Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
FrA10.4
Dissipativity-based Switching Adaptive Control Tengfei Liu, David J. Hill, and Cong Wang
Abstract— This paper presents a new switching adaptive control strategy for nonlinear systems with uncertainties on discrete finite sets. Differently to multiple model supervisory control, switching adaptive control needs no estimators for switching decisions. The switching logic is designed based on the relationship between passivity (or more generally dissipativity) and the adaptive systems, such that in the process of switching control, the transient boundary can be guaranteed by appropriately switching the parameter estimate. This makes it possible to apply the strategy in nonlinear systems modelled in local regions in the state space. Simulation will be employed to show the effectiveness of the approach.
I. I NTRODUCTION Transient boundedness is quite important for both theoretical systems synthesis and practical implementation of the controllers in applications. For instance, in nonlinear systems identification and modeling, because of the inherent nonlinear property, it is usually hard to completely identify the uncertainties. So, the models of nonlinear systems we often use for control design are locally valid along the experienced trajectories in state space and sometimes depend on the operating situation and environments. If the model is only valid in the region Ωζ in state space, the controlled state trajectory ϕ should not leave Ωζ . More examples can be found in different research areas, e.g., biological neuroscience [16] and physics [22]. However, the transient boundary is often hard to guarantee in adaptive control of systems with large uncertainties. Although researchers aimed to improve the transient performance of adaptive control systems, it still seems hard to design adaptive control algorithms for specialized plants and control objectives without trial-and-error [3]. One of the reasons is that the estimation in adaptive control is often designed by considering the unknown parameters on continuums. Actually, for a large number of practical systems, the unknown model parameters belong to discrete sets and there is no need to embed these sets in continuums [24]. The idea of using switching to improve the performance of adaptive control can be traced back to [1], in which candidate controllers and parallel estimators are designed for a plant to track deterministic set point in the presence of unmodeled dynamics. Then, the approaches known as supervisory This research was supported under the Australian Research Councils Discovery funding scheme (project number FF0455875). Tengfei Liu and David J. Hill are with Department of Information Engineering, Research School of Information Sciences & Engineering, The Australian National University, ACT 0200, Canberra, Australia
[email protected] (Tengfei Liu);
[email protected] (David J. Hill) Cong Wang is with School of Automation Science & Engineering, South China University of Technology, Guangzhou 510640, China
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
control or multiple model adaptive control were developed to overcome the limitations of conventional adaptive control (e.g.,[2] [5] [6] [4] [11] [12] [13] and [14]). A supervisory control system usually consists of three modules: a plant, candidate controllers and a supervisor. A family of candidate controllers are designed to cope with different situations depending on the uncertainties. Switching among candidate controllers is orchestrated by the high-level decision maker, called a supervisor. The supervisor is constructed with estimators corresponding to the candidate controllers. Roughly speaking, one candidate controller is selected for control if the estimation error of the corresponding estimator is the smallest one. Several switching logics have also been derived to improve the performance of supervisory control [24]. Even though switching of candidate controllers is realized for performance improvement, the transient boundary cannot be guaranteed to be small enough in complex situations due to the delay of the estimator-based switching decision, which is a dynamical process. The concept of dissipativity was originally derived from electrical networks. In systems theory, a dissipative system is defined to be one for which the supply rate (input power) and storage function (stored energy) can be found, with the property that energy is always dissipated [21]. As a special case of dissipativity, passivity has been proved to be quite effective for adaptive control. Through passivity theory, the adaptive control design problem can be ultimately solved by appropriately designing a passive block (known as the parameter estimation subsystem) to connect the estimation error of the unknown parameters to the control error of the system. Furthermore, a large class of nonlinear systems can be rendered passive via state feedback [8], based on which a passivity aimed nonlinear adaptive control was developed [17]. In this paper, we will develop a new control strategy named switching adaptive control for uncertain systems by using both the ideas of supervisory control and passivity based adaptive control. The uncertainties are not considered on continuums but on discrete sets such that candidate controllers can be designed to conquer different situations. By analyzing the relation between dissipativity/passivity and adaptive control, we will design a logic-based switching law for the candidate controllers such that transient boundary and ultimate convergence can be achieved by appropriately switching the candidate controllers. No estimation errors will be used in this new framework. The rest of the paper is organized as follows. Section 2 is problem formulation. In Section 3, we will discuss the switching adaptive control in passive and dissipative systems.
