Dyadic Perturbation Observer Framework for Control of ... - IEEE Xplore
53rd IEEE Conference on Decision and Control December 15-17, 2014. Los Angeles, California, USA
Dyadic Perturbation Observer Framework for Control of a Class of Nonlinear PDE/ODE Systems Aditya A. Paranjape and Soon-Jo Chung Abstract— This paper presents the general theory of the dyadic perturbation observer framework as a generic method for controlling a class of nonlinear systems described by partial and/or ordinary differential equations. The method is particularly applicable to boundary control problems for systems described by partial differential equations. Conditions for closed-loop stability and robustness are derived using finitegain L stability theory, and the results are further specialized for finite dimensional systems.
I. I NTRODUCTION The Dyadic Perturbation Observer (DPO) framework was first presented, under the title of perturbation observerbased control, and demonstrated successfully in experiments involving control of beam bending in [17], [18]. It was designed primarily for controlling the following class of PDE systems: wt (t, x) ,
∂w(t, x) = Aw(t, x) + f (t, x, w(t, x)), ∂t
(1)
where w(t, x) is the state of the system, x is the spatial coordinate, and w denotes the vector consisting of w and some of its partial spatial derivatives. The problem formulation will be made precise later in the paper. The operator A is linear and time-invariant, while the nonlinearities are all captured by the function f (·). We denote by wx the partial derivative ∂w/∂x, additional subscripts denote higher order partial derivatives with respect to those variables. The objective of this note is to present the general theory of the DPO framework as a generic method for controlling a class of systems described by ordinary and partial differential equations (ODEs and PDEs), including combinations of both, and derive conditions for closed-loop stability and robustness. There are several well-established families of methods for designing controllers for finite dimensional systems, such as dynamic inversion, backstepping, gain scheduling, and Lyapunov function-based approaches [11]. The important feature of all these methods is that they are applicable to any given class of ODEs, although it may not always be convenient to apply one particular method to a given system. A large body of work on the control of PDE systems has focussed on approximating the PDE system by ODEs, and using any of the rich assortment of the aforementioned Aditya Paranjape is with the Department of Mechanical Engineering at McGill University. Soon-Jo Chung is with the Department of Aerospace Engineering and the Coordinated Science Laboratory at the University of Illinois at Urbana-Champaign. Email:[email protected]; [email protected]. This research was supported by NSF (IIS-1253758) and ARO (W911NF-10-1-0296).
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ODE control techniques [1], [3], [8]. Control using ODE approximations, however, is known to be vulnerable to the so-called spillover instabilities [2], [6] which arise due to the PDE being approximated with an insufficient number of modes. On the other hand, inclusion of a large number of modes leads to systems with large orders. In comparison, although PDE control methods that leave the PDE intact yield stability and performance guarantees on par with the ODE techniques, the control design approaches and the control architectures tend to require considerable information about, the structure of the PDE [9], [12], [13], [14], [20], [21], [22]. The development of the DPO method was motivated by practical systems described by a combination of ODEs and PDEs, such as flexible aircraft wings [16] (flexible wing structure combined with the rigid body aircraft dynamics), robotic surgical systems (a multi-segmented flexible robotic arm), temperature control systems (heat diffusion and mass flow of air), etc. Importantly, the DPO control design method is dimension-independent; i.e., it can be applied to finite as well as infinite dimensional systems (or a combination thereof). The DPO does not prescribe any particular form for the control signal; rather, the architecture only requires that the control signal satisfy certain boundness properties. The DPO framework decomposes the system in (1) into two halves for the purpose of control design: one half (called the particular half) only accommodates the nonlinearity (f (·) in (1)), while the other half (called the homogeneous half) accommodates only the control input (which could be a boundary condition of a PDE). The DPO architecture has been illustrated in Fig. 1. The control signal is designed to ensure that the output of the homogeneous half tracks the desired reference signal minus the output of the particular half, thereby ensuring that the output of the two halves put together tracks the reference signal. The stability of the closed loop is verified explicitly using the small gain theorem in the sense of finite-gain L stability. Note that the observer is used to identify the particular and homogeneous state variables. The DPO framework can accommodate modeling and parametric uncertainties as well as external disturbances. The paper is organized as follows. Mathematical preliminaries are recapitulated in Sec. II, followed by the problem formulation in Sec. III. The DPO control architecture has been presented in Sec. IV, while a finite dimensional analogue is discussed in Sec. V. Simulation results illustrating the DPO framework are presented in Sec. VI.
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define T ? , limt→∞ T C (t), and the induced norm kT ?ki , limt→∞ kT C (t)ki . III. P ROBLEM F ORMULATION We consider a class of systems described by PDEs of the form
Fig. 1. A block diagram of the DPO framework, with the subscripts p and h denoting the particular and homogeneous components, respectively.
II. P RELIMINARIES Let Z be Ra Hilbert space with the inner product defined by L hz1 , z2 i = 0 z1T z2 dx for any z1 , z2 ∈ Z, where L denotes the total span of the spatial qRdimension. The standard norm L > z z dx. on Z is given by kzkZ = 0 Definition 1: We define the space W consisting of variables w(t, x) ∈ Rn , with x ∈ [0, L] and t ∈ R+ , satisfying w(t) , w(t, ·) ∈ Z ∀ t ≥ 0 and ess supt≥0 kw(t)kZ < ∞. The space W is a Banach space with the norm s Z L w(t, x)T w(t, x)dx (2) kwkW = ess sup
wt = Aw + f (t, x, w), Bw = u(t), N w = 0, y(t) = Cw (3) where x ∈ [0, L] for some L > 0 (without any loss of generality), w , w(t, x); u(t), y(t) ∈ Rm (m < n), and A is the infinitesimal generator of a C0 semigroup T (t). The set ¯ such that wx ⊆ w, ¯ where wx = {∂q/∂x | q ∈ w}. w⊂w The operators N and B capture, respectively, the homogeneous boundary conditions and the boundary control input. The operator C is called the output operator. The domains of the boundary operators B, N , and C are assumed to contain (albeit not necessarily strictly) D(A). The right-hand side, f (t, x, w), is a nonlinear function of its arguments and in the form f (t, x, w) =
N X
αj (t) φ(x, w),
(4)
j=1
where φ(x, w) are known, smooth functions of their arguments, while the coefficients αj (t) ∈ Rn are assumed to with the truncated norm given by be unknown, but with known bounds kαj kL∞ < να and s kα˙ j kL∞ < να˙ ∀ j. We have chosen the same bounding value Z L for all j only for brevity. We assume that the state w(t) is kwkW,τ = ess sup w(t, x)T w(t, x)dx 0≤t≤τ 0 measured at all times. The control objective is to ensure that Definition 2 (L∞ and L1 norms): Given p(t) ∈ Rn with the output y(t) tracks a reference signal r(t) ∈ R [14], [18], components pi (t) (1 ≤ i ≤ n), we define [20]. Assumption 1: The C0 semigroup T (t) (whose infinitesikp(t)k∞ = max |pi (t)| 1≤i≤n mal generator is A in (3)) satisfies kT (t)ki ≤ M e−ωt , ∀ t ≥ kpkL∞ = ess sup kp(t)k∞ , kpkL∞ ,τ = ess sup kp(t)k∞ 0, where M, ω ∈ R+ . Moreover, kT ? ki is bounded. t≥0 0≤t≤τ Assumption 2: We set the initial condition w(0) = 0. n If kpkL∞ < ∞, then we denote p ∈ L∞ . The L1 norm of This assumption does not alter the fundamental nature of a linear operator F : Ln∞ 7→ Lm the stability result because it only introduces exponentially ∞ is defined as kFkL1 = supkpkL∞ =1 kFpkL∞ , p ∈ Ln∞ decaying terms, under Assumption 1. Let us denote the induced norm of an operator A by kAki , Thus, the system dynamics can be viewed as the sum of and the domain of A by D(A). When the operator A : a linear, exponentially stable, well-posed operator and an W → W is bounded, it can be checked that kAki = external nonlinear forcing term. The control design method kσmax,x (A(t))kL1 , where σmax,x (A(t)) is a time-varying can be used for systems of the form w˙ = Ag w + f (t, x, w), signal and the maximum singular value is taken over all where Ag need not be stable. The system be rewritten as w˙ = values of x at each instant in time. Given w(t) ∈ D(A), let Aw+h(t, x, w), h(t, x, w) = f (t, x, w)+Ag w−Aw, with ¯ x) denote a vector of w and its partial spatial derivatives A exponentially stable, as in the above problem formulation, w(t, with orders less than that of the highest order in A. This and realized through observer feedback (see the simulation straight-forward extension allows us to accommodate forcing example in Section VI). functions that depend on the partial derivatives of w. Assumption 3: For every ρ > 0, there exist positive conDefinition 3 ([19], Definition 1.1, Ch. 6): Consider a stants νφ,1 (ρ) and νφ,2 (ρ) such that if kwkW,τ < ρ for some system w˙ = Aw + f (t, x, w), w(t = 0) = w0 ∈ D(A), τ > 0, then kφj (x, w)kZ ≤ νφ,1 (ρ)kwkW,τ + νφ,2 (ρ) ∀ j. In where A is the infinitesimal generator of a C0 semigroup general, νφ,1 (ρ) and νφ,2 (ρ) are class K functions of ρ. It ¯ The T (t) and w ⊂ w. R t mild solution w(t) is given by follows that kf (t, x, w)kW,τ ≤ ν1 (ρ)kwkW,τ + ν2 (ρ), where w(t) = T (t)w0 + 0 T (t − τ )f (τ, x, w(τ )) dτ, where ν1 (ρ) = N να νφ,1 (ρ) and ν2 (ρ) = N να νφ,2 (ρ) w0 = w(0, ·). Assumption 3 essentially implies that the spatial derivatives Definition 4 (Convolution): Given a semi-group T (t), we of w(t, x) enter through bounded functions. define the operator T C (t) :R W 7→ W so that ∀ q(t) ∈ W Assumption 4: The output operator C in (3) is bounded; t and ∀ t > 0, T C (t)q(t) = 0 T (t − τ )q(τ ) dτ. We further i.e., kykL∞ = kCw(t, x)kL∞ ≤ KkwkW for some K > 0. t≥0
0
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RL For example, let y(t) = 0 w(t, x)dx ∈ R. Then, the Cauchy-Schwarz inequality, we get |y(t)| = using √ √ R L ≤ w(t, x)dx Lkw(t)k LkwkW . 0 Z =⇒ kykL∞ ≤ Assumption 4 is relaxed in Sec. IV-C.
We define the observation error w ˜=w ˆp + w ˆh − w. Then, from (3), (5), and (6), we conclude that the dynamics of w ˜ are given by w ˜t = Aw ˜ + fˆ(t, x, w) − f (t, x, w), Bw ˜ = Nw ˜ = 0, y˜(t) = C w(t, ˜ x).
IV. C ONTROL D ESIGN USING DPO A. Design of the Perturbation Observers We use the symbol “∧” to denote observer states, and the subscripts p and h to denote states of the particular and the homogeneous halves, respectively. The dynamics of the two halves are given by w ˆp,t = Aw ˆp + fˆ(t, x, w), B w ˆp = N w ˆp = 0, yˆp = C w ˆp (5) w ˆh,t = Aw ˆh , B w ˆh = u(t), N w ˆh = 0, yˆh = C w ˆh (6)
(9)
We will now show that it is possible to ensure that the observation error is uniformly bounded. Lemma 2: The observation error kw(t)k ˜ Z is uniformly bounded ∀ t > 0, and the bound can be made arbitrarily small. Moreover, the error in the output, y˜(t), is uniformly bounded and can be made arbitrarily small.
Proof: the Lyapunov function V = hw(t), ˜ P w(t)i+ ˜ PN Consider 1 ˜ j (t)> αj (t) where α ˜ j (t) = α ˆ j (t) − αj (t). Using j=1 α γ the fact that α ˆ˙ is found using the projection operator, and P N 1 We will design a projection-based law for calculat- (7), we get V˙ (t) ≤ −λ hw, ˜ −γ α ˜ j> (t)α˙ j (t) . P ˜ P wi j=1 P ing fˆ(·), which we write in the form fˆ(t, x, w) = P By adding and subtracting (λP /γ) j α ˜ j (t)> α ˜ j (t), N ˆ j (t)φj (x, w), 1 ≤ j ≤ N . For the system z(t) ˙ = P j=1 α N > ˙ V ≤ −λp V + (λP /γ) j=1 α ˜ j (t) α ˜ j (t) − Az(t), with z(t) ∈ Z and Bz = N z = 0, we postulate the we get PN > (1/γ) α ˜ (t) α ˙ (t). Since α ˙ , α ˆ and α are bounded existence of a self-adjoint operator P > 0 and a constant j j j j j=1 j for all t with known bounds, it follows readily that λP > 0 such that for all t, kwk ˜ W ∝ γ1 , and the constant of proportionality depends d hz(t), Pz(t)i ≤ −λP hz(t), Pz(t)i ∀ z(t) ∈ D(A)∩ker(B). on λP , να , and να˙ (see the proof of Thm. 4 in [18] as dt (7) an example). The boundedness of of y˜(t) follows from Moreover, we postulate that there exists another constant Assumption 4. Note that the bounds can be made arbitrarily µp > 0 such that µp kz(t)kZ ≤ hz(t), Pz(t)i This assump- small by choosing a suitably large value of γ. tion is, in fact, quite weak: it can be ensured by including B. DPO Control Synthesis and Closed-Loop Stability Analµp kz(t)kZ as one of the elements of the positive definite ysis function. The standard projection operator is used to design We design the control signal u(t) to ensure that the α ˆ j (t): output of the homogeneous half, yˆh (t) in (6), tracks a α ˆ˙ j,i (t) = −γProj (α ˆ j,i , hP w(t), ˜ φj (x, w)ei i) , |ˆ α(t)| < να reference signal r(t) − yˆp (t), where r(t) is the reference (8) signal for the original system (3), while yˆp is the output of where αj,i (1 ≤ i ≤ n) is the ith component of αj , ei the particular half (5). Lemma 2 would then ensure that the denotes the ith column of the n × n identity matrix, and output y(t) of the system (3) tracks the reference signal r(t), γ > 0 is called the adaptation gain. We now prove that as desired. We can write the input-output dynamics of the kw ˆp kW is bounded. linear, exponentially stable homogeneous half in (6) in the Lemma 1: If kwkW,τ < ρ for some τ > 0, then there exist Laplace domain [5]: yˆh (s) = Gc (s)u(s), where the transfer constants κ0 ≡ κ0 (ρ) and κ1 ≡ κ1 (ρ) such that kw ˆp kW,τ ≤ function Gc (s) depends on A and the boundary conditions, κ0 kwkW,τ + κ1 as explained in [5]. We design u(t) in the Laplace domain as follows: ˆ Proof: Using the projection operator for obtaining f (·), one can ensure that there exist constants κ01 (ρ) and κ02 (ρ) such u(s) = H(s)(r(s) − yˆp (s)) = H(s)(r(s) − C w ˆp (s)) (10) that kfˆ(t, x, w)kZ ≤ κ01 (ρ)kw(t)kZ +κ02 (ρ), ∀ t ≤ τ. Next, using the formula for the mild solution w ˆp (Definition 3) and The ideal choice for H(s) would then be 1/Gc (s) which is, the definition of the convolution operator (Definition 4), we in general, neither proper nor stable. Therefore, we choose H(s) to satisfy H(0)Gc (0) = 1 and impose further condiget tions for closed-loop stability (see Lemma 3 and Theorem 1). w ˆp = T (t) ? fˆ(t, x, w) Interestingly, a polynomial approximation of the state w ˆh yields a low-pass filter H(s) which ensures that u(t) is ˆ =⇒ kw ˆp (t)kZ ≤ kT ? ki kf (t, x, wkZ ∀ t sufficiently differentiable, and all derivatives are bounded =⇒ kw ˆp kW ≤ kT ? ki κ01 (ρ)kw(t)kW + κ02 (ρ) , ∀ t ≤ τ. [17], [18]. Lemma 3: If kwkW < ρ, then the control law in (10) If we define κj (ρ) = kT ? ki κ0j (ρ), for j = 1, 2, we get the desired result. The uniqueness of the mild solution is ensures that there exist constants δir ≡ δir (ρ, H(s)), δiw ≡ δiw (ρ, H(s)), and δiu ≡ δiu (ρ, H(s)) such that ku(i) kL∞ ≤ guaranteed by Theorem 6.1.4 in [19]. di u(t) We note that, since yˆp = C w ˆp , it follows from the δir krkL∞ + δiw kwkW + δiu , where u(i) (t) = dti and boundedness of C (i.e., Assumption 4) that kˆ yp k is bounded. u(0) (t) = u(t). 2067
Proof: From (10), we get kukL∞ ≤ kH(s)kL1 (krkL∞ + kC w ˆp kL∞ ). The proof follows from Assumption 4 and Lemma 1. We will prove the bounded-input-bounded-output (BIBO) stability of the closed-loop (in the sense of L∞ boundedness of signals) using the small gain theorem. We first state a known result which asserts that the boundary control system in (3) can be recast into a PDE system with homogeneous boundary conditions. Lemma 4 (Theorem 3.3.3, [4]): The coordinate transform v = w − βu, where the operator β satisfies Bβu = u and N βu = 0, transforms the system (3) into the form v˙ = Av + Aβu − β u˙ + f (t, x, w), Bv = N v = 0 (11) Assumption 5 (Small-gain condition): We assume that there exists a constant ρ and an arbitrarily small > 0 which satisfy the following property: ∆1 krkL∞ + ∆2 ≤ρ− 1 − ∆0
(12)
where ∆0 = kT ? ki δ0 + kβki δ0w , ∆1 = kT ? ki δ1 + kβki δ0r , ∆2 = kT ? ki δ2 + kβki δ0u , and δ0 = ν1 (ρ) + kAβki δ0w + kβki δ1w , δ1 = kAβki δ0r + kβki δ1r , and δ2 = ν2 (ρ) + kAβki δ0u + kβki δ1u . We are now ready to prove the main result. Theorem 1: The closed-loop system (3), (5), (6), (8), and (10) is BIBO stable in the sense of L∞ if there exists a stable, strictly proper H(s), a constant ρ and an arbitrarily small > 0 satisfying the small-gain condition in (12). Proof: We will prove this result by contradiction. Suppose that kw(τ )kZ = ρ for some time τ > 0 and that kw(t)kZ < ρ, ∀ 0 ≤ t < τ < ∞. The system in (3) is recast into (11) using Lemma 4. We use the variation of constants formula to obtain the mild solution, as in Definition 3. Thereafter, using Assumption 3 and Lemma 3, we get kvkW,τ ≤ kT ? ki (δ0 kwkW,τ + δ1 krkL∞ + δ2 ) where δ0 , δ1 , and δ2 are defined in Assumption 4. Since w = v + βu, using the triangle inequality and Lemma 3, we get kwkW,τ ≤ kT ?ki (δ0 kwkW,τ +δ1 krkL∞ + δ2 ) + kβki (δ0w kwkW,τ + δ0r krkL∞ + δ0u ), which implies ∆ krkL∞ +∆2 that kwkW,τ ≤ 1 1−∆ ≤ ρ − , from the small-gain 0 condition in (12). This contradicts our initial assumption that kw(t)kZ = ρ, and shows that kwkW < ρ. Well-posedness (i.e., existence and uniqueness of the mild solution for all t) follows from Theorem 6.1.4 in [19]. This completes the proof. C. Generalization to Unbounded Output Operators It is possible to relax Assumption 4 by deriving a stability condition of the form in Theorem 1 for a system consisting ¯ rather than just w. Recall that w ¯ is the of the dynamics of w vector of w and its partial spatial derivatives. The resulting stability condition is cumbersome and difficult to verify in a practical setting; its representation in a more tractable form
