On inf-convolution-based robust practical stabilization under computational uncertainty
aa r X i v : . [ ee ss . S Y ] F e b THIS WORK HAS BEEN ACCEPTED FOR PUBLICATION IN THE IEEE TRANSACTIONS ON AUTOMATIC CONTROL, DOI: 10.1109/TAC.2021.3052747 1
On inf-convolution-based robust practical stabilization undercomputational uncertainty
Patrick Schmidt , Pavel Osinenko , , Stefan Streif Abstract —This work is concerned with practical stabilization ofnonlinear systems by means of inf-convolution-based sample-and-holdcontrol. It is a fairly general stabilization technique based on a genericnon-smooth control Lyapunov function (CLF) and robust to actuatoruncertainty, measurement noise, etc. The stabilization technique itselfinvolves computation of descent directions of the CLF. It turns out thatnon-exact realization of this computation leads not just to a quantitative,but also qualitative obstruction in the sense that the result of thecomputation might fail to be a descent direction altogether and thereis also no straightforward way to relate it to a descent direction.Disturbance, primarily measurement noise, complicate the describedissue even more. This work suggests a modified inf-convolution-basedcontrol that is robust w. r. t. system and measurement noise, as wellas computational uncertainty. The assumptions on the CLF are mild,as, e. g., any piece-wise smooth function, which often results from anumerical LF/CLF construction, satisfies them. A computational studywith a three-wheel robot with dynamical steering and throttle undervarious tolerances w. r. t. computational uncertainty demonstrates therelevance of the addressed issue and the necessity of modifying theused stabilization technique. Similar analyses may be extended to othermethods which involve optimization, such as Dini aiming or steepestdescent.
Index Terms —Nonlinear systems, Stability of nonlinear systems, Com-putational methods, Computational uncertainty
I. I
NTRODUCTION
Since not every nonlinear system can be asymptotically stabilizedby a static continuous feedback [10], a great amount of research hasbeen conducted in the search for alternative methods which includetime-varying, dynamical and discontinuous control laws [2], [3], [15],[18], [19], [26], [30]. In this work, we focus specifically on discon-tinuous control laws due to their relatively simple design (cf. sliding-mode control) as compared to the case of time-varying or dynamicalcontrols whose design might be somewhat involved (compare, e. g.,[7] with [33]). Since a discontinuous control law leads, in general, toa closed-loop dynamical system with a discontinuous right-hand side,special attention must be paid to the treatment of system trajectories.A good overview of generalized notions of the system trajectoryin such cases was done by Cortes [16]. One may implement thediscontinuous control law in the sample-and-hold (SH) manner, inwhich the control actions are held constant during predefined timesamples. This enables “standard” Carath´eodory system trajectories atthe cost of given up asymptotic stability for practical stability whichdescribes convergence to any predefined vicinity of the equilibriumwithin finite time [11]. Practical stability, although being a weakerform of stability than the asymptotic one, is still widely applicable. ©2021 IEEE. Personal use of this material is permitted. Permission fromIEEE must be obtained for all other uses, in any current or future media,including reprinting/republishing this material for advertising or promotionalpurposes, creating new collective works, for resale or redistribution to serversor lists, or reuse of any copyrighted component of this work in other works.This work was partially funded by the European Union, European SocialFund ESF, Saxony. Technische Universit¨at Chemnitz, Automatic Control and System Dynam-ics Lab, 09126 Chemnitz, Germany. Computational and Data Science and Engineering Center, Skolkovo Insti-tute of Science and Technology, 143026 Moscow, Russia.
Corresponding author: Stefan Streif ([email protected]).
This work addresses practical stabilization with the use of acontrol Lyapunov function (CLF). The latter can be obtained byvarious techniques [4], [5], [6], [21], [28]. The resulting CLF isoften nonsmooth (in general, this is the case when the system failsto satisfy Brockett’s condition) [10]. This property differentiatesthe current work from other existing ones, such as [17], wherelocal differentiability is assumed. Stabilizing control actions canbe determined from the CLF in different ways [9], e. g., steepestdescent, infimum convolution (InfC), Dini aiming [22], [23] andoptimization-based feedback. Robustness properties of some of theseSH stabilizing controls were extensively studied [12], [13], [35]. It ismainly the measurement noise that might complicate the stabilizationdue to the phenomenon called “chattering” [35] whereas the modeland actuator uncertainty can be addressed straightforwardly. Theissue may be tackled by various means, such as, e. g., the so called“internal tracking controller” [25]. On the other hand, the InfC controlpossesses a natural robustness with regards to the measurement noise[35]. In this work, we focus specifically on this kind of control. Themain challenge is that the optimization problems, which are involvedin the computation of the InfC stabilizing control actions, cannot ingeneral be solved exactly . This non-exactness can be understood asa computational uncertainty . The importance of addressing it wasstated in several works, e. g., [8, Problem 8.4], [20].This works starts with a nominal system under the InfC feedback κ in the SH mode. The transition from a system ˙ x = f ( x, κ ( x )) tothe one ˙ x = f ( x, ˜ κ ( x )) , where ˜ κ denotes the InfC feedback in theSH mode under non-exact computation, was addressed in [31]. Thegoal of this work is to fuse the result of [31] with robustness w. r. t.measurement noise and system disturbance, which is a challengingtask.
Furthermore, the aim of the paper is a verified analysis ofnonlinear systems extended by a measurement error and systemdisturbance. Verified here means that an algorithm is derived whichenables computing necessary bounds on the sampling time, at leastin principle. The central result, namely, a theorem on robust practicalstabilization by InfC under computational uncertainty is presented inSection III, followed by a case study in Section IV.The core text will list technical lemmas and the main theoremwith its proof sketch, while the detailed proofs are provided in theappendix.