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FrA10.4 Section 4 will show the effectiveness of the approach with a simulation. Section 5 is conclusion. II. P ROBLEM F ORMULATION Switching adaptive control problem is formulated as an extension of traditional adaptive control in this section (see [17] and references therein). Uncertain nonlinear systems can generally be represented in the following form: x˙ = F (x, u, θ ∗ , v) y = H (x, u, θ ∗ , v)
(1)
where x ∈ Rn , u ∈ Rm , and y ∈ Rl are the plant state, input, and output vectors, respectively; θ ∗ ∈ Θ ⊂ R p is an unknown parameter vector; v (t) is an external signal that can be interpreted either as a reference or a disturbance. Definition 1 (Switching adaptive control problem): Consider the plant (1). Assume that the unknown parameter vector θ ∗ belongs to an a priori known discrete finite set Θ, denoted as Θ = θ 1 , . . . , θ M ⊂ R p . Given the following control objective Qt ≤ 0; t ≥ t∗ (2) where Qt = Qt (x (s) , u (s) , v (s) ; 0 ≤ s ≤ t, θ ∗ ) is the objective functional; t∗ > 0 is a time. The switching adaptive design is to find a control algorithm in the following form u (t) = ut x (s) , u (s) , θˆ (s) , v (s) ; 0 ≤ s < t (3) θˆ (t) |t ≤t 0. III. S WITCHING A DAPTIVE C ONTROL A. Strictly Passive Systems Many physical systems (e.g., Lagrange systems [21]) can be redesigned to be passive, based on which, control can be realized by constructing system composed of two passive subsystems connected by negative feedback. Note that in passivity-based adaptive control design, the parameter estimate law is often designed as a passive block which
connects the unknown parameters to the control error of the system [20]. From this standpoint, we study directly switching adaptive control of the control error system with parameter estimate error as input. The system is in the following form: x˙ = f (x) + g (x) ΦT (t) θˆ − θ ∗ (6) y = h (x) where x ∈ Rn , y ∈ Rm , Φ : [0, ∞) → Rm×M , and θ ∗ ∈ RM represents the unknown parameter vector which belongs to a finite discrete set Θ; f (·), g (·) and h (·) are smooth functions, satisfying f (0) = h (0) = 0. The system is strictly passive with u = ΦT (t) θˆ − θ ∗ as input, y = h (x)as output and V (x) ∈ C1 as storage function. Thus, using the nonlinear Kalman-Yakubovich-Popov (KYP) Lemma (see [7]), we have ∂V x˙ = −S (x) + uT y (7) V˙ = ∂x where S (x) > 0 is the positive definite dissipation term. Assumption 1: Assume there exist two class K∞ functions α1 and α2 such that V satisfies
α1 (kxk) ≤ V (x) ≤ α2 (kxk) ;
∀x ∈ Ω
(8)
where Ω is a compact set containing 0. Assumption 2: The time-varying function Φ (t) is continuous with respect to t. This assumption guarantees the existence and uniqueness of solutions of system (6). Note that if Φ is a function of both t and x, then it should be assumed that Φ is continuous in t and locally Lipschitz in x uniformly over t. Note that the control objective is to restrict the state in a local region in the state space. It is enough that the Lipschitz condition is satisfied locally in the region. It will also be shown in the following discussions that a continuous Φ (t) guarantees effective switching. Assumption 3: Let xθ˜ (t) represent the state trajectory of the system with θ˜ = θˆ − θ ∗ . For any θ i ∈ Θ and θ j ∈ Θ (θ i 6= θ j ), there exist positive constants Ti j > 0 and 0 < V < V , such that if the initial state x (0) satisfies V x(θ i −θ j ) (0) ≤ V (9) the trajectory of the system satisfies max0≤t≤Ti j V x(θ i −θ j ) (t) ≥ V limt→∞V x(θ i −θ j ) (t) ≥ V