¯ dynamics are is an open problem. It must be noted that the w constructed purely as an analytical tool to verify the stability of the closed loop system (3), (5), (6), (8), and (10). We start by showing a number of properties. Given the system (3), we can differentiate the state variable w(t, x) with respect to x and recast it into the form ¯ ¯ t = A(w) ¯ + f¯(t, x, w) ¯ w w ¯ ¯ ¯ = u(t), N w ¯ = 0, y(t) = C¯w ¯ Bw
(13)
The expression for A¯ is not unique, as we show in the next example, and needs to be determined on a case-by-case basis. Example 1: Consider the PDE system ξt + ξxx = sin(ξx ) + x2 , with the boundary control ξ(t, 0) = u(t). The array ξ = [ξ, ξx ], while ξ¯ = [ξ, ξx , ξxx ]. From the above equation, we have that ξxt + ξxxx = cos(ξx )ξxx + 2 x, and 2 ξxxt + ξxxxx = cos(ξx )ξxxx + 2 − sin(ξx )ξxx so that, using the notation from (13), ∂2 0 0 − ∂x 2 ∂ ∂2 ¯ A(ξ) = 0 cos(ξx ) ∂x − ∂x 0 2 2 2 ∂ ∂ 0 cos(ξx ) ∂x − ∂x 2 2 ¯ ¯ B ξ = ξ(t, 0), ¯ = [sin(ξx ) + x2 , 2 x, 2 − sin(ξx )ξ 2 ]> . f¯(t, x, ξ) xx ¯ Note that the operator A(ξ) is the sum of the original operator A along the diagonal and additional terms that arise due to partial derivatives of ξ on the right-hand side. Note that the representation is not unique, in that 0 0 −1 ∂2 ∂2 ¯ is also an − ∂x A(ξ) = cos(ξx ) ∂x 2 2 ∂2 ∂2 0 cos(ξx ) ∂x2 − ∂x2 admissible representation. It may be possible to use convex combinations of admissible representations to get better global convergence properties, as in [7]. We prove the boundness of C w ˆp , which is essential to ensure the boundedness of the control signal u(t) in (10). This result is a direct generalization of Lemma 1. Assumption 6: The output C w ˆp (t) is bounded if kw ˆ p kZ and kw ˆp,x kZ are bounded. This is in the spirit of Agmon’s inequality (Lemma 2.4, [12]) for unbounded operators such as C w ˆp (t) = w ˆp (t, 0) (i.e., the value at a given spatial location). We impose further structure on the basis functions φj in addition to Assumption 3. Assumption 7: If there exists a constant ρ1 such that ¯ )kZ ≤ ρ1 for some time τ , then there exist constants kw(τ νp0 (ρ1 ), νp1 (ρ1 ) > 0 such that
∂φj
∂φj
∂x + ∂w wx (τ ) ≤ νp0 (ρ1 )kwkZ + νp1 (ρ1 ). Z Lemma 5: If there exists a time τ and a constant ¯ W,τ ≤ ρ1 , then there exist conρ1 such that kwk stants κ ¯0 ≡ κ ¯ 0 (ρ1 ) and κ ¯1 ≡ κ ¯ 1 (ρ1 ) such that max(kw ˆp kW,τ , kw ˆp,x kW,τ ) < κ ¯ 0 kwkW,τ + κ ¯ 1 so that |ˆ yp (t)|2 = |C w ˆp (t)|2 ≤ 2(κ0 kw(t)kZ + κ1 ), ∀ t ≤ τ. Proof: The dynamics of w ˆp,x can be written as w ˆp,xt =
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Aw ˆp,x + g(t, x, w), where g(t, x, w) =
N X j=1
α ˆ j (t)
and w ˜=w ˆ − w. The dynamics of the two halves are given by w ˆ˙ p = Am w ˆp + fˆ(w), yˆp = C w ˆp (15) ˙ w ˆ h = Am w ˆh + Bu, yˆh = C w ˆh (16)
∂φj ∂φj ¯ + wx , wx ⊆ w. ∂x ∂w
The mild solution w ˆp,x is found using Definition 3: ¯ Thus, kw w ˆp,x (t) = T (t) ? g(t, x, w). ˆp,x (t)kZ ≤ kT ? ¯ Z . Since φj (x, w)’s satisfies Assumption 7, ki kg(t, x, w)k and since maxj |ˆ αj (t)| < να (from the projection law in (8)), it follows that kw ˆp,x kW,τ ≤ κ00 (ρ1 )kwkW,τ + κ01 , where κ00 (ρ1 ) = N kT ? ki να νp0 (ρ1 ) and κ01 (ρ1 ) = N kT ? ki να νp1 (ρ1 ). Moreover, from Lemma 1, we deduce that there exist constants κ0 (ρ1 ) and κ1 (ρ1 ) such that kw ˆp kW,τ ≤ κ0 (ρ1 )kwkW,τ + κ1 (ρ1 ). Setting κ ¯ 0 = max(κ0 , κ00 ) and κ ¯ 1 = max(κ1 , κ01 ), we get max(kw ˆp kW,τ , kw ˆp,x kW,τ ) ≤ κ ¯ 0 (ρ1 )kwkW,τ + κ ¯ 1 (ρ1 ). The bound on the output y(t) follows from Assumption 6. We state the main result and allude to the fact that its proof is identical to Theorem 1. Theorem 2: Consider the system (13). Suppose that a ¯ stable representation A(w) exists and furthermore, there exists an operator A0 which gives a lower bound on the ¯ decay rate of trajectories corresponding to A(w). Then there exists a semi-group T0 (t) corresponding to A0 satisfying the growth bound similar to Assumption 1. Furthermore, there exists a BIBO (in the sense of L∞ ) stabilizing controller given by (5), (6), (8), and (10) provided the small gain condition of Theorem 1 is satisfied (with the operator T (t) replaced by T0 (t)). V. I NSIGHTS FROM F INITE D IMENSIONAL ODE S YSTEMS In this section, we will specialize Theorem 1 to finite dimensional nonlinear systems. The resulting controller and the small gain stability condition will be similar to those obtained in L1 adaptive control [10]. The conditions obtained in this section would be important while using DPO in a practical setting, where hardware implementation and verification and validation protocols would run in a finite dimensional setting. Recently, [15] presented an extension of the L1 adaptive control theory to a class of semilinear PDEs with matched nonlinearities. In contrast, the DPO can handle unmatched nonlinearities (through the particular half), which makes it applicable to realistic system models and practical problems, such as those discussed in [18]. We consider systems of the form w˙ = Am w + Bu + f (w), y = Cw (14) where w ∈ Rn , f : Rn → Rn , and u, y ∈ R. The nonlinearity f (w) is generally not matched to u(t), i.e., it is not always possible to write f (w) = Bg(w) for some nonlinear g(w). The system could be nonlinear and of the form w˙ = f0 (w) + Bu, in which case, in (14), f (w) = f0 (w)−Am w. Clearly, Am is also a design element, constrained by the stability and performance requirements. 1) DPO Controller: Let w ˆp and w ˆh denote the states of the particular and the homogeneous half. Let w ˆ=w ˆp + w ˆh
where fˆ(w) is the predicted value of f (w). This can be done using a wide range of methods, such as the projection operator and we retain this notation for simplicity. As in (10), we choose a control signal u(t) whose Laplace transform is given by u(s) = H(s)(r(s) − yˆp (s))
(17)
−1
so that yˆh (s) = C(sI − Am ) BH(s)(r(s) − yˆp (s)), and we prescribe H(0) = −1/(CAm B). 2) Closed-Loop Stability: In order to show stability of the closed-loop, we need to introduce some notation, following Sec. III. First, we assume that the nonlinearity is semiglobally Lipschitz; i.e., if for some τ > 0, kwkL∞ ,τ < ρb , where ρb ∈ R+ , then there exist constants δ0 , δ(ρb ) and δ1 , δ1 (ρb ), with δ0 , δ1 ∈ R+ such that kf (w)kL∞ ,τ ≤ δ0 kwkL∞ ,τ + δ1 . The next assumption concerns fˆ(w) in (15). Using techniques such as the projection operator to design fˆ(w), we ensure that kfˆ(w) − f (w)k∞ is globally uniformly bounded. Specifically, if for some τ > 0, kwkL∞ ,τ < ρb , where ρb ∈ R+ , then there exist constants κ0 , κ0 (ρb ) and κ1 , κ1 (ρb ), with κ0 , κ1 ∈ R+ such that kfˆ(w) − f (w)kL∞ ,τ ≤ κ0 kwkL∞ ,τ + κ1 . Finally, we state a small-gain design condition akin to (12). We assume that there exists a low pass filter H(s) and a scalar Γ > 0 such that 1−k(sI − Am )−1 BH(s)kL1 k(sI − Am )−1 kL1 kCk∞ (κ0 +δ0 ) − k(sI −Am )−1 kL1 δ0 > Γ.