Notation : B R ( x ) describes a ball with radius R at x , i. e., B R ( x ) := { x : k x k ≤ R } and B R means that x = 0 ; co ( X ) denotesthe closure of the convex hull of a set X ; k•k denotes the Euclideannorm; R > , R ≥ are the sets of positive, respectively, non-negativereal numbers. II. P RELIMINARIES
A. System description and assumptions
This work addresses practical stabilization of an uncertain nonlin-ear system in the following form: ˙ x = f ( x, κ (ˆ x )) + q, (1)where x, ˆ x ∈ R n denote the state and, respectively, its measurement, q : R ≥ → R n is a (time-varying) disturbance, κ : R n → R m THIS WORK HAS BEEN ACCEPTED FOR PUBLICATION IN THE IEEE TRANSACTIONS ON AUTOMATIC CONTROL, DOI: 10.1109/TAC.2021.3052747 is a control law that only has access to the measured state ˆ x . Weassume that the admissible control actions are in some compact inputconstraint set U .The following is assumed about (1). Assumption 1 (System properties) . • (disturbance boundedness) there exist numbers ¯ e, ¯ q s. t. ∀ t ≥ k x ( t ) − ˆ x ( t ) k ≤ ¯ e and k q ( t ) k ≤ ¯ q ; • (Lipschitz property) for any z ∈ R n and ω > there exists L f = L f ( z, ω ) > such that for all x, y ∈ B ω ( z ) and for all u ∈ U , k f ( x, u ) − f ( y, u ) k ≤ L f k x − y k . (2)Notice that a system with a bounded actuator uncertainty p ( t ) , p : R ≥ → R m of the form ˙ x = f ( x, κ (ˆ x ) + p ( t )) (3)can be transformed into the form (1), using (2), and so we omitactuator uncertainty from now on. B. Controller description
Firstly, as discussed in the introduction, we implement the controllaw κ in the SH mode as follows: ˙ x = f ( x, u k ) + q,t ∈ [ kδ, ( k + 1) δ ] , u k ≡ κ (ˆ x ( kδ )) , k ∈ N , (4)where δ is the sampling time (for simplicity of further derivationsassumed constant). The starting point of practical stabilization is aproper, positive-definite, locally Lipschitz continuous control Lya-punov function (CLF) V : R n → R that satisfies the followingcondition [13]: for each compact set X ⊆ R n , there exists a compact U ( X ) ⊆ U such that ∀ x ∈ X inf θ ∈ co ( f ( x, U ( X ))) D θ V ( x ) ≤ − w ( x ) , (5)where w : R n → R is a continuous non-negative function with x = 0 = ⇒ w ( x ) > . In (5), D θ V ( x ) denotes the generalizeddirectional lower derivative in a direction θ ∈ R n , defined by D θ V ( x ) , lim inf µ → + V ( x + µθ ) − V ( x ) µ . (6)Practical stabilization is defined in the following way: Definition 1 (Practical stabilization) . Consider a system (4) with e ≡ and q ≡ . Then, a control law u = κ ( x ) practically stabilizes (4) in the sample-and-hold mode, if for all r, R with R > r > ,there exists a sufficiently small sampling time δ > such that anyclosed-loop trajectory x ( t ) with x (0) ∈ B R , is bounded and entersand remains in B r after a time T depending uniformly on r and R . To practically stabilize the system (4), the control action u k iscomputed at each time step k ∈ N . There are different techniquesfor this task as discussed in the introduction, and we focus on InfC.First, consider the following inf-convolution [14] of V : V α ( x ) := inf y ∈ R n (cid:26) V ( y ) + 12 α k y − x k (cid:27) , α ∈ (0 , . (7)The above equation is also known as Moreau-Yosida regularization[27]. For a y α ( x ) , a corresponding minimizer for (7), the vector ζ α ( x ) := x − y α ( x ) α (8)happens to be a proximal subgradient of V at x in the sense that V ( z ) ≥ V ( y α ( x )) + h ζ α ( x ) , z − y α ( x ) i − k z − y α ( x ) k α (9) holds for all z ∈ R n .The core of the InfC control under exact optimization is thefollowing property: h ζ, θ i ≤ D θ V ( x ) , (10)which holds for all proximal subgradients ζ of V at each point x and for any direction θ . The corresponding control algorithm can befound, e. g., in [13]. Namely, at each time step t k = δk , compute y α ( x k ) and ζ α ( x k ) based on the current state x k . Then, determinethe control action u k by u k ∈ U k , U k := arg min u ∈ U h ζ α ( x k ) , f ( x k , u ) i . (11)Now, under computational uncertainty , the minimizer y α ( x ) has tobe substituted with an approximate minimizer y εα ( x ) , which, for someoptimization accuracy ε x > (that may depend on x ), yields: ∀ x ∈ R n : V ( y εα ( x )) + 12 α k y εα ( x ) − x k ≤ V α ( x ) + ε x . (12)The control action κ ηx also yields merely an approximate conditionof the form h ζ εα ( x ) , f ( y εα ( x ) , κ ηx ) i ≤ inf u ∈ U ( Y ) h ζ εα ( x ) , f ( y εα ( x ) , u ) i + η x , (13)where η x > denotes the respective optimization accuracy and U ( Y ) ⊆ U is the set of admissible control actions for a given compactset Y containing y εα ( x ) , so that (5) holds for all y ∈ Y . Notice thatthe vector ζ εα ( x ) := x − y εα ( x ) α (14)is not , in general, a proximal subgradient. Consequently, the property(10), which is absolutely crucial in InfC, cannot be used directlyunder computational uncertainty .In this work, we are concerned with computational uncertainty anddo not assume exact knowledge of y α ( x ) for given α and x . Instead,we use approximate minimizers in the sense of the following: Lemma 1.
Let
R > , α ∈ (0 , and ε > . Then, for all x ∈ B R there exists an ε -minimizer y εα ( x ) for (7) satisfying k y εα ( x ) − x k ≤ (2 ¯ V ) / α, (15) where ¯ V := sup k x k≤ R V ( x ) . The inf-convolution has the following approximation propertyunder approximate minimizers:
Lemma 2.
Under the conditions of Lemma 1, for any ε > , an ε > and an α ∈ (0 , can be chosen for y εα ( x ) so as to satisfy,for all x ∈ B R , the following property: V α ( x ) ≤ V ( x ) ≤ V α ( x ) + ε . (16)In the following, we refer to the control law, whose control actionsare determined via (12) and (13) as uInfC , a shorthand for InfCcontrol under computational uncertainty . We subsequently pursue robust practical stabilization under computational uncertainty in thefollowing sense (cf. [25]):
Definition 2 (Semiglobal robust practical stabilization by uInfC) . AnuInfC is said to robustly practically stabilize (1) in the SH mode (4) if, for each R and r ∈ (0 , R ) , there exist numbers ˜ e = ˜ e ( r, R ) > , ˜ q = ˜ q ( r, R ) > , ˜ η = ˜ η ( r, R, x ) > , ˜ ε = ˜ ε ( r, R, x ) > , ˜ δ = ˜ δ ( r, R ) > , depending uniformly on r, R and x ∈ R n , such that if the followingproperties hold: • the sampling time satisfies δ ≤ ˜ δ ; CHMIDT, OSINENKO, STREIF: ROBUST PRACTICAL STABILIZATION OF NONLINEAR SYSTEMS UNDER COMPUTATIONAL UNCERTAINTY 3 • the accuracies in (12) and (13) are bounded as ε ˆ x k ≤ ˜ ε , η ˆ x k ≤ ˜ η , where ˆ x k is the sampled measured state at a step k ∈ N ; • the bounds on the measurement error and disturbance satisfy ¯ e ≤ ˜ e and ¯ q ≤ ˜ q , respectively;then, any closed-loop trajectory x ( t ) , t ≥ , x (0) = x ∈ B R isbounded and there exists T s. t. x ( t ) ∈ B r , ∀ t ≥ T . Remark 1.