(10) (11)
This assumption guarantees that with specific initial state x (0) only θˆ = θ ∗ can drive x to zero. Note that we use the storage function directly to describe the property. The satisfaction of this assumption can be verified by the behavior of x (t) based on Assumption 1. This assumption would be restrictive for application, and will be relaxed in next subsection with tracking control design. The following theorem provides a switching logic with a parameter V ∗ > 0 for θˆ in system (6) such that whenever the
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FrA10.4 state x (t) hits the boundary V = V ∗ , the derivative of V is rendered negative, where the storage function is a Lyapunov function. The following theorem shows the implementation of switching adaptive control for strictly passive systems. Theorem 1: Consider system (6). The following switching adaptive control logic is designed if V = V ∗ and Φ (t ∗ ) y (t ∗ ) 6= 0 θˆ − = θˆ T ∗ ∗ θˆ = arg n minθ ∈Θ θ Φ(tT ) y (t ) o ¯ = θ ∈ Θ θ − θˆ − Φ (t ∗ ) y (t ∗ ) ≥ 0 Θ ¯ Θ = Θ − θˆ − − Θ end
(12)
where V ∗ is a parameter of the switching logic which evaluates the transient boundary; t ∗ represents the time when V = V ∗ . It is designed that V ∗ satisfies V (x (0)) < V ∗ . With Assumptions 1 and 2 satisfied, the switching logic (12) guarantees 1) x (t) is bounded, i.e., kx (t)k ≤ Rx , where Rx depends on V ∗ ; 2) If Assumption 3 is satisfied, with V ∗ < V there exists a time T ∗ such that when t ≥ T ∗ , θˆ = θ ∗ , and x (t) converges to zero. Proof: Taking the derivative of V along the trajectory of (6), we have ∂V V˙ = x˙ = −S (x) + hT (x) ΦT (t) θˆ − θ ∗ ∂x
(13)
If θˆ = θ ∗ , then due to the positive definite S (x), V never diverges to V ∗ with V (x (0)) < V ∗ . Suppose there exists a time t ∗ when V (x (t ∗ )) = V ∗ . In this case, it is clear that θˆ 6= θ ∗ . We now consider the behavior of V at and after t ∗ . For the finite set Θ, because θ ∗ ∈ Θ, min θ T Φ (t ∗ ) y (t ∗ ) ≤ θ ∗T Φ (t ∗ ) y (t ∗ ) (14) θ ∈Θ
Considering the switching law (12), we have
θˆ T Φ (t ∗ ) y (t ∗ ) ≤ θ ∗T Φ (t ∗ ) y (t ∗ )
(15)
i.e.,
θˆ − θ ∗
T
Φ (t ∗ ) y (t ∗ ) ≤ 0
(16)
Because h (x) is continuous with respect to x, and x (t) is continuous with respect to t, we have h (x (t)) is continuous with respect to t. Assumption 2 guarantees that Φ (t) is continuous with respect to t. Thus, hT (x (t)) ΦT (t) is continuous T with respect to t. If θˆ − θ ∗ Φ (t ∗ ) y (t ∗ ) = 0, the problem is essentially one of free system stability and V˙ = −S (x (t ∗ )). If T θˆ − θ ∗ Φ (t ∗ ) y (t ∗ ) < 0 (17) i.e.,
hT (x (t ∗ )) ΦT (t ∗ ) θˆ − θ ∗ < 0