(18)
We define a constant ρ = N/D, where N and D are given by N = k(sI − Am )−1 BH(s)kL1 (krkL∞ +k(sI − Am )−1 kL1 kCk∞ (δ1 + κ1 )) + k(sI − Am )−1 kL1 δ1 D = 1 − k(sI − Am )−1 kL1 δ0
(19)
−k(sI − Am )−1 BH(s)kL1 k(sI − Am )−1 kL1 kCk∞ (δ0 +κ0 ) We state the main result, similar to Lemma 1 and Theorem 1, and omit the proofs for brevity. Theorem 3: Consider the particular half of the observer in (15). If kwkL∞ < ρ = N/D, with N and D defined in (19), then the observer state w ˆp satisfies the bound kw ˆp kL∞ ≤ k(sI − Am )−1 kL1 ((δ0 + κ0 )kwkL∞ + (δ1 + κ1 )). Moreover, if the small gain condition (18) is satisfied, then the closedloop system (14), (15), (16), and (17) is BIBO stable and kwkL∞ < ρ. VI. S IMULATIONS Consider the forced wave equation θtt (t, x) − 0.1θtxx (t, x) − 2θyy (t, x) = 1000θ(t, x) (20) Z 0.1 θx (t, L = 0.1) = 0, θ(t, 0) = u(t), Cθ(t, x) = θ(t, x)dx
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ACKNOWLEDGEMENT
1
∫L0 θ(t,y) dy
Actual Reference
We gratefully acknowledge the inspiration and the encouragement from Prof. Miroslav Krstic. We thank the anonymous reviewers for the meticulous and thorough comments which helped improve the paper to its current form.
0.5
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(a) Polynomial design
∫L0 θ(t,y) dy
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(b) Finite state approximation design Fig. 2.
DPO illustrated with two different control synthesis methods.
Note that the dynamics in (20) are unstable. The DPO is designed as follows: θˆp,tt − 0.1θˆp,txx − 2θˆp,xx = Fˆ (t, x) − µ(0.1θˆp,t + 2θˆp ) θˆh,tt − 0.1θˆh,txx − 2θˆh,xx = −µ(0.1θˆh,t + 2θˆh ) (21) ˆ ˆ ˆ ˆ θp (t, 0) = θp,x (t, 0.1) = 0, θh (t, 0) = u(t), θh,x (t, 0.1) = 0, where µ is a design parameter. Simulation results for two controllers (both in the DPO framework) are shown in Fig. 2. The first controller is designed by approximating the homogeneous half of the observer in (21) by a polynomial, as explained in [18]. The second controller, of the form (17), is derived for a finite state representation of (20) obtained using Galerkin’s method. In both cases, the closed-loop system is stable and the tracking error is almost negligible when µ ∈ [450, 550] (approximately). The polynomial expansion-based controller presented in this paper has also been demonstrated in experiments on the bending vibrations of a beam [17], [18]. The closed-loop stability criterion does not necessarily inform what a good choice for the RHS would be, and an optimum design of the RHS remains an open problem. VII. C ONCLUSION We presented a novel control method for finite as well infinite dimensional systems based on a two-stage perturbation observer. The control signal is designed for the homogeneous half of the observer, while “perturbation prediction” is accomplished by putting together both halves. The robustness of the controller was proved using the small gain theorem. In particular, we showed that the L∞ -norm of the system state as well as the control inputs is uniformly bounded. Simulations were performed to demonstrate the effectiveness of the controller.
[1] A. Armaou and P. D. Christofides, “Wave suppression by nonlinear finite-dimensional control,” Chem. Eng. Sci., vol. 55, pp. 2627 – 2640, 2000. [2] M. J. Balas, “Feedback control of flexible systems,” IEEE Transactions on Automatic Control, vol. 23, no. 4, pp. 673 – 679, 1978. [3] P. D. Christofides and P. Daoutidis, “Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds,” in 36th IEEE Conf. Dec. Control, 1997, pp. 1068–1073. [4] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, 1995. [5] R. Curtain and K. Morris, “Transfer functions of distributed parameter systems: A tutorial,” Automatica, vol. 45, pp. 1101 – 1116, 2009. [6] R. F. Curtain and K. Glover, “Robust stabilization of infinite dimensional systems by finite dimensional controllers,” Systems and Control Letters, vol. 7, no. 1, pp. 41 – 47, 1986. [7] A. P. Dani, S.-J. Chung, and S. Hutchinson, “Observer design for stochastic nonlinear systems via contraction-based incremental stability,” IEEE Trans. on Autom. Control, 2013, conditionally accepted. [8] D. Dochain, N. Tali-Maamar, and J. P. Babary, “On modelling, monitoring, and control of fixed bed bioreactors,” Computers and Chemical Engineering, vol. 21, no. 11, pp. 1255 – 1266, 1997. [9] W. He, S. S. Ge, B. V. E. How, Y. S. Choo, and K. S. Hong, “Robust adaptive boundary control of a flexible marine riser with vessel dynamics,” Automatica, vol. 47, pp. 722 – 732, 2011. [10] N. Hovakimyan and C. Cao, L1 Adaptive Control Theory: Guaranteed Robustness with Fast Adaptation, ser. Advances in Design and Control. SIAM, 2010. [11] H. K. Khalil, Nonlinear Systems, 3rd ed. Pearson Education, Upper Saddle, NJ, 2000. [12] M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs. Advances in Design and Control, SIAM, 2008. [13] ——, Adaptive Control of PDEs. Princeton University Press, 2010. [14] T. Meurer and A. Kugi, “Tracking control for boundary controlled parabolic pdes with varying parameters: Combining backstepping and differential flatness,” vol. 45, 2009, pp. 1182 – 1194. [15] V. Natarajan and J. Bentsman, “Adaptive observers with projection operator and L1 adaptive controllers for infinite dimensional systems with full-state measurement,” IEEE Transactions on Automatic Control, vol. 59, no. 3, pp. 585 – 598, 2014. [16] A. A. Paranjape, S.-J. Chung, H. H. Hilton, and A. Chakravarthy, “Dynamics and performance of tailless micro aerial vehicle with flexible articulated wings,” AIAA J., vol. 50, no. 5, pp. 1177 – 1188, 2012. [17] A. A. Paranjape, J. Guan, S.-J. Chung, and M. Krstic, “PDE boundary control for Euler-Bernoulli beam using a two stage perturbation observer,” in IEEE Conf. Dec. Control, Maui, HI, 2012, pp. 4442 – 4448. [18] ——, “PDE boundary control for flexible articulated wings on a robotic aircraft,” IEEE Trans. Robot., vol. 29, no. 3, pp. 625 – 640, 2013. [19] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York. [20] A. A. Siranosian, M. Krstic, A. Smyshlyaev, and M. Bememt, “Motion planning and tracking for tip displacement and deflection angle for flexible beams,” Journal of Dynamic Systems, Measurement, and Control, vol. 131, no. 031009, 2009. [21] J.-M. Wang, B. Ren, and M. Krstic, “Stabilization and Gevrey regularity of a Schr¨odinger equation in boundary feedback with a heat equations,” IEEE Trans. on Autom. Control, vol. 57, no. 1, pp. 179 – 185, 2012. [22] J. J. Winkin, D. Dochain, and P. Ligarius, “Dynamical analysis of distributed parameter tubular reactors,” Automatica, vol. 36, pp. 349 – 361, 2000.