The considered optimization accuracy bounds ˜ ε, ˜ η inDefinition 2 depend on the current sampled measured state ˆ x k ata sample step k ∈ N . The derived results of this work allow alsoa uniform choice of ˜ ε, ˜ η , i. e., independent of the current sampledmeasured state (see Remark 4). The next section presents the main theorem on practical robuststabilization under computational uncertainty.III. R
OBUST PRACTICAL STABILIZATION UNDERCOMPUTATIONAL UNCERTAINTY
The work [31] showed practical stabilization by InfC using acertain additional assumption on the given CLF. Here, we relax thisassumption to the following version:
Assumption 2.
For all compact sets Y , F ⊂ R n and for all ν, χ > there exist ˜ Y ⊆ Y , µ ≥ such that: for each ˜ y ∈ ˜ Y , θ ∈ F and ∀ µ ′ ∈ (0 , µ ] it holds that (cid:12)(cid:12)(cid:12)(cid:12) V (˜ y + µ ′ θ ) − V (˜ y ) µ ′ − D θ V (˜ y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ν ; (17)2) for each y ∈ Y there exists ˜ y ∈ ˜ Y such that k y − ˜ y k ≤ χ. (18) Remark 2.
The first part in Assumption 2 contains a local homogene-ity condition for all points ˜ y ∈ ˜ Y , i. e., V is globally lower Dini dif-ferentiable and the lim inf in (6) is locally uniform, as stated in [31].The second part in Assumption 2 covers all points in Y , which donot satisfy (17) . On the contrary, Assumption 1 in [31] contains onlypart 1 of Assumption 2. Nevertheless, stabilization is also possible,if (17) does not hold for all y ∈ Y but rather ˜ Y ⊂ A ⊂ R n , wherethe complement of A , denoted by A := R n \ A , is given as a setwith measure zero and ˜ Y
6⊂ B χ ( A ) := { y ∈ R n : k y − A k ≤ χ } .In Assumption 2, Y \ ˜ Y is such a set of measure zero, and part2 secures a global stabilization result. If y εα (ˆ x ) lies in such a set,Assumption 1 in [31] would not be satisfied. It can be shown that,for instance, any piece-wise affine function satisfies this assumption(a small demonstrative example is given in the appendix). SuchCLFs arise, e. g., in triangulation-based numerical constructions ofLyapunov functions [5]. Therefore, the above assumption is fulfilledby a larger set of CLFs, than Assumption 1 in [31], namely by allCLFs with countable number of sets of zero measure. Assumption2 is interpreted algorithmically in the sense that we can always beprovided with a point ˜ y for a χ that is specified later. We can now state the main result.
Theorem 1.
Consider the system (1) and let Assumption 1 hold.Let V be a CLF satisfying (5) and Assumption 2. Then, (1) can bepractically robustly stabilized by uInfC control in the SH mode (4) in the sense of Definition 2. Remark 3.
Theorem 1 ensures robust practical stability of (4) upto prescribed precision in terms of the parameters R and r , if thebounds on sampling time, system disturbance, measurement error andoptimization accuracy are fulfilled. Since the proof is constructive, thederived bounds on the sampling time can be computed, at least inprinciple, though might be conservative depending on the system, given CLF and decay rate. Nevertheless, they can be adapted toobtain more suitable bounds. Some ideas are discussed in SectionIV. Now, a sketch of the proof is presented. The whole proof can befound in the appendix. It is also the basis for the presented algorithm.
Proof. (Sketch)
The first part of the proof is concerned with derivingsome a priori bounds based on the given starting and target ballradii, say, R and r . Among these bounds, is the one on the trajectoryovershoot and, most importantly, the one on the guaranteed decayrate of V α until the state reaches the target ball. As one can see, inInfC, we work effectively with the inf-convolution V α instead of theoriginal CLF V .In the second part, to actually show sample-to-sample decay of V α ,we need to derive particular bounds on the optimization accuracies η ˆ x k and ε ˆ x k with special care. This process is complicated by the factthat we do not have access to the true state, but to only an estimatethereof, the ˆ x .In the third part, we use a property of V α analogous to Taylorseries expansion in smooth analysis (keep in mind, we work withnon-smooth tools all along). Expressing some bounds on the inter-sample system trajectory, we can show that V α decays sample-to-sample to a limit that guarantees that the true state x enters and neverleaves the target ball B r provided that some additional conditionson the sampling time and optimization accuracies hold. This partis somewhat tedious, but made possible by exploiting Assumption2. Algorithm 1 summarizes the uInfC control procedure. Algorithm 1 uInfC
Input:
System ˙ x = f ( x, u ) + q, ˆ x = x + e and a CLF V ( x ) Set:
Sampling time δ At t k = δk : Measure ˆ x k Compute y ε x α (ˆ x k ) via InfC (12) with accuracy at least ε ˆ x k Compute control action ˜ κ η x ˆ x k by (13) with accuracy at least η ˆ x k using ˜ y ε x α (ˆ x k ) from Assumption 2 Apply ˜ κ η x ˆ x k to the system and hold constant until the next sample k + 1 In the following section, we study robust practical stabilization byuInfC of the so-called extended nonholonomic dynamic integrator(ENDI) which is essentially a model of a three-wheel robot withdynamical steering and throttle. Such a model is a prototype of manyreal-world machines.IV. C
ASE STUDY : E
XTENDED NONHOLONOMIC INTEGRATOR