then, there exists a ∆t > 0, such that hT (x (t)) ΦT (t) θˆ − θ ∗ < 0; ∀t ∈ (t ∗ ,t ∗ + ∆t]
(18)
(19)
Thus, from (13), we have V (t) < V (t ∗ ) −
Zt t∗
S (x (s)) ds; ∀t ∈ (t ∗ ,t ∗ + ∆t]
(20)
where S (x (t)) is positive definite. This implies that the switching logic (12) makes V˙ < 0 once V = V ∗ . Because the initial state x (0) satisfies V (x (0)) < V ∗ , V is restricted in the region V ≤ V ∗ . With Assumption 1, there exists a Rx > 0 such that kxk ≤ Rx . With Assumption 3, V ∗ can be designed satisfying V ∗ < V such that there is only one parameter candidate in Θ that can restrict x (t) in the region V ≤ V ∗ . In this case if θˆ 6= θ ∗ , there exist a finite time t ∗ when V = V ∗ . It is obvious that switching will not stop until θˆ = θ ∗ . If T θˆ − − θ ∗ Φ (t ∗ ) y (t ∗ ) < 0, then there exists a ∆t ′ such T that θˆ − − θ ∗ Φ (t) y (t) < 0; ∀t ∈ (t ∗ − ∆t ′,t ∗ ]. Then, due to (13), V (x (t ∗ )) = V ∗ is impossible. So we have T θˆ − − θ ∗ Φ (t ∗ ) y (t ∗ ) ≥ 0. Using algorithm (12), for θ ∈ Θ, T if θ − θˆ − nΦ (t ∗ ) y (t ∗ ) > 0, then (θ − θ ∗)T Φ o (t ∗ ) y (t ∗ ) > T 0. So, ∀θ ∈ θ ∈ Θ θ − θˆ − Φ (t ∗ ) y (t ∗ ) > 0 , we have θ 6= θ ∗ . Because Θ is a finite set and switching once eliminates at least one element from Θ, θ ∗ will ultimately be selected and then the convergence of x (t) can be guaranteed by free system stability. Remark 1: It is possible that when V = V ∗ there exist more than one candidate parameters θ i ’s in Θ (e.g., a non-convex parameter set Θ may cause this), such that θ iT Φ (t ∗ ) y (t ∗ ) = minθ ∈Θ θ T Φ (t ∗ ) y (t ∗ ). In this case, the value of θˆ can be selected arbitrarily from the θ i ’s. Remark 2: This approach can be easily extended to systems with nonlinearly parameterized uncertainties. As the uncertainties we considered are time-varying and only continuity property is needed for effective switching, the regressors of the uncertainties can also be state-dependent. In this case, the regressor should be locally Lipschitz with respect to the state uniformly over t such that the uniqueness of the solution of the system can be guaranteed. Remark 3: From the proof, it can be seen that the boundary of x depends on V ∗ , while V ∗ depends on the initial state x (0). Theoretically, if x (0) = 0, V ∗ can be chosen arbitrarily small. But from the viewpoint of applications, a larger V ∗ is helpful for disturbance rejection and robustness of the system. B. Adaptive Switching for Tracking Control In the previous subsection, we developed switching algorithms to guarantee the transient boundary of x. If Assumption 3 is satisfied, by choosing V ∗ < V there exists a time T ∗ such that when t ≥ T ∗ , θˆ = θ ∗ and x converges to zero. However, this condition seems restrictive in some situations. If the initial state x (0) is large, then V ∗ should be designed large enough to satisfy V ∗ > V (x (0)). But in this case, it is difficult to make sure V ∗ < V , i.e., there is only one element in Θ that can restrict the motion in the region V ≤ V ∗ . On the other hand, without Assumption 3, algorithm (12) can guarantee only boundedness.