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Dyadic Perturbation Observer Framework for Control of a Class of Nonlinear PDE/ODE Systems Aditya A. Paranjape and Soon-Jo Chung Abstract— This paper presents the general theory of the dyadic perturbation observer framework as a generic method for controlling a class of nonlinear systems described by partial and/or ordinary differential equations. The method is particularly applicable to boundary control problems for systems described by partial differential equations. Conditions for closed-loop stability and robustness are derived using finitegain L stability theory, and the results are further specialized for finite dimensional systems.
I. I NTRODUCTION The Dyadic Perturbation Observer (DPO) framework was first presented, under the title of perturbation observerbased control, and demonstrated successfully in experiments involving control of beam bending in [17], [18]. It was designed primarily for controlling the following class of PDE systems: wt (t, x) ,
∂w(t, x) = Aw(t, x) + f (t, x, w(t, x)), ∂t
(1)
where w(t, x) is the state of the system, x is the spatial coordinate, and w denotes the vector consisting of w and some of its partial spatial derivatives. The problem formulation will be made precise later in the paper. The operator A is linear and time-invariant, while the nonlinearities are all captured by the function f (·). We denote by wx the partial derivative ∂w/∂x, additional subscripts denote higher order partial derivatives with respect to those variables. The objective of this note is to present the general theory of the DPO framework as a generic method for controlling a class of systems described by ordinary and partial differential equations (ODEs and PDEs), including combinations of both, and derive conditions for closed-loop stability and robustness. There are several well-established families of methods for designing controllers for finite dimensional systems, such as dynamic inversion, backstepping, gain scheduling, and Lyapunov function-based approaches [11]. The important feature of all these methods is that they are applicable to any given class of ODEs, although it may not always be convenient to apply one particular method to a given system. A large body of work on the control of PDE systems has focussed on approximating the PDE system by ODEs, and using any of the rich assortment of the aforementioned Aditya Paranjape is with the Department of Mechanical Engineering at McGill University. Soon-Jo Chung is with the Department of Aerospace Engineering and the Coordinated Science Laboratory at the University of Illinois at Urbana-Champaign. Email:[email protected]; [email protected]. This research was supported by NSF (IIS-1253758) and ARO (W911NF-10-1-0296).
978-1-4673-6090-6/14/$31.00 ©2014 IEEE
ODE control techniques [1], [3], [8]. Control using ODE approximations, however, is known to be vulnerable to the so-called spillover instabilities [2], [6] which arise due to the PDE being approximated with an insufficient number of modes. On the other hand, inclusion of a large number of modes leads to systems with large orders. In comparison, although PDE control methods that leave the PDE intact yield stability and performance guarantees on par with the ODE techniques, the control design approaches and the control architectures tend to require considerable information about, the structure of the PDE [9], [12], [13], [14], [20], [21], [22]. The development of the DPO method was motivated by practical systems described by a combination of ODEs and PDEs, such as flexible aircraft wings [16] (flexible wing structure combined with the rigid body aircraft dynamics), robotic surgical systems (a multi-segmented flexible robotic arm), temperature control systems (heat diffusion and mass flow of air), etc. Importantly, the DPO control design method is dimension-independent; i.e., it can be applied to finite as well as infinite dimensional systems (or a combination thereof). The DPO does not prescribe any particular form for the control signal; rather, the architecture only requires that the control signal satisfy certain boundness properties. The DPO framework decomposes the system in (1) into two halves for the purpose of control design: one half (called the particular half) only accommodates the nonlinearity (f (·) in (1)), while the other half (called the homogeneous half) accommodates only the control input (which could be a boundary condition of a PDE). The DPO architecture has been illustrated in Fig. 1. The control signal is designed to ensure that the output of the homogeneous half tracks the desired reference signal minus the output of the particular half, thereby ensuring that the output of the two halves put together tracks the reference signal. The stability of the closed loop is verified explicitly using the small gain theorem in the sense of finite-gain L stability. Note that the observer is used to identify the particular and homogeneous state variables. The DPO framework can accommodate modeling and parametric uncertainties as well as external disturbances. The paper is organized as follows. Mathematical preliminaries are recapitulated in Sec. II, followed by the problem formulation in Sec. III. The DPO control architecture has been presented in Sec. IV, while a finite dimensional analogue is discussed in Sec. V. Simulation results illustrating the DPO framework are presented in Sec. VI.
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define T ? , limt→∞ T C (t), and the induced norm kT ?ki , limt→∞ kT C (t)ki . III. P ROBLEM F ORMULATION We consider a class of systems described by PDEs of the form
Fig. 1. A block diagram of the DPO framework, with the subscripts p and h denoting the particular and homogeneous components, respectively.
II. P RELIMINARIES Let Z be Ra Hilbert space with the inner product defined by L hz1 , z2 i = 0 z1T z2 dx for any z1 , z2 ∈ Z, where L denotes the total span of the spatial qRdimension. The standard norm L > z z dx. on Z is given by kzkZ = 0 Definition 1: We define the space W consisting of variables w(t, x) ∈ Rn , with x ∈ [0, L] and t ∈ R+ , satisfying w(t) , w(t, ·) ∈ Z ∀ t ≥ 0 and ess supt≥0 kw(t)kZ < ∞. The space W is a Banach space with the norm s Z L w(t, x)T w(t, x)dx (2) kwkW = ess sup
wt = Aw + f (t, x, w), Bw = u(t), N w = 0, y(t) = Cw (3) where x ∈ [0, L] for some L > 0 (without any loss of generality), w , w(t, x); u(t), y(t) ∈ Rm (m < n), and A is the infinitesimal generator of a C0 semigroup T (t). The set ¯ such that wx ⊆ w, ¯ where wx = {∂q/∂x | q ∈ w}. w⊂w The operators N and B capture, respectively, the homogeneous boundary conditions and the boundary control input. The operator C is called the output operator. The domains of the boundary operators B, N , and C are assumed to contain (albeit not necessarily strictly) D(A). The right-hand side, f (t, x, w), is a nonlinear function of its arguments and in the form f (t, x, w) =
N X
αj (t) φ(x, w),
(4)
j=1
where φ(x, w) are known, smooth functions of their arguments, while the coefficients αj (t) ∈ Rn are assumed to with the truncated norm given by be unknown, but with known bounds kαj kL∞ < να and s kα˙ j kL∞ < να˙ ∀ j. We have chosen the same bounding value Z L for all j only for brevity. We assume that the state w(t) is kwkW,τ = ess sup w(t, x)T w(t, x)dx 0≤t≤τ 0 measured at all times. The control objective is to ensure that Definition 2 (L∞ and L1 norms): Given p(t) ∈ Rn with the output y(t) tracks a reference signal r(t) ∈ R [14], [18], components pi (t) (1 ≤ i ≤ n), we define [20]. Assumption 1: The C0 semigroup T (t) (whose infinitesikp(t)k∞ = max |pi (t)| 1≤i≤n mal generator is A in (3)) satisfies kT (t)ki ≤ M e−ωt , ∀ t ≥ kpkL∞ = ess sup kp(t)k∞ , kpkL∞ ,τ = ess sup kp(t)k∞ 0, where M, ω ∈ R+ . Moreover, kT ? ki is bounded. t≥0 0≤t≤τ Assumption 2: We set the initial condition w(0) = 0. n If kpkL∞ < ∞, then we denote p ∈ L∞ . The L1 norm of This assumption does not alter the fundamental nature of a linear operator F : Ln∞ 7→ Lm the stability result because it only introduces exponentially ∞ is defined as kFkL1 = supkpkL∞ =1 kFpkL∞ , p ∈ Ln∞ decaying terms, under Assumption 1. Let us denote the induced norm of an operator A by kAki , Thus, the system dynamics can be viewed as the sum of and the domain of A by D(A). When the operator A : a linear, exponentially stable, well-posed operator and an W → W is bounded, it can be checked that kAki = external nonlinear forcing term. The control design method kσmax,x (A(t))kL1 , where σmax,x (A(t)) is a time-varying can be used for systems of the form w˙ = Ag w + f (t, x, w), signal and the maximum singular value is taken over all where Ag need not be stable. The system be rewritten as w˙ = values of x at each instant in time. Given w(t) ∈ D(A), let Aw+h(t, x, w), h(t, x, w) = f (t, x, w)+Ag w−Aw, with ¯ x) denote a vector of w and its partial spatial derivatives A exponentially stable, as in the above problem formulation, w(t, with orders less than that of the highest order in A. This and realized through observer feedback (see the simulation straight-forward extension allows us to accommodate forcing example in Section VI). functions that depend on the partial derivatives of w. Assumption 3: For every ρ > 0, there exist positive conDefinition 3 ([19], Definition 1.1, Ch. 6): Consider a stants νφ,1 (ρ) and νφ,2 (ρ) such that if kwkW,τ < ρ for some system w˙ = Aw + f (t, x, w), w(t = 0) = w0 ∈ D(A), τ > 0, then kφj (x, w)kZ ≤ νφ,1 (ρ)kwkW,τ + νφ,2 (ρ) ∀ j. In where A is the infinitesimal generator of a C0 semigroup general, νφ,1 (ρ) and νφ,2 (ρ) are class K functions of ρ. It ¯ The T (t) and w ⊂ w. R t mild solution w(t) is given by follows that kf (t, x, w)kW,τ ≤ ν1 (ρ)kwkW,τ + ν2 (ρ), where w(t) = T (t)w0 + 0 T (t − τ )f (τ, x, w(τ )) dτ, where ν1 (ρ) = N να νφ,1 (ρ) and ν2 (ρ) = N να νφ,2 (ρ) w0 = w(0, ·). Assumption 3 essentially implies that the spatial derivatives Definition 4 (Convolution): Given a semi-group T (t), we of w(t, x) enter through bounded functions. define the operator T C (t) :R W 7→ W so that ∀ q(t) ∈ W Assumption 4: The output operator C in (3) is bounded; t and ∀ t > 0, T C (t)q(t) = 0 T (t − τ )q(τ ) dτ. We further i.e., kykL∞ = kCw(t, x)kL∞ ≤ KkwkW for some K > 0. t≥0
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RL For example, let y(t) = 0 w(t, x)dx ∈ R. Then, the Cauchy-Schwarz inequality, we get |y(t)| = using √ √ R L ≤ w(t, x)dx Lkw(t)k LkwkW . 0 Z =⇒ kykL∞ ≤ Assumption 4 is relaxed in Sec. IV-C.
