuming that b0 = 2, a0 = 5, a1 = −10 and k = 1 and for r = E1 sin ω1 t + E2 sin ω2 t, with E1 ≠ 0, E2 ≠ 0, and ω1 ≠ ω2 ≠ 0 . AA (DICII) RAC - Adaptive Estimation and Control 88 / 102 MRAC for SISO Systems The relative degree one assumption is key to be able to exploit the SPR design method for the parameter identification. In general, regardless of the value of the relative degree, one always obtains the error equation e1 = Wm (s)ρ(u − θ⊺ ω). To exploit the SPR design one rewrites this equation as e1 = Wm (s)(s + p0 ) ⋯ (s + prd−1 )ρ(uf − θ⊺ ωf ), where uf = 1 u, (s + p0 ) ⋯ (s + prd−1 ) ωf = 1 ω, (s + p0 ) ⋯ (s + prd−1 ) and p0 , . . . , prd−1 are positive constants to be selected either to render Wm (s)(s + p0 ) ⋯ (s + prd−1 ) SPR or to allow applicability of the SPR adaptive estimation algorithm for bilinearly parameterized models. AA (DICII) RAC - Adaptive Estimation and Control 89 / 102 Adaptive Backstepping Adaptive integrator backstepping is a powerful adaptive control design methodology which allows solving linear and nonlinear adaptive control problems in a modular way. While we do not discuss in-depth such a methodology, we provide some examples to understand its basic principles and the class of problems to which it can be applied. Consider, to begin with, the adaptive stabilization problem for the system ẋ1 = x2 + ϕ1 (x1 ), ẋ2 = θϕ2 (x) + u, where θ is an unknown constant, x1 (t) ∈ IR, x2 (t) ∈ IR, and u(t) ∈ IR. In addition ϕ1 (0) = 0 and ϕ2 (0) = 0, that is the origin is an equilibrium of the system for u = 0. If θ were known we could proceed as follows. Pretend that x2 is a virtual control and design the partial stabilizer x2⋆ = −x1 − ϕ1 (x1 ) = α1 (x1 ). This suggests that a way to address the problem is to control x2 to be as close as possible to x2⋆ by using the control u. This can be achieved selecting the Lyapunov function V (x1 , x2 ) = x12 (x2 − α1 (x1 ))2 + , 2 2 which penalizes the distance of x1 from zero and of x2 from x2⋆ . AA (DICII) RAC - Adaptive Estimation and Control 90 / 102 Adaptive Backstepping Along the trajectories of the system one has = x1 (x2 + ϕ1 (x1 )) +(x2 − α1 (x1 ))(θϕ2 (x) + u − α̇1 (x1 )) ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ −x12 + x1 (x2 − α1 (x1 )) +(x2 − α1 (x1 ))(θϕ2 (x) + u − α̇1 (x1 )) = V̇ = which suggests the selection u = −θϕ2 (x) + α̇1 (x1 ) − x1 − (x2 − α1 (x1 )), yielding V̇ = −x12 − (x2 − α1 (x1 ))2 = −2V . Note that α̇1 (x1 ) is known, since it is a function of x1 and x2 . Replacing the known parameter controller with u = −θ̂ϕ2 (x) + α̇1 (x1 ) − x1 − (x2 − α1 (x1 )) and using V as a storage function yields V̇ = −2V − θ̃(x2 − α1 (x1 ))ϕ2 (x), which reveals a passivity property from θ̃ to −(x2 − α1 (x1 ))ϕ2 (x). AA (DICII) RAC - Adaptive Estimation and Control 91 / 102 Adaptive Backstepping Exploiting the passivity property we conclude that the update law θ̃˙ = θ̂˙ = γ(x2 − α1 (x1 ))ϕ2 (x) = γ [ 0 ϕ2 (x) ] [ x1 ] x2 − α1 (x1 ) with γ > 0, yields a closed-loop adaptive system such that x1 ∈ L∞ ∩ L2 , x2 ∈ L∞ ∩ L2 , θ̃ ∈ L∞ , ẋ1 ∈ L∞ , ẋ2 ∈ L∞ , hence lim x1 (t) = 0 and lim x2 (t) = 0. t→∞ t→∞ This adaptive design is simple because the system satisfies the so-called matching condition: the parametric uncertainty is matched with the control, that is it can be directly canceled by the control signal, provided all parameters are known. We now illustrate how the proposed ideas can be extended to the case of the so-called extended matching, that is to the case in which the parametric uncertainty is one integrator away from the controller. AA (DICII) RAC - Adaptive Estimation and Control 92 / 102 Adaptive Backstepping Consider the system ẋ1 = x2 + θϕ1 (x1 ), ẋ2 = u, with ϕ1 (0) = 0 and θ an unknown constant. If θ were known we could repeat the previous procedure and construct a stabilizing controller on the basis of x2⋆ = −x1 − θϕ1 (x1 ) = α1 (x1 , θ). However, since θ is not know, we cannot complete the procedure as α1 is also a function of θ. To circumvent this issue we proceed as follow. Pretend, again, that x2 is the control signal and note that an adaptive controller for the system ẋ1 = x2 + θϕ1 (x1 ), is given by x2⋆ = −x1 − θ̂1 ϕ1 (x1 ) = α1 (x1 , θ̂1 ), θ̂˙1 = γx1 ϕ1 (x1 ), with γ > 0, and where we have used the notation θ̂1 to indicate that this is the estimate of θ at the first step of backstepping. This is a dynamic controller that has to be backstepped through one integrator. AA (DICII) RAC - Adaptive Estimation and Control 93 / 102 Adaptive Backstepping Consider now the differential equation for the variable z2 = x2 − α1 (x1 , θ̂1 ), that is ż2 = u − ∂α1 ∂α1 (x2 + θϕ1 (x1 )) − (γx1 ϕ1 (x1 )) = u − Z20 (x1 , z2 , θ̂1 ) − Z21 (x1 , θ̂1 )θ, ∂x1 ∂ θ̂1 and the candidate storage function S(x1 , z2 , θ̂1 ) = x12 z22 (θ̂1 − θ)2 + + 2 2 2γ yielding Ṡ = −x12 + z2 (u + x1 − Z20 (x1 , z2 , θ̂1 ) − Z21 (x1 , θ̂1 )θ). One could attempt to use the estimate θ̂1 to eliminate the θ dependent term in Ṡ by means of the control law u = −x1 − z2 + Z20 (x1 , z2 , θ̂1 ) + Z21 (x1 , θ̂1 )θ̂1 yielding Ṡ = −x12 − z22 − z2 Z21 (x1 , θ̂1 )(θ̂1 − θ), which contains a term upon which we have no control. AA (DICII) RAC - Adaptive Estimation and Control 94 / 102 Adaptive Backstepping There are two ways in which this issue can be resolved. The simplest possible approach is to consider a second estimate θ̂2 for the parameter θ and the control law u = −x1 − z2 + Z20 (x1 , z2 , θ̂1 ) + Z21 (x1 , θ̂1 )θ̂2 yielding Ṡ = −x12 − z22 + z2 Z21 (x1 , θ̂1 )(θ̂2 − θ), that is a passivity property for the x1 , z2 , θ̂1 system from the input θ̂2 − θ to the output z2 Z21 (x1 , θ̂1 ), from which we can design the update law θ̂˙2 = −γz2 Z21 (x1 , θ̂1 ). This approach has the clear disadvantage that one has to estimate θ twice, that is one has to use the so-called over-parameterization. The second solution relies on the fact that one could use the storage function S(x1 , z1 , θ1 ) = x12 z22 (θ̂ − θ)2 + + 2 2 2γ without committing the update law for θ̂ at the first step. Interestingly, the resulting update law is the sum of the two update laws designed in the over-parameterization approach, that is θ̂˙ = γ(ϕ1 (x1 )x1 − Z21 (x1 , θ̂)z2 ). AA (DICII) RAC - Adaptive Estimation and Control 95 / 102 Congelation of Variables via Adaptive Backstepping: An Example Consider again the example that we have used to introduce adaptive backstepping, that is the system ẋ1 = x2 + θϕ1 (x1 ), ẋ2 = u, where θ is an unknown parameter, x1 (t) ∈ IR, x2 (t) ∈ IR, and u(t) ∈ IR. Assume that ϕ1 (0) = 0 and that ϕ is a smooth function. Contrary to what assumed while studying adaptive backstepping, the parameter θ is not constant, that is θ = θ(t). However, consistently with the congelation of variables method, we assume that there exists an unknown constant lθ such that ∣θ(t) − lθ ∣ ≤ ∆θ , for some known constant ∆θ > 0. The objective is to achieve adaptive stabilization by combining the congelation of variables method with adaptive backstepping. AA (DICII) RAC - Adaptive Estimation and Control 96 / 102 Congelation of Variables via Adaptive Backstepping: An Example Consistently with the backstepping design methodology, pretend that x2 is the control signal and select the virtual control law x2⋆ = −(1 + ζ1 (x1 ))x1 − θ̂ϕ1 (x1 ) = α1 (x1 , θ̂), where ζ1 (x1 ) = 1 2 ∆θ ( + ϵθ ∣ϕ̄1,1 (x1 )∣2 ) > 0, 2 ϵθ ϵθ > 0 is a balancing coefficient, and ϕ̄1,1 (x1 ) is such that ϕ1 (x1 ) = ϕ̄1,1 (x1 )x1 . Note that ϕ̄1,1 (x1 ) is well-defined since ϕ1 (0) = 0 and ϕ1 is a smooth function. The term ζ1 (x1 ) is a nonlinear damping term which has been introduced to counteract the effect of parameter variation. AA (DICII) RAC - Adaptive Estimation and Control 97 / 102 Congelation of Variables via Adaptive Backstepping: An Example Consider now the candidate storage function S1 = 1 2 (θ̂ − lθ )2 x1 + 2 2γ and note that its time derivative along the trajectories of the system is Ṡ1 (x1 , z2 , θ̂) = −(1 + ζ1 )x12 + x1 ϕ1 (θ − lθ ) +x1 z2 − (θ̂ − lθ )(γ −1 θ̂˙ − τ1 ) ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ≤ ³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ·¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ 1 1 −(1 + + ∆θ ( x12 + ϵθ ∣ϕ1 ∣2 ) +x1 z2 − (θ̂ − lθ )(γ −1 θ̂˙ − τ1 ) 2 ϵθ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ≥ ζ1 )x12 = = ³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹·¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ ∆θ ˙ − (1 + ) x12 +x1 z2 − (θ̂ − lθ )(γ −1 θ̂ − τ1 ), 2ϵθ where τ1 (x1 ) = x1 ϕ1 (x1 ), and z2 (x1 , x2 , θ̂) = x2 − α1 (x1 , θ̂). If x2 were the control signal then z2 would be equal to zero for all t ≥ 0, hence the parameter update law θ̂˙ = γτ1 (x1 ) would complete the design. AA (DICII) RAC - Adaptive Estimation and Control 98 / 102 Congelation of Variables via Adaptive Backstepping: An Example However, since x2 is controlled through an integrator, hence a step of backstepping is required, to avoid over-parameterization, we do not select the update law and proceed with the second step. To this end consider the dynamics of z2 , which are given by ż2 = u − ∂α1 ∂α1 ˙ (x2 + θϕ1 (x1 )) − θ̂. ∂x1 ∂ θ̂ and the candidate storage function 1 S2 (x1 , z2 , θ̂) = S1 (x1 , θ̂) + z22 . 2 Its time derivative along the trajectories of the system is such that Ṡ2 = Ṡ1 + z2 ż2 ≤ − (1 + ∆θ ˙ ) x12 + x1 z2 − (θ̂ − lθ )(τ1 + w2 z2 − γ −1 θ̂) 2ϵθ +z2 u − z2 where w2 (x1 , θ̂) = − AA (DICII) ∂α1 ∂α1 ˙ x2 + θ̂w2 z2 + (θ − lθ )w2 z2 − z2 θ̂ ∂x1 ∂ θ̂ ∂α1 ϕ1 (x1 ). ∂x1 RAC - Adaptive Estimation and Control 99 / 102 Congelation of Variables via Adaptive Backstepping: An Example The design is completed by selecting θ̂˙ = γ(τ1 + w2 z2 ) = γτ2 (x1 , z2 , θ̂), which is a standard passivity-based update law, and u = α2 (x1 , z2 , θ̂) = −x1 − (1 + ζ2 )z2 − θ̂w2 + ∂α1 ∂ θ̂ γτ2 , where ζ2 (x1 , θ̂) = 1 ∆θ ϵθ ∣w̄2,1 (x1 , θ̂)∣2 , 2 w̄2,1 (x1 , θ̂) = − ∂α1 ϕ̄1,1 (x1 ). ∂x1 Noting that w2 (x1 , θ̂) = w̄2,1 (x1 , θ̂)x1 , and substituting into Ṡ2 , yields Ṡ2 ≤ ≤ ∆θ 2 ϵθ ∆θ ˙ x1 − ∣w̄2,1 ∣2 z22 + (θ − lθ )w2 z2 − (θ − lθ )(τ2 − γ −1 θ̂) 2ϵθ 2 −x12 − z22 ≤ 0. −x12 − z22 − AA (DICII) RAC - Adaptive Estimation and Control 100 / 102 Congelation of Variables via Adaptive Backstepping: An Example Note that if θ were constant, the selection lθ = θ would give ∆θ = 0, which means that the adaptive backstepping controller designed via the congelation of variables method reduces to the classical adaptive backstepping controller designed for the case of constant parameter. Exercise Discuss how one could modify the proposed adaptive controller for the case in which ∆θ is unknown. AA (DICII) RAC - Adaptive Estimation and Control 101 / 102 Conclusions We have solved adaptive parameter estimation problems introducing several alternative methods which differ in terms of the underlying structure of the model and of the information pattern. We have also seen how these methods can be modified to accommodate unbounded signals. We have studied simultaneous state and parameter estimation problems introducing an adaptive version of Luenberger observer and discussing the I&I methodology on a specific example. We have discussed in detail the MRAC problem providing examples, highlighting the impact that non-constant parameters have on the analysis and design, and proposing three design approaches: the classical MRAC design, the congelation of variable design, and the I&I design. We have presented the simplest possible embodiment of adaptive integrator backstepping and discussed its combination with the congelation of variables method. We have also discussed how adaptive backstepping may result in over-parameterization and how this shortcoming can be resolved. AA (DICII) RAC - Adaptive Estimation and Control 102 / 102