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FrA10.4 In traditional adaptive control, the control error system can usually be transformed into the following form: x˙ = A (t) x + B (t) ΦT (t) θˆ − θ ∗ (21) y = C (t) x where x ∈ Rn , A (t) ∈ Rn×n , B (t) ∈ Rn×m , C (t) ∈ Rm×n , and Φ (t) ∈ Rm×M . State x usually represents the control error, and θ˜ = θˆ − θ ∗ with θˆ being parameter estimate and θ ∗ being real value is parameter estimate error. Φ (t) satisfies Assumption 2. The following assumption is the time-varying version of the KYP lemma [21] [15]. Assumption 4: The matrices A (t), B (t) and C (t) satisfy the condition of linear KYP Lemma, i.e., there exist symmetric matrices P(t) and Q (t) such that ˙ = −Q (t) P(t) A (t) + AT (t) P (t) + P(t) C (t) = BT (t) P (t)
(22) (23)
Furthermore, there exist positive constants p1 , p2 , q1 and q2 such that p1 I ≤ P (t) ≤ p2 I
(24)
q1 I ≤ Q (t) ≤ q2 I
(25)
If θ ∗ is known, by letting θˆ = θ ∗ , system (21) is exponentially stable. Motivated by this, our objective in this subsection is to design a switching logic for system (21) such that the behavior of x can approximate the ideal behavior of θˆ = θ ∗ . Theorem 2: Consider system (21). Define V = xT P (t) x. Convergence of x, i.e., limt→∞ x (t) = 0 can be achieved with the following parameter switching logic if V = µ (t) and Φ (t ∗ ) y (t ∗ ) 6= 0 θˆ − = θˆ T ∗ ∗ θˆ = arg o n minθ ∈Θ θ Φ(tT ) y (t ) (26) ¯ = θ ∈ Θ θ − θˆ − Φ (t ∗ ) y (t ∗ ) ≥ 0 Θ − ¯ Θ = Θ − θˆ − Θ end q1 where µ (t) = V (0) exp − p (1+ t with δ > 0; t ∗ repreδ ) 2 sents the time when V (t ∗ ) = µ (t ∗ ). Proof: Using Assumption 4, we have (27) V˙ = −xT Q (t) x + 2yT ΦT (t) θˆ − θ ∗
In the case that θˆ = θ ∗ , system (21) is exponentially stable. We have q1 (28) V˙ = −xT Q (t) x ≤ −q1 xT x ≤ − V p2
Thus, q1 V (t2 ) ≤ V (t1 ) exp − (t2 − t1 ) ; p2
∀t2 ≥ t1 ≥ 0
(29)
If θˆ = θ ∗ , V never diverges to µ (t). Suppose that there exists a time t ∗ when V (t ∗ ) = µ (t ∗ ). In this case, it is clear
that θˆ 6= θ ∗ . We now consider the behavior of V at and after t ∗ . For the finite set Θ, because θ ∗ ∈ Θ, we have min θ T Φ (t ∗ ) y (t ∗ ) ≤ θ ∗T Φ (t ∗ ) y (t ∗ ) (30) θ ∈Θ
Considering the switching logic, we have
θˆ T Φ (t ∗ ) y (t ∗ ) ≤ θ ∗T Φ (t ∗ ) y (t ∗ ) i.e.,
θˆ − θ ∗
T
Φ (t ∗ ) y (t ∗ ) ≤ 0
(31) (32)
Because y (t) and Φ (t) are continuous with respect to t, there exists a ∆t > 0 such that T q1 q1 V˙ ≤ − V + θˆ − θ ∗ Φ (t) y (t) ≤ − V ; ∀t ∈ (t ∗ ,t ∗ + ∆t] p2 p2 (33) Thus, we have q1 V (t) ≤ V (t ∗ ) exp − (t − t ∗ ) p2 q1 t = µ (t) ; < V (0) exp − p2 (1 + δ ) ∀t ∈ (t ∗ ,t ∗ + ∆t] (34) Thus, V (t) is restricted in the region V (t) ≤ µ (t). If θˆ − − θ ∗ Φ (t ∗ ) y(t ∗ ) < 0, then there exists a ∆t ′ > 0 such that θˆ − − θ ∗ Φ (t) y (t) < 0, ∀t ∈ (t ∗ − ∆t ′,t ∗ ]. If this is true, it is impossible that V (t ∗ ) = µ (t ∗ ). This is a contradiction. So we have θˆ − − θ ∗ Φ (t ∗ ) y (t ∗ ) ≥ 0. Using T algorithm (26), for θ ∈ Θ, if θ − θˆ − Φ (t ∗ ) y (t ∗ ) > 0, then (nθ − θ ∗) T Φ (t ∗ ) y (t ∗ ) > 0. Thus, weohave θ ∗ is not in the set T θ ∈ Θ θ − θˆ − Φ (t ∗ ) y (t ∗ ) > 0 . This can effectively reduce the switching times of the logic. Using p1 I ≤ P (t) ≤ p2 I, we have V (t) ≤ µ (t) ⇒ lim kxk = 0 t→∞
(35)