We define the observation error w ˜=w ˆp + w ˆh − w. Then, from (3), (5), and (6), we conclude that the dynamics of w ˜ are given by w ˜t = Aw ˜ + fˆ(t, x, w) − f (t, x, w), Bw ˜ = Nw ˜ = 0, y˜(t) = C w(t, ˜ x).
IV. C ONTROL D ESIGN USING DPO A. Design of the Perturbation Observers We use the symbol “∧” to denote observer states, and the subscripts p and h to denote states of the particular and the homogeneous halves, respectively. The dynamics of the two halves are given by w ˆp,t = Aw ˆp + fˆ(t, x, w), B w ˆp = N w ˆp = 0, yˆp = C w ˆp (5) w ˆh,t = Aw ˆh , B w ˆh = u(t), N w ˆh = 0, yˆh = C w ˆh (6)
(9)
We will now show that it is possible to ensure that the observation error is uniformly bounded. Lemma 2: The observation error kw(t)k ˜ Z is uniformly bounded ∀ t > 0, and the bound can be made arbitrarily small. Moreover, the error in the output, y˜(t), is uniformly bounded and can be made arbitrarily small.
Proof: the Lyapunov function V = hw(t), ˜ P w(t)i+ ˜ PN Consider 1 ˜ j (t)> αj (t) where α ˜ j (t) = α ˆ j (t) − αj (t). Using j=1 α γ the fact that α ˆ˙ is found using the projection operator, and P N 1 We will design a projection-based law for calculat- (7), we get V˙ (t) ≤ −λ hw, ˜ −γ α ˜ j> (t)α˙ j (t) . P ˜ P wi j=1 P ing fˆ(·), which we write in the form fˆ(t, x, w) = P By adding and subtracting (λP /γ) j α ˜ j (t)> α ˜ j (t), N ˆ j (t)φj (x, w), 1 ≤ j ≤ N . For the system z(t) ˙ = P j=1 α N > ˙ V ≤ −λp V + (λP /γ) j=1 α ˜ j (t) α ˜ j (t) − Az(t), with z(t) ∈ Z and Bz = N z = 0, we postulate the we get PN > (1/γ) α ˜ (t) α ˙ (t). Since α ˙ , α ˆ and α are bounded existence of a self-adjoint operator P > 0 and a constant j j j j j=1 j for all t with known bounds, it follows readily that λP > 0 such that for all t, kwk ˜ W ∝ γ1 , and the constant of proportionality depends d hz(t), Pz(t)i ≤ −λP hz(t), Pz(t)i ∀ z(t) ∈ D(A)∩ker(B). on λP , να , and να˙ (see the proof of Thm. 4 in [18] as dt (7) an example). The boundedness of of y˜(t) follows from Moreover, we postulate that there exists another constant Assumption 4. Note that the bounds can be made arbitrarily µp > 0 such that µp kz(t)kZ ≤ hz(t), Pz(t)i This assump- small by choosing a suitably large value of γ. tion is, in fact, quite weak: it can be ensured by including B. DPO Control Synthesis and Closed-Loop Stability Analµp kz(t)kZ as one of the elements of the positive definite ysis function. The standard projection operator is used to design We design the control signal u(t) to ensure that the α ˆ j (t): output of the homogeneous half, yˆh (t) in (6), tracks a α ˆ˙ j,i (t) = −γProj (α ˆ j,i , hP w(t), ˜ φj (x, w)ei i) , |ˆ α(t)| < να reference signal r(t) − yˆp (t), where r(t) is the reference (8) signal for the original system (3), while yˆp is the output of where αj,i (1 ≤ i ≤ n) is the ith component of αj , ei the particular half (5). Lemma 2 would then ensure that the denotes the ith column of the n × n identity matrix, and output y(t) of the system (3) tracks the reference signal r(t), γ > 0 is called the adaptation gain. We now prove that as desired. We can write the input-output dynamics of the kw ˆp kW is bounded. linear, exponentially stable homogeneous half in (6) in the Lemma 1: If kwkW,τ < ρ for some τ > 0, then there exist Laplace domain [5]: yˆh (s) = Gc (s)u(s), where the transfer constants κ0 ≡ κ0 (ρ) and κ1 ≡ κ1 (ρ) such that kw ˆp kW,τ ≤ function Gc (s) depends on A and the boundary conditions, κ0 kwkW,τ + κ1 as explained in [5]. We design u(t) in the Laplace domain as follows: ˆ Proof: Using the projection operator for obtaining f (·), one can ensure that there exist constants κ01 (ρ) and κ02 (ρ) such u(s) = H(s)(r(s) − yˆp (s)) = H(s)(r(s) − C w ˆp (s)) (10) that kfˆ(t, x, w)kZ ≤ κ01 (ρ)kw(t)kZ +κ02 (ρ), ∀ t ≤ τ. Next, using the formula for the mild solution w ˆp (Definition 3) and The ideal choice for H(s) would then be 1/Gc (s) which is, the definition of the convolution operator (Definition 4), we in general, neither proper nor stable. Therefore, we choose H(s) to satisfy H(0)Gc (0) = 1 and impose further condiget tions for closed-loop stability (see Lemma 3 and Theorem 1). w ˆp = T (t) ? fˆ(t, x, w) Interestingly, a polynomial approximation of the state w ˆh yields a low-pass filter H(s) which ensures that u(t) is ˆ =⇒ kw ˆp (t)kZ ≤ kT ? ki kf (t, x, wkZ ∀ t sufficiently differentiable, and all derivatives are bounded =⇒ kw ˆp kW ≤ kT ? ki κ01 (ρ)kw(t)kW + κ02 (ρ) , ∀ t ≤ τ. [17], [18]. Lemma 3: If kwkW < ρ, then the control law in (10) If we define κj (ρ) = kT ? ki κ0j (ρ), for j = 1, 2, we get the desired result. The uniqueness of the mild solution is ensures that there exist constants δir ≡ δir (ρ, H(s)), δiw ≡ δiw (ρ, H(s)), and δiu ≡ δiu (ρ, H(s)) such that ku(i) kL∞ ≤ guaranteed by Theorem 6.1.4 in [19]. di u(t) We note that, since yˆp = C w ˆp , it follows from the δir krkL∞ + δiw kwkW + δiu , where u(i) (t) = dti and boundedness of C (i.e., Assumption 4) that kˆ yp k is bounded. u(0) (t) = u(t). 2067
Proof: From (10), we get kukL∞ ≤ kH(s)kL1 (krkL∞ + kC w ˆp kL∞ ). The proof follows from Assumption 4 and Lemma 1. We will prove the bounded-input-bounded-output (BIBO) stability of the closed-loop (in the sense of L∞ boundedness of signals) using the small gain theorem. We first state a known result which asserts that the boundary control system in (3) can be recast into a PDE system with homogeneous boundary conditions. Lemma 4 (Theorem 3.3.3, [4]): The coordinate transform v = w − βu, where the operator β satisfies Bβu = u and N βu = 0, transforms the system (3) into the form v˙ = Av + Aβu − β u˙ + f (t, x, w), Bv = N v = 0 (11) Assumption 5 (Small-gain condition): We assume that there exists a constant ρ and an arbitrarily small > 0 which satisfy the following property: ∆1 krkL∞ + ∆2 ≤ρ− 1 − ∆0
(12)
where ∆0 = kT ? ki δ0 + kβki δ0w , ∆1 = kT ? ki δ1 + kβki δ0r , ∆2 = kT ? ki δ2 + kβki δ0u , and δ0 = ν1 (ρ) + kAβki δ0w + kβki δ1w , δ1 = kAβki δ0r + kβki δ1r , and δ2 = ν2 (ρ) + kAβki δ0u + kβki δ1u . We are now ready to prove the main result. Theorem 1: The closed-loop system (3), (5), (6), (8), and (10) is BIBO stable in the sense of L∞ if there exists a stable, strictly proper H(s), a constant ρ and an arbitrarily small > 0 satisfying the small-gain condition in (12). Proof: We will prove this result by contradiction. Suppose that kw(τ )kZ = ρ for some time τ > 0 and that kw(t)kZ < ρ, ∀ 0 ≤ t < τ < ∞. The system in (3) is recast into (11) using Lemma 4. We use the variation of constants formula to obtain the mild solution, as in Definition 3. Thereafter, using Assumption 3 and Lemma 3, we get kvkW,τ ≤ kT ? ki (δ0 kwkW,τ + δ1 krkL∞ + δ2 ) where δ0 , δ1 , and δ2 are defined in Assumption 4. Since w = v + βu, using the triangle inequality and Lemma 3, we get kwkW,τ ≤ kT ?ki (δ0 kwkW,τ +δ1 krkL∞ + δ2 ) + kβki (δ0w kwkW,τ + δ0r krkL∞ + δ0u ), which implies ∆ krkL∞ +∆2 that kwkW,τ ≤ 1 1−∆ ≤ ρ − , from the small-gain 0 condition in (12). This contradicts our initial assumption that kw(t)kZ = ρ, and shows that kwkW < ρ. Well-posedness (i.e., existence and uniqueness of the mild solution for all t) follows from Theorem 6.1.4 in [19]. This completes the proof. C. Generalization to Unbounded Output Operators It is possible to relax Assumption 4 by deriving a stability condition of the form in Theorem 1 for a system consisting ¯ rather than just w. Recall that w ¯ is the of the dynamics of w vector of w and its partial spatial derivatives. The resulting stability condition is cumbersome and difficult to verify in a practical setting; its representation in a more tractable form