A three-wheel robot with dynamical actuators of the driving andsteering torques is described as follows [1], [32], [34]: ˙ ϕ = η ˙ ϕ = η ˙ ϕ = ϕ η − η ϕ ˙ η = u ˙ η = u . (ENDI)The ENDI is essentially the Brockett’s nonholonomic integrator ˙ ϕ = − ϕ | {z } =: g ( ϕ ) ω + ϕ | {z } =: g ( ϕ ) ω (NI) THIS WORK HAS BEEN ACCEPTED FOR PUBLICATION IN THE IEEE TRANSACTIONS ON AUTOMATIC CONTROL, DOI: 10.1109/TAC.2021.3052747 with additional integrators before the control inputs. A (locallysemiconcave) CLF for (ENDI) can be computed via non-smoothbackstepping as per [29]. Namely, we set the state vector as x = (cid:0) ϕ ⊤ η ⊤ (cid:1) ⊤ and V ( x ) = min θ ∈ [0 , π ) (cid:26) ˜ F ( ϕ ; θ ) + 12 k η − κ ( ϕ ; θ ) k (cid:27) , (19)where ˜ F ( ϕ ; θ ) = ϕ + ϕ + 2 ϕ − ϕ ( ϕ cos θ + ϕ sin θ ) , (20)and κ ( ϕ ; θ ) = − (cid:18) h ζ ( ϕ ; θ ) , g ( ϕ ) ih ζ ( ϕ ; θ ) , g ( ϕ ) i (cid:19) , ζ ( ϕ ; θ ) = ∇ ϕ ˜ F ( ϕ ; θ ) . (21)Note that for the minimizer θ ⋆ of (20), ˜ F ( ϕ, θ ) reduces to the CLFgiven in [9] as ˜ F ( ϕ ; θ ⋆ ) = ˜ V ( ϕ ) = ϕ + ϕ + 2 ϕ − | ϕ | q ϕ + ϕ . (22)The results of simulation under different accuracies anddisturbance bounds are presented in the following. The initialcondition is set to x = (cid:0) − . . . . (cid:1) ⊤ andthe set of admissible controls is given as U = [ − , .Furthermore, we set α = 0 . and δ = 10 − .In the first simulation, the influence of the optimization accuracy onthe state convergence is studied. Fig. 1 shows the CLF behavior andthe norm of the states along with the controls for different values of ε ˆ x and η ˆ x , namely ε ˆ x = η ˆ x ∈ { − , − , − , − } , and ¯ q = ¯ e =0 . · − . It can be observed that insufficient accuracy ( ε ˆ x = η ˆ x =10 − or ε ˆ x = η ˆ x = 10 − ) leads to the loss of practical stability.Higher accuracies lead to ever smaller vicinities of the origin thatthe state converges into. This clearly demonstrates that computationaluncertainty must be taken into account in practical stabilization.In the second simulation, the influence of ¯ e and ¯ q is investigated.We set ¯ e = ¯ q ∈ { . · − , . · − , . · − , . · − } , and ε ˆ x = η ˆ x = 10 − . From Fig. 2 it can be observed, that the trajectoryconverges faster to the origin for smaller measurement errors anddisturbance bounds. For ¯ e = ¯ q = 0 . · − , the algorithm fails tostabilize the system.Finally, it can be observed that the results only have smallimprovements for much higher restrictions on optimization accuracyand error bounds. Based on the algorithm derived from the proof ofTheorem 1, i. e., Algorithm 2, an upper bound for the sampling timecan be stated as ¯ δ = 0 . · − and for the optimization accuracyas ε ˆ x = 0 . · − . Thus, the computation of a verified boundon the sampling time is plausible, but rather conservative (whichis somewhat expected). The computed bounds might be relaxedprovided with some physical insight into the given system, such asmaximum velocity of the respective differential equation, for instance.A more detailed discussion on this requires future work and goesbeyond the scope of the current one. Algorithm 2
Upper bounds for sampling time, optimization accura-cies and error bounds based on the proof of Theorem 1
Input:
System ˙ x = f ( x, u ) + q, ˆ x = x + e and CLF V ( x ) Set: R , r , ¯ e , ¯ q , U Compute α ( x ) , α ( x ) and w ( x ) such that α ( x ) ≤ V ( x ) ≤ α ( x ) and (27) hold. Define ̺ V ( x ) = α ( x ) and λ V ( x ) = α − ( x ) . Compute ˆ V , ˆ R ∗ , ˆ v , ˆ r ∗ , ¯ f , L V , L f , and ¯ w according to part 1in the proof of Theorem 1. Compute upper bounds for ε , α , η x , δ , ε ˆ x , χ , ¯ e , T α based on(42)-(45) and (53). V ( ˆ x k ) . . k ˆ x k k ε = η = 10 − ε = η = 10 − ε = η = 10 − ε = η = 10 − Fig. 1. Norm of the state k ˆ x k k and Lyapunov function V (ˆ x k ) for differentoptimization accuracies. V ( ˆ x k ) . . k ˆ x k k ¯ q = ¯ e = 0 . · − ¯ q = ¯ e = 0 . · − ¯ q = ¯ e = 0 . · − ¯ q = ¯ e = 0 . · − Fig. 2. Norm of the state k ˆ x k k and Lyapunov function V (ˆ x k ) for differenterror and disturbance bounds. V. C
ONCLUSION
This work was concerned with practical robust stabilization ofnonlinear systems under computational uncertainty related to non-exact optimization. We showed that, under a mild assumption on theCLF, the InfC controller can robustly practically stabilize the givensystem even if the computations involved are merely approximate.The result should be seen as complementary to the existing oneswhich are only concerned with robustness regarding system and mea-surement noise. Summarizing, in addressing practical stabilization,computational uncertainty should be considered along with otheruncertainties, especially in the cases where safety is crucial.VI. A
PPENDIX
A. Demonstration of Assumption 2
Example 1.
Consider V ( x ) = | x | . Let Y , F be given and choose ˜ Y = Y \ [ − χ / , χ / ] . Without loss of generality, y > is considered(the other cases are treated analogously). Since F is compact, thereexist bounds such that for all θ ∈ F : θ min ≤ θ ≤ θ max . Furthermore,let µ be bounded by µ < χ θ max . Then, there are two possible cases. CHMIDT, OSINENKO, STREIF: ROBUST PRACTICAL STABILIZATION OF NONLINEAR SYSTEMS UNDER COMPUTATIONAL UNCERTAINTY 5 • Case 1: y > and y + µθ > :Since y + µθ > ⇔ y > − µθ > − µθ max > − χ θ max θ max = χ ,this case means, that y ∈ ˜ Y . Here, we obtain D θ V ( y ) = lim inf µ → y + µθ − yµ = θ. Furthermore, (17) holds, since (cid:12)(cid:12)(cid:12)(cid:12) y + µ ′ θ − yµ ′ − D θ V ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 ≤ ν. Thus, (17) holds for all y > χ / . • Case 2: y > and y + µθ ≤ :In this case, y ∈ (0 , χ / ] and (17) does not hold, since D θ V (˜ y ) = − θ + lim inf µ → − y / µ can not be computed, butbased on (18) , a point ˜ y ∈ ˜ Y can be chosen. Then, this pointsatisfies (17) , since ˜ y > χ / is just case 1.B. Proof of Lemma 1Proof. Define R := (2 ¯ V ) / α . Then, inf k x − y k≤ R (cid:26) V ( y ) + 12 α k y − x k (cid:27) ≤ V ( x ) ≤ ¯ V holds for all x ∈ B R . Furthermore, for any R > R , inf R ≤k x − y k≤ R (cid:26) V ( y ) + 12 α k y − x k (cid:27) ≥ α R ≥ ¯ V holds as well. Therefore, inf y ∈ R n (cid:26) V ( y ) + k y − x k α (cid:27) = inf k x − y k≤ R (cid:26) V ( y ) + k y − x k α (cid:27) . C. Proof of Lemma 2Proof.