Remark 4: With the converging switching parameter µ (t), Assumption 3 is not needed, as the elements in Θ that cannot drive x (t) to zero will definitely be ultimately eliminated from Θ. Remark 5: The basic idea of algorithm (26) can be extended to strictly passive system (6) because of the inherent asymptotic stability of strictly passive systems. In this case, the converging switching parameter µ (t) equals a class KL function β (V (0) ,t) satisfying β (β (V (0) ,t1 ) , (t2 − t1 )) = β (V (0) ,t2 ) for all 0 ≤ t1 ≤ t2 . The proof is easy following that of Theorem 2 and omitted here. C. General Dissipative Systems In this subsection, we further extend switching adaptive control of passive systems to general dissipative nonlinear systems with uncertainties in the following form: x˙ = f (x) + g (x) ΦT (t) θˆ − θ ∗ (36) y = h (x) + j (x) ΦT (t) θˆ − θ ∗ where x ∈ Rn , y ∈ Rm , Φ : [0, ∞) → Rm×M , and θ ∗ ∈ RM represents the unknown parameter vector; f (·), g (·), h (·) and j (·)
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FrA10.4 are smooth functions, satisfying f (0) = h (0) = 0. Assume that the system (36) is dissipative with u = ΦT (t) θˆ − θ ∗ as input, V (x) satisfying Assumption 1 as storage function and w (u, y) = yT Qy + yT Su + uT Ru as supply rate. It is also assumed that the system is reachable from the origin and zero-state detectable. Our objective is to design a switching law of θˆ such that the state x converges to zero. With the existence of uT Ru in the supply rate, the switching adaptive logic cannot be realized similarly. Note that u = ΦT (t) θˆ − θ ∗ contains the parameter estimate error, and cannot be adjusted directly. As uT Ru is quadratic in u, it is uncertain whether the adjustment of θˆ leads to a negative definite V˙ . To overcome this problem, we will use the KYP type lemma for general dissipative systems. The following theorem shows the implementation of switching adaptive control in dissipative systems with supply rate containing uT Ru. Theorem 3: Consider system (36), which is dissipative with w (u, y) as supply rate, with Q ≤ 0. The following switching adaptive control logic is designed if V = V ∗ and ∇T V (x (t ∗ )) g (x (t ∗ )) ΦT (t ∗ ) 6= 0 θˆ − = θˆ T (t ∗ )) g (x (t ∗ )) ΦT (t ∗ ) θˆ − − θ θˆ = arg minθ ∈ΘT ∇ V (x ¯ = θ ∈ Θ ∇ V (x (t ∗ )) g (x (t ∗ )) ΦT (t ∗ ) θ − θˆ − ≥ 0 Θ ¯ Θ = Θ − θˆ − − Θ end (37) where V ∗ is a parameter of the switching logic which evaluates the transient boundary; t ∗ represents the time when V = V ∗ . It is designed that V ∗ satisfies V (x (0)) < V ∗ . The switching logic (37) guarantees 1) x (t) is bounded, i.e., kx (t)k ≤ Rx , where Rx depends on V ∗ ; 2) If Assumption 3 is satisfied, with V ∗ < V there exists a time T ∗ such that when t ≥ T ∗ , θˆ = θ ∗ , and x (t) converges to zero. Sketch of proof: Using nonlinear KYP Lemma [21] [7], along the trajectory of (36) with Q ≤ 0, taking the derivative of V , we have V˙
= =
∇T V (x) x˙ ∇T V (x) f (x) + ∇T V (x) g (x) u
≤
−LT (x) L (x) + ∇T V (x) g (x) u
where x = [x1 , x2 ]T ∈ R2 and u ∈ R are the state variable and system input, respectively. f k (x) represents a family of unknown nonlinearities. The control task is tracking control of the system state x (t) in one of the unknown environments to a periodic reference orbit xd (t), generated from the following reference model x˙d1 = xd2 (40) x˙d2 = fd (xd ) where xd = [xd1 , xd2 ]T ∈ R2 is the system state and fd (xd ) is a known smooth nonlinear function. In this simulation M = 3 and (41) f 1 (x) = 1 + x22 x1 2 2 f (x) = −x1 + 1.5 1 + x1 x2 (42) 3 2 (43) f (x) = x1 − 1.5 1 + x1 x2