¯ dynamics are is an open problem. It must be noted that the w constructed purely as an analytical tool to verify the stability of the closed loop system (3), (5), (6), (8), and (10). We start by showing a number of properties. Given the system (3), we can differentiate the state variable w(t, x) with respect to x and recast it into the form ¯ ¯ t = A(w) ¯ + f¯(t, x, w) ¯ w w ¯ ¯ ¯ = u(t), N w ¯ = 0, y(t) = C¯w ¯ Bw
(13)
The expression for A¯ is not unique, as we show in the next example, and needs to be determined on a case-by-case basis. Example 1: Consider the PDE system ξt + ξxx = sin(ξx ) + x2 , with the boundary control ξ(t, 0) = u(t). The array ξ = [ξ, ξx ], while ξ¯ = [ξ, ξx , ξxx ]. From the above equation, we have that ξxt + ξxxx = cos(ξx )ξxx + 2 x, and 2 ξxxt + ξxxxx = cos(ξx )ξxxx + 2 − sin(ξx )ξxx so that, using the notation from (13), ∂2 0 0 − ∂x 2 ∂ ∂2 ¯ A(ξ) = 0 cos(ξx ) ∂x − ∂x 0 2 2 2 ∂ ∂ 0 cos(ξx ) ∂x − ∂x 2 2 ¯ ¯ B ξ = ξ(t, 0), ¯ = [sin(ξx ) + x2 , 2 x, 2 − sin(ξx )ξ 2 ]> . f¯(t, x, ξ) xx ¯ Note that the operator A(ξ) is the sum of the original operator A along the diagonal and additional terms that arise due to partial derivatives of ξ on the right-hand side. Note that the representation is not unique, in that 0 0 −1 ∂2 ∂2 ¯ is also an − ∂x A(ξ) = cos(ξx ) ∂x 2 2 ∂2 ∂2 0 cos(ξx ) ∂x2 − ∂x2 admissible representation. It may be possible to use convex combinations of admissible representations to get better global convergence properties, as in [7]. We prove the boundness of C w ˆp , which is essential to ensure the boundedness of the control signal u(t) in (10). This result is a direct generalization of Lemma 1. Assumption 6: The output C w ˆp (t) is bounded if kw ˆ p kZ and kw ˆp,x kZ are bounded. This is in the spirit of Agmon’s inequality (Lemma 2.4, [12]) for unbounded operators such as C w ˆp (t) = w ˆp (t, 0) (i.e., the value at a given spatial location). We impose further structure on the basis functions φj in addition to Assumption 3. Assumption 7: If there exists a constant ρ1 such that ¯ )kZ ≤ ρ1 for some time τ , then there exist constants kw(τ νp0 (ρ1 ), νp1 (ρ1 ) > 0 such that
∂φj
∂φj
∂x + ∂w wx (τ ) ≤ νp0 (ρ1 )kwkZ + νp1 (ρ1 ). Z Lemma 5: If there exists a time τ and a constant ¯ W,τ ≤ ρ1 , then there exist conρ1 such that kwk stants κ ¯0 ≡ κ ¯ 0 (ρ1 ) and κ ¯1 ≡ κ ¯ 1 (ρ1 ) such that max(kw ˆp kW,τ , kw ˆp,x kW,τ ) < κ ¯ 0 kwkW,τ + κ ¯ 1 so that |ˆ yp (t)|2 = |C w ˆp (t)|2 ≤ 2(κ0 kw(t)kZ + κ1 ), ∀ t ≤ τ. Proof: The dynamics of w ˆp,x can be written as w ˆp,xt =
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Aw ˆp,x + g(t, x, w), where g(t, x, w) =
N X j=1
α ˆ j (t)
and w ˜=w ˆ − w. The dynamics of the two halves are given by w ˆ˙ p = Am w ˆp + fˆ(w), yˆp = C w ˆp (15) ˙ w ˆ h = Am w ˆh + Bu, yˆh = C w ˆh (16)
∂φj ∂φj ¯ + wx , wx ⊆ w. ∂x ∂w
The mild solution w ˆp,x is found using Definition 3: ¯ Thus, kw w ˆp,x (t) = T (t) ? g(t, x, w). ˆp,x (t)kZ ≤ kT ? ¯ Z . Since φj (x, w)’s satisfies Assumption 7, ki kg(t, x, w)k and since maxj |ˆ αj (t)| < να (from the projection law in (8)), it follows that kw ˆp,x kW,τ ≤ κ00 (ρ1 )kwkW,τ + κ01 , where κ00 (ρ1 ) = N kT ? ki να νp0 (ρ1 ) and κ01 (ρ1 ) = N kT ? ki να νp1 (ρ1 ). Moreover, from Lemma 1, we deduce that there exist constants κ0 (ρ1 ) and κ1 (ρ1 ) such that kw ˆp kW,τ ≤ κ0 (ρ1 )kwkW,τ + κ1 (ρ1 ). Setting κ ¯ 0 = max(κ0 , κ00 ) and κ ¯ 1 = max(κ1 , κ01 ), we get max(kw ˆp kW,τ , kw ˆp,x kW,τ ) ≤ κ ¯ 0 (ρ1 )kwkW,τ + κ ¯ 1 (ρ1 ). The bound on the output y(t) follows from Assumption 6. We state the main result and allude to the fact that its proof is identical to Theorem 1. Theorem 2: Consider the system (13). Suppose that a ¯ stable representation A(w) exists and furthermore, there exists an operator A0 which gives a lower bound on the ¯ decay rate of trajectories corresponding to A(w). Then there exists a semi-group T0 (t) corresponding to A0 satisfying the growth bound similar to Assumption 1. Furthermore, there exists a BIBO (in the sense of L∞ ) stabilizing controller given by (5), (6), (8), and (10) provided the small gain condition of Theorem 1 is satisfied (with the operator T (t) replaced by T0 (t)). V. I NSIGHTS FROM F INITE D IMENSIONAL ODE S YSTEMS In this section, we will specialize Theorem 1 to finite dimensional nonlinear systems. The resulting controller and the small gain stability condition will be similar to those obtained in L1 adaptive control [10]. The conditions obtained in this section would be important while using DPO in a practical setting, where hardware implementation and verification and validation protocols would run in a finite dimensional setting. Recently, [15] presented an extension of the L1 adaptive control theory to a class of semilinear PDEs with matched nonlinearities. In contrast, the DPO can handle unmatched nonlinearities (through the particular half), which makes it applicable to realistic system models and practical problems, such as those discussed in [18]. We consider systems of the form w˙ = Am w + Bu + f (w), y = Cw (14) where w ∈ Rn , f : Rn → Rn , and u, y ∈ R. The nonlinearity f (w) is generally not matched to u(t), i.e., it is not always possible to write f (w) = Bg(w) for some nonlinear g(w). The system could be nonlinear and of the form w˙ = f0 (w) + Bu, in which case, in (14), f (w) = f0 (w)−Am w. Clearly, Am is also a design element, constrained by the stability and performance requirements. 1) DPO Controller: Let w ˆp and w ˆh denote the states of the particular and the homogeneous half. Let w ˆ=w ˆp + w ˆh
where fˆ(w) is the predicted value of f (w). This can be done using a wide range of methods, such as the projection operator and we retain this notation for simplicity. As in (10), we choose a control signal u(t) whose Laplace transform is given by u(s) = H(s)(r(s) − yˆp (s))
(17)
−1
so that yˆh (s) = C(sI − Am ) BH(s)(r(s) − yˆp (s)), and we prescribe H(0) = −1/(CAm B). 2) Closed-Loop Stability: In order to show stability of the closed-loop, we need to introduce some notation, following Sec. III. First, we assume that the nonlinearity is semiglobally Lipschitz; i.e., if for some τ > 0, kwkL∞ ,τ < ρb , where ρb ∈ R+ , then there exist constants δ0 , δ(ρb ) and δ1 , δ1 (ρb ), with δ0 , δ1 ∈ R+ such that kf (w)kL∞ ,τ ≤ δ0 kwkL∞ ,τ + δ1 . The next assumption concerns fˆ(w) in (15). Using techniques such as the projection operator to design fˆ(w), we ensure that kfˆ(w) − f (w)k∞ is globally uniformly bounded. Specifically, if for some τ > 0, kwkL∞ ,τ < ρb , where ρb ∈ R+ , then there exist constants κ0 , κ0 (ρb ) and κ1 , κ1 (ρb ), with κ0 , κ1 ∈ R+ such that kfˆ(w) − f (w)kL∞ ,τ ≤ κ0 kwkL∞ ,τ + κ1 . Finally, we state a small-gain design condition akin to (12). We assume that there exists a low pass filter H(s) and a scalar Γ > 0 such that 1−k(sI − Am )−1 BH(s)kL1 k(sI − Am )−1 kL1 kCk∞ (κ0 +δ0 ) − k(sI −Am )−1 kL1 δ0 > Γ.