The first inequality follows directly from the definition of theInfC according to (7). Lemma 1 implies k y εα ( x ) − x k ≤ (2 ¯ V ) / ,since α < . Choose ε such that (2 ¯ V ) / ≤ ε L V . Then, byLipschitzness of V , | V ( x ) − V ( y εα ( x )) | ≤ L V k x − y εα ( x ) k ≤ ε / follows, and also V ( x ) − V ( y εα ( x )) ≤ ε / ⇔ V ( x ) ≤ V ( y εα ( x )) + ε / . Furthermore, (12) yields V ( y εα ( x )) ≤ V α ( x )+ ε ≤ V α ( x )+ ε / .Combining these two inequalities yields the desired result. D. Proof of Theorem 1Proof.
The proof is split into four parts. Preliminary settings aremade in the first part. In the second one, a relaxed decay conditionof the CLF and InfC is presented. The actual decay is demonstratedin the third part and in the last part, the parameters for the decay aredetermined.
Part 1: Preliminaries
Let B r be the target and B R the starting ball for x , respectively.Construct two non-decreasing functions ̺ V and λ V with theproperties ∀ x ∈ R n , r, v > V ( x ) ≤ ̺ V ( r ) = ⇒ k x k ≤ r (23)and V ( x ) ≥ v = ⇒ k x k ≥ λ V ( v ) . (24)By Lemma 4.3 in [24], there exist two class K ∞ functions α and α s. t. V ( x ) can be bounded via α ( k x k ) ≤ V ( x ) ≤ α ( k x k ) , ∀ x ∈ R n . Taking ̺ V ( r ) as α ( r ) and λ V ( r ) as α − ( r ) yield the above properties. Due to Lemma 2, (16) holds for any ε ∈ R . It follows that V α ( x ) ≤ ̺ V ( r ) − ε = ⇒ V ( x ) ≤ ̺ V ( r ) = ⇒ k x k ≤ r (25)and V α ( x ) ≥ v = ⇒ V ( x ) ≥ v = ⇒ k x k ≥ λ V ( v ) . (26)Let q and e be bounded from above by ¯ q ≤ r and, respectively, ¯ e ≤ r for all t ≥ according to Assumption 1.Define ˆ R := R + ¯ e + ¯ q , which is given as the radius of the startingball for ˆ x and set ˆ V := sup k x k≤ ˆ R V ( x ) . Choose ˆ R ∗ and define Θ such that ˆ V ≤ Θ := ̺ V ( ˆ R ∗ ) holds. If V (ˆ x ) ≤ ̺ V ( ˆ R ∗ ) , then k ˆ x k ≤ ˆ R ∗ and, furthermore, k x k ≤ R ∗ := ˆ R ∗ + ¯ e . Thus, ˆ R ∗ yieldsan overshoot bound for the measured state ˆ x and R ∗ is given as anovershoot bound for the real state x . Define ˆ V ∗ := sup k x k≤ ˆ R ∗ V ( x ) .Let ˆ r := r − ¯ e − ¯ q be the radius of the target ball for ˆ x and define ˆ v := ̺ V (ˆ r ) . Then, V (ˆ x ) ≤ ̺ V (ˆ r ) implies k ˆ x k ≤ ˆ r and k x k ≤ r .Set ˆ r ∗ := λ V ( ˆ v / ) , which is denoted as the radius of a ball, neverbe entered by ˆ x ( t ) .It follows, that V (ˆ x ) ≥ ˆ v / implies k ˆ x k ≥ ˆ r ∗ and k x k ≥ r ∗ :=ˆ r ∗ − ¯ e .Let U ∗ ⊆ U be the compact set corresponding to B ˆ R ∗ + √ V ∗ in(5). Then, ∀ x ∈ B ˆ R ∗ + √ V ∗ : inf θ ∈ co ( f ( x, U ∗ )) D θ V ( x ) ≤ − w ( x ) . (27)Let L f be the Lipschitz constant of f on B ˆ R ∗ + √ V ∗ . Finally, set ¯ f := sup x ∈B ˆ R ∗ + √ V ∗ u ∈ U ∗ k f ( x, u ) k , ¯ w := inf ˆ r ∗ ≤k x k≤ ˆ R ∗ + √ V ∗ w ( x ) , (28)and consider the Lipschitz condition for the CLF with | V ( y ) − V ( x ) | ≤ L V k y − x k , ∀ x, y ∈ B ˆ R ∗ + √ V ∗ . Part 2: Establishing decay rate
Consider x ∈ B ˆ R ∗ + √ V ∗ . Let y ε α (ˆ x ) be an approximate mini-mizer of V α ( x ) satisfying (12) and define the corresponding proximal ε x -subgradient ζ ε α (ˆ x ) as ζ ε α (ˆ x ) := ˆ x − y ε α (ˆ x ) α . The minimizer y ε α (ˆ x ) must not necessarily satisfy (17), but based on (18), a point ˜ y ε α (ˆ x ) in a ball of radius χ centered at the minimizer can be found s. t. (17)holds. It means, that ˜ y ε α (ˆ x ) ∈ B χ ( y ε α (ˆ x )) , i. e., that this point iswithin a χ -ball of the respective approximate minimizer. It is usedto define ˜ ζ ε α (ˆ x ) := ˆ x − ˜ y ε α (ˆ x ) α . In the following, a decay conditionwill be established for the scalar product D ˜ ζ ε α (ˆ x ) , f (ˆ x, ˜ κ η ˆ x ) E , where ˜ κ η ˆ x ∈ U ∗ is given as a control law satisfying (13) for a given η ˆ x .The parameters ε ˆ x and η ˆ x will be determined later.With the help of the Lipschitz constant L f , equations (2) and (13),the following inequality holds for any ˆ x ∈ B ˆ R ∗ \ B ˆ r ∗ : D ˜ ζ ε α (ˆ x ) , f (ˆ x, ˜ κ η ˆ x ) E = D ˜ ζ ε α (ˆ x ) , f (˜ y ε α (ˆ x ) , ˜ κ η