The real value of k in the plant is unknown. The objective is to design a switching adaptive controller for the plant such that the state of the plant can accurately track the reference model (40) where fd (xd ) = −xd1 + 1.2 1 − x2d1 xd2 . Through learning from neural control [9], the dynamics f k (x) of the plant can be represented by Gaussian neural network W¯ kT S (x) along the system state trajectory x (t), i.e., W¯ kT S (x) = f k (x) + ε for x ∈ Ωζ , with Ωζ representing the neighborhood along x (t) in which neural network approximation is not larger that ε . Correspondingly, the neural network weights are denoted with W¯ 1 , W¯ 2 and W¯ 3 . It is obviously that the neural network approximation is only locally valid. The following neural networks based controller can thus be constructed using backstepping: ˆ¯ T S (x) + α˙ (44) u = −z − c z − W 1
IV. S IMULATION In this section, we use a neural learning based control system to show the working procedure of switching adaptive control. We employ the following Brunovsky system to illustrate the implementation of switching adaptive control in neural learning control systems. x˙1 = x2 (39) x˙2 = f k (x) + u; k = 1, . . . , M
1
where z1 = x1 − xd1 , z2 = x2 − α1 , α1 = −c1 z1 + xd2 and fd (xd ) with c α˙1 = −c1 (−c1 z1 + z1 )+ 1 > 0 and c2 > 0 being control gains; Wˆ¯ ∈ W = W¯ 1 , W¯ 2 , W¯ 3 . From (39), (40), and (44), we have z˙1 = −c1 z1 + z2 (45) ˆ¯ T S (x) + f k (x) z˙2 = −z1 − c2 z2 − W In the region where the learned knowledge is valid, we have
(38)
By using the zero-state detectability, the free system asymptotic stability with θˆ = θ ∗ can be guaranteed. The basic idea of the proof is similar with that of Theorem 1 and the major approaches are omitted here.
2 2
z˙1
= −c1 z1 + z2
z˙2
ˆ¯ T S (x) + W ¯ kT S (x) + ε = −z1 − c2z2 − W T ˆ¯ − W ¯ k S (x) + ε = −z1 − c2z2 − W
(46) (47)
Because the approximation error is locally small, the influence of ε is neglected for control design. The switching sequence of f k (x) is 1 → 3 → 2. The initial states are x (0) = [1.8, −0.6]T , xd (0) = [1.8120, −0.6443]T and initially Wˆ¯ = W¯ 2 . Design V = zT z and V ∗ = 0.5. The control gains are c1 = 3 and c2 = 5. The switching sequence of the neural weights is shown in Fig. 1. The control performance is shown in Fig. 2 and 3. It is clear that the switching logic responses to the switching of the plant effectively, and the motion of the state is bounded and ultimate convergence is achieved. The knowledge stored in neural networks is effectively employed for control as expected in [9].
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FrA10.4 V. C ONCLUSION This paper introduced a novel switching adaptive control strategy for nonlinear systems with uncertainties on discrete sets. Differently to all the existing results, the switching logic for the parameter estimate is designed based on the relationship between dissipativity/passivity and adaptive control, and no estimation errors are required. Due to the guaranteed transient boundary, this result is quite useful for the nonlinear systems modelled in local regions in the state space. Further research includes robustness analysis and switching adaptive control in dynamical networks.
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Plant dynamics switching. 4
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3 2 1 0
0
5
0
5
10 15 20 time/s Candidate controllers switching.
25
30
25
30
4 3 2 1 0
10
15 time/s
20
Fig. 1. Switching sequences of the internal dynamics of the plant and the candidate controllers with switching adaptive control.
Tracking performance of x . 1
3
2
1
0
−1
−2
−3
0
5
10
15 time/s
20
25
30
Fig. 2. Tracking performance of x1 with switching adaptive control (‘–’: x1 ; ‘- -’: xd1 ; ‘-.’: x1 − xd1 ).
Tracking performance of x . 2
4 3 2 1 0 −1 −2 −3 −4
0
5
10
15 time/s
20
25
30
Fig. 3. Tracking performance of x2 with switching adaptive control (‘–’: x2 ; ‘- -’: xd2 ; ‘-.’: x2 − xd2 ).
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