(18)
We define a constant ρ = N/D, where N and D are given by N = k(sI − Am )−1 BH(s)kL1 (krkL∞ +k(sI − Am )−1 kL1 kCk∞ (δ1 + κ1 )) + k(sI − Am )−1 kL1 δ1 D = 1 − k(sI − Am )−1 kL1 δ0
(19)
−k(sI − Am )−1 BH(s)kL1 k(sI − Am )−1 kL1 kCk∞ (δ0 +κ0 ) We state the main result, similar to Lemma 1 and Theorem 1, and omit the proofs for brevity. Theorem 3: Consider the particular half of the observer in (15). If kwkL∞ < ρ = N/D, with N and D defined in (19), then the observer state w ˆp satisfies the bound kw ˆp kL∞ ≤ k(sI − Am )−1 kL1 ((δ0 + κ0 )kwkL∞ + (δ1 + κ1 )). Moreover, if the small gain condition (18) is satisfied, then the closedloop system (14), (15), (16), and (17) is BIBO stable and kwkL∞ < ρ. VI. S IMULATIONS Consider the forced wave equation θtt (t, x) − 0.1θtxx (t, x) − 2θyy (t, x) = 1000θ(t, x) (20) Z 0.1 θx (t, L = 0.1) = 0, θ(t, 0) = u(t), Cθ(t, x) = θ(t, x)dx
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0
ACKNOWLEDGEMENT
1
∫L0 θ(t,y) dy
Actual Reference
We gratefully acknowledge the inspiration and the encouragement from Prof. Miroslav Krstic. We thank the anonymous reviewers for the meticulous and thorough comments which helped improve the paper to its current form.
0.5
0
−0.5 0
5
10 Time [s]
15
R EFERENCES
20
(a) Polynomial design
∫L0 θ(t,y) dy
1
Actual Reference
0.5
0
−0.5 0
5
10 Time [s]
15
20
(b) Finite state approximation design Fig. 2.
DPO illustrated with two different control synthesis methods.
Note that the dynamics in (20) are unstable. The DPO is designed as follows: θˆp,tt − 0.1θˆp,txx − 2θˆp,xx = Fˆ (t, x) − µ(0.1θˆp,t + 2θˆp ) θˆh,tt − 0.1θˆh,txx − 2θˆh,xx = −µ(0.1θˆh,t + 2θˆh ) (21) ˆ ˆ ˆ ˆ θp (t, 0) = θp,x (t, 0.1) = 0, θh (t, 0) = u(t), θh,x (t, 0.1) = 0, where µ is a design parameter. Simulation results for two controllers (both in the DPO framework) are shown in Fig. 2. The first controller is designed by approximating the homogeneous half of the observer in (21) by a polynomial, as explained in [18]. The second controller, of the form (17), is derived for a finite state representation of (20) obtained using Galerkin’s method. In both cases, the closed-loop system is stable and the tracking error is almost negligible when µ ∈ [450, 550] (approximately). The polynomial expansion-based controller presented in this paper has also been demonstrated in experiments on the bending vibrations of a beam [17], [18]. The closed-loop stability criterion does not necessarily inform what a good choice for the RHS would be, and an optimum design of the RHS remains an open problem. VII. C ONCLUSION We presented a novel control method for finite as well infinite dimensional systems based on a two-stage perturbation observer. The control signal is designed for the homogeneous half of the observer, while “perturbation prediction” is accomplished by putting together both halves. The robustness of the controller was proved using the small gain theorem. In particular, we showed that the L∞ -norm of the system state as well as the control inputs is uniformly bounded. Simulations were performed to demonstrate the effectiveness of the controller.
[1] A. Armaou and P. D. Christofides, “Wave suppression by nonlinear finite-dimensional control,” Chem. Eng. Sci., vol. 55, pp. 2627 – 2640, 2000. [2] M. J. Balas, “Feedback control of flexible systems,” IEEE Transactions on Automatic Control, vol. 23, no. 4, pp. 673 – 679, 1978. [3] P. D. Christofides and P. Daoutidis, “Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds,” in 36th IEEE Conf. Dec. Control, 1997, pp. 1068–1073. [4] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, 1995. [5] R. Curtain and K. Morris, “Transfer functions of distributed parameter systems: A tutorial,” Automatica, vol. 45, pp. 1101 – 1116, 2009. [6] R. F. Curtain and K. Glover, “Robust stabilization of infinite dimensional systems by finite dimensional controllers,” Systems and Control Letters, vol. 7, no. 1, pp. 41 – 47, 1986. [7] A. P. Dani, S.-J. Chung, and S. Hutchinson, “Observer design for stochastic nonlinear systems via contraction-based incremental stability,” IEEE Trans. on Autom. Control, 2013, conditionally accepted. [8] D. Dochain, N. Tali-Maamar, and J. P. Babary, “On modelling, monitoring, and control of fixed bed bioreactors,” Computers and Chemical Engineering, vol. 21, no. 11, pp. 1255 – 1266, 1997. [9] W. He, S. S. Ge, B. V. E. How, Y. S. Choo, and K. S. Hong, “Robust adaptive boundary control of a flexible marine riser with vessel dynamics,” Automatica, vol. 47, pp. 722 – 732, 2011. [10] N. Hovakimyan and C. Cao, L1 Adaptive Control Theory: Guaranteed Robustness with Fast Adaptation, ser. Advances in Design and Control. SIAM, 2010. [11] H. K. Khalil, Nonlinear Systems, 3rd ed. Pearson Education, Upper Saddle, NJ, 2000. [12] M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs. Advances in Design and Control, SIAM, 2008. [13] ——, Adaptive Control of PDEs. Princeton University Press, 2010. [14] T. Meurer and A. Kugi, “Tracking control for boundary controlled parabolic pdes with varying parameters: Combining backstepping and differential flatness,” vol. 45, 2009, pp. 1182 – 1194. [15] V. Natarajan and J. Bentsman, “Adaptive observers with projection operator and L1 adaptive controllers for infinite dimensional systems with full-state measurement,” IEEE Transactions on Automatic Control, vol. 59, no. 3, pp. 585 – 598, 2014. [16] A. A. Paranjape, S.-J. Chung, H. H. Hilton, and A. Chakravarthy, “Dynamics and performance of tailless micro aerial vehicle with flexible articulated wings,” AIAA J., vol. 50, no. 5, pp. 1177 – 1188, 2012. [17] A. A. Paranjape, J. Guan, S.-J. Chung, and M. Krstic, “PDE boundary control for Euler-Bernoulli beam using a two stage perturbation observer,” in IEEE Conf. Dec. Control, Maui, HI, 2012, pp. 4442 – 4448. [18] ——, “PDE boundary control for flexible articulated wings on a robotic aircraft,” IEEE Trans. Robot., vol. 29, no. 3, pp. 625 – 640, 2013. [19] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York. [20] A. A. Siranosian, M. Krstic, A. Smyshlyaev, and M. Bememt, “Motion planning and tracking for tip displacement and deflection angle for flexible beams,” Journal of Dynamic Systems, Measurement, and Control, vol. 131, no. 031009, 2009. [21] J.-M. Wang, B. Ren, and M. Krstic, “Stabilization and Gevrey regularity of a Schr¨odinger equation in boundary feedback with a heat equations,” IEEE Trans. on Autom. Control, vol. 57, no. 1, pp. 179 – 185, 2012. [22] J. J. Winkin, D. Dochain, and P. Ligarius, “Dynamical analysis of distributed parameter tubular reactors,” Automatica, vol. 36, pp. 349 – 361, 2000.
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