ˆ x ) E + D ˜ ζ ε α (ˆ x ) , f (ˆ x, ˜ κ η ˆ x ) − f (˜ y ε α (ˆ x ) , ˜ κ η ˆ x ) E ≤ inf u ∈ U ∗ D ˜ ζ ε α (ˆ x ) , f (˜ y ε α (ˆ x ) , u ) E + η ˆ x + (cid:13)(cid:13)(cid:13) ˜ ζ ε α (ˆ x ) (cid:13)(cid:13)(cid:13) L f (cid:13)(cid:13)(cid:13) ˆ x − ˜ y ε α (ˆ x ) (cid:13)(cid:13)(cid:13) . (29)Notice that y ε α (ˆ x ) is an ε x -minimizer for the inf-convolution (7). Thecontrol actions are determined in an approximate format characterizedby η ˆ x . For now, using the relations (12), (14), (18), and the definition THIS WORK HAS BEEN ACCEPTED FOR PUBLICATION IN THE IEEE TRANSACTIONS ON AUTOMATIC CONTROL, DOI: 10.1109/TAC.2021.3052747 ˆ R ∗ + p V ∗ ˆ R ∗ ˆ R ˆ r ˆ r ∗ x k ˆ x k ˜ y ε α (ˆ x k ) y ε α (ˆ x k ) ˜ Y k ⊆ Y k Y k := B√ V ∗ α (ˆ x k ) ≤ χ x k True state ˆ x k Measured state y ε α (ˆ x k ) Approximative ˜ y ε α (ˆ x k ) A pointminimizer of (7) satisfying (18) B R ∗ Overshoot B ˆ R ∗ Overshootbound ( x ) bound ( ˆ x ) B ˆ R Starting ball ( ˆ x ) B R Starting ball ( x ) B r Target ball ( x ) B ˆ r Target ball ( ˆ x ) B ˆ r ∗ / Core ball ( ˆ x ) B r ∗ / Core ball ( x ) Fig. 3. A schematic picture of the geometric setting of the proof. of ˜ ζ ε α (ˆ x ) , an upper bound for (cid:13)(cid:13)(cid:13) ˜ ζ ε α (ˆ x ) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ y ε α (ˆ x ) − ˆ x (cid:13)(cid:13)(cid:13) in (29) canbe determined by (cid:13)(cid:13)(cid:13) ˜ ζ ε α (ˆ x ) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ y ε α (ˆ x ) − ˆ x (cid:13)(cid:13)(cid:13) = 1 α (cid:13)(cid:13)(cid:13) ˜ y ε α (ˆ x ) − ˆ x (cid:13)(cid:13)(cid:13) ≤ α (cid:16)(cid:13)(cid:13)(cid:13) ˜ y ε α (ˆ x ) − y ε α (ˆ x ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) y ε α (ˆ x ) − ˆ x (cid:13)(cid:13)(cid:13)(cid:17) ≤ α (cid:18)(cid:13)(cid:13)(cid:13) ˜ y ε α (ˆ x ) − y ε α (ˆ x ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) y ε α (ˆ x ) − ˆ x (cid:13)(cid:13)(cid:13) (cid:19) ≤ α ( χ + 2 α ( V (ˆ x ) − V ( y ε α (ˆ x )) + ε x )) . (30)The second inequality in (30) result from the fact, that for any a, b > , it holds that ( a + b ) = a + b + 2 ab ≤ a + 2 b , since ≤ ( a − b ) = a + b − ab ⇔ ab ≤ a + b . Combining (29) and (30)and choosing α such that p V ∗ α ≤ n ˆ r ∗ , ε L V o holds, the followingequation can be obtained, which holds for all ˆ x ∈ B ˆ R ∗ \ B ˆ r ∗ : D ˜ ζ ε α (ˆ x ) , f (ˆ x, ˜ κ η ˆ x ) E ≤ inf u ∈ U ∗ D ˜ ζ ε α (ˆ x ) , f (˜ y ε α (ˆ x ) , u ) E + η ˆ x + L f (cid:18) α ( χ + 2 α ( V (ˆ x ) − V ( y ε α (ˆ x )) + ε x )) (cid:19) ≤ inf u ∈ U ∗ D ˜ ζ ε α (ˆ x ) , f (˜ y ε α (ˆ x ) , u ) E + η ˆ x + L f (cid:18) α ( χ + 2 α ( ε + ε x )) (cid:19) . (31)Furthermore, observe that (cid:13)(cid:13)(cid:13) y ε α (ˆ x ) (cid:13)(cid:13)(cid:13) ∈ [ ˆ r ∗ / , ˆ R ∗ + p V ∗ ] , since ˆ x ∈ B ˆ R ∗ \ B ˆ r ∗ . Part 3: Deriving decay along system trajectories
Consider an arbitrary ˆ x ∈ X , where X ⊂ R n is compact. For itssubgradient ζ ε α (ˆ x ) , the following condition holds for any ˆ x ∈ X and h ∈ R , θ ∈ R n : V α (ˆ x + hθ ) ≤ V α (ˆ x ) + h D ζ ε α (ˆ x ) , θ E + h k θ k α + ε x . (32)Note that (32) does not hold for ˜ ζ ε α (ˆ x ) instead of ζ ε α (ˆ x ) , since it isnot even an approximative proximal subgradient. Therefore, observethat for all ˆ x ∈ B ˆ R ∗ + √ V ∗ : ζ ε α (ˆ x ) = ˆ x − y ε α (ˆ x ) α = ˆ x − ˜ y ε α (ˆ x ) + ˜ y ε α (ˆ x ) − y ε α (ˆ x ) α = ˜ ζ ε α (ˆ x ) + ˜ y ε α (ˆ x ) − y ε α (ˆ x ) α (33)holds. Furthermore, based on (18), (cid:13)(cid:13)(cid:13) ˜ y ε α (ˆ x ) − y ε α (ˆ x ) (cid:13)(cid:13)(cid:13) ≤ χ holds aswell. Consider Taylor expansion (32) and (33). Then, the followinginequalities hold: V α (ˆ x + hθ ) ≤ V α (ˆ x ) + h D ζ ε α (ˆ x ) , θ E + h k θ k α + ε x ≤ V α (ˆ x ) + h D ˜ ζ ε α (ˆ x ) , θ E + h k θ k α + ε x + h χα k θ k . (34)Assume now that the trajectory of (4) exists locally on the samplingperiod [ kδ, ( k +1) δ ] and that V α (ˆ x k ) ≤ ˆ V holds. To see that it existson the entire sampling period, observe that, based on Lemma 2 with V α (ˆ x ) ≤ V (ˆ x ) ≤ V α (ˆ x ) + ε , ∀ ˆ x ∈ B ˆ R ∗ , (35)the following inequalities hold for t ∈ [ kδ, ( k + 1) δ ] : V (ˆ x ( t )) ≤ V α (ˆ x ( t )) + ε ≤ ˆ V + ε . (36)Inequality (36) is used to show that the trajectory ˆ x ( t ) exists on theentire sampling period and it can be also used to find a bound for ε to satisfy V (ˆ x ( t )) ≤ Θ which implies k ˆ x ( t ) k ≤ ˆ R ∗ and that means,that the overshoot is bounded, ˆ x ( t ) ∈ B ˆ R ∗ and x ( t ) ∈ B R ∗ , forall t ≥ . It is shown in the following steps, that V α (ˆ x k ) can onlydecrease to a prescribed limit sample-wise, i. e., V α (ˆ x k +1 ) ≤ V α (ˆ x k ) for k ∈ N until V α (ˆ x k ) ≤ ˆ v . This ensures the boundedness of thetrajectory at each sampling period.Now, consider the following cases. Case 1: V α (ˆ x k ) ≥ ˆ v (Outside the core ball)The trajectory ˆ x ( t ) can be expressed as ˆ x ( t ) = ˆ x k + Z tkδ f (ˆ x ( τ ) , ˜ κ η ˆ x k ) + q ( τ ) d τ = ˆ x k + δ δ (cid:18)Z tkδ f (ˆ x ( τ ) , ˜ κ η ˆ x k ) + q ( τ ) d τ (cid:19)| {z } =: F k . (37)Furthermore, the following inequality can be obtained using (34): V α (ˆ x ( t )) − V α (ˆ x k ) = V α (ˆ x k + δF k ) − V α (ˆ x k ) ≤ δ D ˜ ζ ε α (ˆ x k ) , F k E + δ k F k k α + ε x k + δ χα k F k k (38)for all t ∈ [ kδ, ( k +1) δ ] with ∆ t := t − kδ . Since F k can be boundedas k F k k ≤ δ ∆ t ( ¯ f + ¯ q ) , it can be re-expressed as F k = ∆ tδ f (ˆ x k , ˜ κ η ˆ x k ) + 1 δ Z tkδ q ( τ ) d τ + 1 δ Z tkδ f (ˆ x ( τ ) , ˜ κ η ˆ x k ) − f (ˆ x k , ˜ κ η ˆ x k ) d τ | {z } =: A (39) CHMIDT, OSINENKO, STREIF: ROBUST PRACTICAL STABILIZATION OF NONLINEAR SYSTEMS UNDER COMPUTATIONAL UNCERTAINTY 7 and k A k ≤ δ ∆ t L f ¯ f , where ¯ q is bounded later. Under Lemma 1,equation (39), inequality (31) and the definition of ˜ ζ ε α (ˆ x ) , it followsthat D ˜ ζ ε α (ˆ x k ) , F k E = (cid:28) ˜ ζ ε α (ˆ x k ) , ∆ tδ f (ˆ x k , ˜ κ η ˆ x k ) (cid:29) + (cid:28) ˜ ζ ε α (ˆ x k ) , A + 1 δ Z tkδ q ( τ ) d τ (cid:29) ≤ ∆ tδ D ˜ ζ ε α (ˆ x k ) , f (ˆ x k , ˜ κ η ˆ x k ) E + (cid:13)(cid:13)(cid:13) ˜ ζ ε α (ˆ x k ) (cid:13)(cid:13)(cid:13) (cid:18) ∆ t δ L f ¯ f + ∆ tδ ¯ q (cid:19) ≤ ∆ tδ (cid:18) inf u ∈ U ∗ D ˜ ζ ε α (ˆ x k ) , f (˜ y ε α (ˆ x k ) , u ) E + η ˆ x k + L f (cid:18) α χ + 4( ε + ε x k ) (cid:19)(cid:19) + p V ∗ α + χα ! (cid:18) ∆ t δ L f ¯ f + ∆ tδ ¯ q (cid:19) . (40)For t = ( k + 1) δ the following inequality can be obtained with (38): V α (ˆ x k +1 ) − V α (ˆ x k ) ≤ δ (cid:20) inf u ∈ U ∗ D ˜ ζ ε α (ˆ x k ) , f (˜ y ε α (ˆ x k ) , u ) E + η ˆ x k + ( δL f ¯ f + ¯ q ) χα + L f (cid:18) α χ + 4( ε + ε x k ) (cid:19) + ( δL f ¯ f + ¯ q ) p V ∗ α + δ ( ¯ f + ¯ q ) α + ε x k + χα δ ( ¯ f + ¯ q ) . (41) Case 2: V α (ˆ x k ) ≤ ˆ v (Inside the target ball)If the sample period size δ satisfies δ ¯ f ≤ ε L V for some ε > ,then V α (ˆ x ( t )) ≤ V α (ˆ x k ) + ε . Choosing ε ≤ ˆ v guarantees that V α (ˆ x ( t )) ≤ v , and ε satisfying V (ˆ x ( t )) ≤ ˆ v ensures k ˆ x ( t ) k ≤ ˆ r and k x ( t ) k ≤ r for all t ≥ . Part 4: Determining parameters for decay
Some of the parameters, e. g., ε and ε , were already determinedin the previous parts. In the following, the different summands of(41) are bounded. With (41) and δ < , ε needs to satisfy L f ε ≤ ¯ w . (42)Note that these bounds influence also ε ˆ x k indirectly. Fix α and setthe following bounds η ˆ x k ≤ ¯ w , δ ¯ w ≤ ˆ v , δ ( ¯ f + ¯ q ) α ≤ ¯ w . (43)Force δ to additionally satisfy ( δL f ¯ f + ¯ q ) √ V ∗ α ≤ ¯ w . From nowon, δ is considered fixed and ε x k is constrained by ε x k ≤ δ ¯ w , L f ε x k ≤ ¯ w . (44)Furthermore, the following inequalities should hold: α L f χ ≤ ¯ w , ( δL f ¯ f + ¯ q ) χα ≤ ¯ w , χα ( ¯ f + ¯ q ) ≤ ¯ w . (45)Now, bounds on the optimization precision ε x k are derived to achieve inf u ∈ U ∗ D ˜ ζ ε α (ˆ x k ) , f (˜ y ε α (ˆ x k ) , u ) E ≤ − w . (46) To this end, observe that based on (34), for all z ∈ R n , V ( z ) ≥ V (˜ y ε α (ˆ x k )) + D ˜ ζ ε α (ˆ x k ) , z − ˜ y ε α (ˆ x k ) E − α (cid:13)(cid:13)(cid:13) z − ˜ y ε α (ˆ x k ) (cid:13)(cid:13)(cid:13) − ε x k − * ˜ y ε α (ˆ x k ) − y ε α (ˆ x k ) α , z − ˜ y ε α (ˆ x k ) +| {z } ≤ χα k z − ˜ y ε α (ˆ x k ) k (47)holds and also, for any θ ∈ R n , V (˜ y ε α (ˆ x k ) + ε ˆ x k θ ) ≥ V (˜ y ε α (ˆ x k )) + ε ˆ x k D ˜ ζ ε α (ˆ x k ) , θ E − α ε x k k θ k − ε x k − χα ε ˆ x k k θ k . (48)This inequality yields the following bound: D ˜ ζ ε α (ˆ x k ) , θ E ≤ V (˜ y ε α (ˆ x k ) + ε ˆ x k θ ) − V (˜ y ε α (ˆ x k )) ε ˆ x k + 12 α ε ˆ x k k θ k + χα k θ k + ε ˆ x k . (49)Using Lemma 1 and Assumption 2 (which ensures ˜ y ε α (ˆ x k ) ∈ ˜ Y k ⊆ Y k := B√ V ∗ α (ˆ x k ) ) enables, for ε x k < µ , the condition (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V (˜ y ε α (ˆ x k ) + ε x k θ ) − V (˜ y ε α (ˆ x k )) ε x k − D θ V (˜ y ε α (ˆ x k )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ¯ w , ∀ θ ∈ co ( f (˜ y ε α (ˆ x k ) , U ∗ )) . (50)With (49) and (50), it yields D ˜ ζ ε α (ˆ x k ) , f (˜ y ε α (ˆ x k ) , u ) E ≤ D f (˜ y ε α (ˆ x k ) ,u ) V (˜ y ε α (ˆ x k )) + ¯ w ε ˆ x k α (cid:13)(cid:13)(cid:13) f (˜ y ε α (ˆ x k ) , u ) (cid:13)(cid:13)(cid:13) + χα (cid:13)(cid:13)(cid:13) f (˜ y ε α (ˆ x k ) , u ) (cid:13)(cid:13)(cid:13) + ε ˆ x k (51)for all u ∈ U ∗ . Consequently, it holds that inf θ ∈ co ( f (˜ y ε α (ˆ x k ) , U ∗ )) D ˜ ζ ε α (ˆ x k ) , θ E ≤ inf θ ∈ co ( f (˜ y ε α (ˆ x k ) , U ∗ )) D θ V (˜ y ε α (ˆ x k )) + ¯ w ε ˆ x k α ¯ f + χα ¯ f + ε ˆ x k ≤ − w ε ˆ x k α ¯ f + χα ¯ f + ε ˆ x k . (52)If ε ˆ x k is bounded from above via ε ˆ x k ≤ ¯ wα / − χ ¯ f α + ¯ f , (53)the desired result follows: inf θ ∈ co ( f (˜ y ε α (ˆ x k ) , U ∗ )) D ˜ ζ ε α (ˆ x k ) , θ E ≤ −
34 ¯ w. (54)This means that an upper bound for the decay at ˜ y ε α (ˆ x k ) isdetermined.The last step is to show that V α ( x k +1 ) ≤ V α ( x k ) for all k ∈ N .For t = ( k + 1) δ an intersample decay rate on V α for the measuredstates with bounds (41), (42)-(45) and (54) can be established as V α (ˆ x k +1 ) − V α (ˆ x k ) ≤ δ (cid:18) −
34 ¯ w + 936 ¯ w (cid:19) = − δ ¯ w. (55) THIS WORK HAS BEEN ACCEPTED FOR PUBLICATION IN THE IEEE TRANSACTIONS ON AUTOMATIC CONTROL, DOI: 10.1109/TAC.2021.3052747
Introduce an upper bound for ¯ e such that ¯ e <
116 ¯ wL V holds. Then,the following inequalities can be obtained V α ( x ( t )) − V α ( x k )= ( V α (ˆ x ( t )) − V α (ˆ x k ))+ ( V α (ˆ x k ) − V α ( x k )) | {z } ≤ L V δ ¯ e + ( V α ( x ( t )) − V α (ˆ x ( t ))) | {z } ≤ L V δ ¯ e ≤ V α (ˆ x ( t )) − V α (ˆ x k ) + 2 L V δ ¯ e, ∀ t ∈ [ kδ, ( k + 1) δ ] . (56)The final result for the decay reads as V α ( x k +1 ) − V α ( x k ) ≤ − δ ¯ w + 2 L V δ ¯ e < − δ ¯ w. (57)This shows that the control action determined in (54), computed using ˆ x , yields a necessary sample-wise decay of V α .The reaching time of Case 2 can be determined as T α = 2 ˆ V ∗ − ˆ v / δ ¯ w .If Case 2 is reached, i. e., V (ˆ x k ) ≤ ˆ v , then two subcases arepossible in the following sampling periods.Either ˆ v ≤ V α (ˆ x k ) ≤ ˆ v (Subcase 2.1) or V α (ˆ x k ) ≤ v (Subcase2.2). If Subcase 2.1 occurs, then V α can either stay in this subcaseduring the next sampling period or, based on the decay condition,transition to the other subcase. If the latter subcase occurs, V α canstay there or move to Case 2. Thus, the trajectory ˆ x ( t ) stays in theball B ˆ r in any subcase for all the subsequent sampling periods. Thisimplies, that x ( t ) stays in the target ball B r after entering it once.This concludes the proof. Remark 4.
In the proof of Theorem 1, bounds for optimizationprecisions ε ˆ x and η ˆ x are derived depending on the current measuredstate. To derive uniform bounds would require, in particular, setting Y = B R and F = f ( Y , U ∗ ) . Such bounds may be, in general, moreconservative than the ones derived in Theorem 1. The next remark discusses the case when the sampling step sizeis fixed and the size of the target ball is to be determined.
Remark 5.
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