Hespanha, J.P. (1998). Logic-Based Switching Algorithms in Control. Ph.D. Thesis. Dept. of Electrical Engineering, Yale University, New Haven, CT.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....simple cases where computing the solution to the optimization problem is actually feasible. A computationally feasible though sub optimal approach to the design of self adjusting controllers is the so called switching control approach originally introduced in [2] and further developed in e.g. [3, 4, 5, 6]. A switching control scheme is typically composed of an inner loop where a candidate controller is connected in closed loop with the system, and an outer loop where a supervisor, based on input output data, decides which controller to place in feedback with the system and when to switch to a ....

J. P. Hespanha, Logic-based switching algorithms in control. Ph.D. thesis, Dept. of Electrical Engineering, Yale University (1998).

.... when the system is not completely known, such an assumption is not realistic) An alternative to dwell time switching control of nonlinear systems is provided by the so called hysteresis switching proposed in [45] and its scale independent versions, which were recently introduced and analyzed in [16], 46] and applied to control of uncertain nonlinear systems in [11] 47] When the uncertainty is purely parametric and there is no measurement noise, switching signals generated by scale independent hysteresis have the property that the switching stops in finite time, whereas in the presence of ....

J.P. Hespanha, Logic-based switching algorithms in control, Ph.D. Thesis, Dept. of Electrical Engineering, Yale University, 1998.

.... have large uncertainty necessitating controllers designed for different admissible models; the hybrid control problem is to switch among these controllers to achieve stabilization [4] 26] Various rules have been proposed for the stabilization of diverse hybrid systems, notably in [37] 49] [55]. As an example of these results, Pettersson and Lennartson [11] 12] 49] proposed the following implementation of Theorem 3.3. Min Projection Strategy: For specific , choose the vector field according to the criterion . Thus, one strategy that may stabilize a hybrid system is to pick the ....

J. P. Hespanha, "Logic-based switching algorithms in control," Ph.D. dissertation, Dept. of Electrical Engineering, Yale Univ., 1998. Applications

....better. The origin of this idea is the concept of certainty equivalence. This is a well known heuristic idea central to the whole adaptive control literature whose role has been formally established for linear plants in [8] and extended to nonlinear systems in the recent paper [9] See also [10]. The performance signals in supervisory control [6] are generated by designing an estimator for each member of R and evaluating their norm squared output estimation error. This task is complicated in the present problem because, besides the 2 For all practical purposes we can relax this ....

....i ; 2.11) as an estimate of fi i ; L g. This is because the pair fi min ; L min g is at least as good as fi i ; L g as far as the values of the corresponding performance signals are concerned. The algorithm used to generate R is the scale independent hysteresis switching logic [10] and can be regarded as the hybrid dynamical system, with state 4 = f 1 ; 2 g 2 N T and output R = R 1 ; 2.12) 5 In general, the optimization in (2.11) can be computed in a closed form because, for each value of i, w i ; is a quadratic function of . 12 de ned by the ....

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Hespanha JP. Logic-Based Switching Algorithms in Control, PhD Thesis, Dec. 1998, Yale University, USA.

.... when the system is not completely known, such an assumption is not realistic) An alternative to dwell time switching control of nonlinear systems is provided by the so called hysteresis switching proposed in [45] and its scale independent versions, which were recently introduced and analyzed in [16], 46] and applied to control of uncertain nonlinear systems in [11] 47] When the uncertainty is purely parametric and there is no measurement noise, switching signals generated by scale independent hysteresis have the property that the switching stops in finite time, whereas in the presence of ....

J. P. Hespanha, Logic-based switching algorithms in control, Ph.D. Thesis, Dept. of Electrical Engineering, Yale University, 1998.

....3. Basically, the choice of the candidate controllers is governed by the resulting properties of this system. This makes sense because E is implemented by the control designer; the validity of such an approach will become clear in view of the results that are discussed next. It was shown in [3, 4] that if the injected system EC q is input to state stable (ISS) with respect to e q , and if the plant P is 0 detectable, then the closed loop system (25) is 0 detectable with respect to 20 PSfrag replacements u Gamma Gamma y y y q C q E P e q Figure 3: The closed loop system ....

J. P. Hespanha, Logic-based switching algorithms in control, Ph.D. Thesis, Dept. of Electrical Engineering, Yale University, 1998.

....3. Basically, the choice of the candidate controllers is governed by the resulting properties of this system. This makes sense because E is implemented by the control designer; the validity of such an approach will become clear in view of the results that are discussed next. It was shown in [4, 5] that if the injected system EC q is input to state stable (ISS) with respect to e q and the process P is 0 detectable, then the closed loop system (25) is 0 detectable with respect to 20 PSfrag replacements u y y y q C q E P e q Figure 3: The closed loop system (25) e q . This ....

J. P. Hespanha, Logic-based switching algorithms in control, Ph.D. Thesis, Dept. of Electrical Engineering, Yale University, 1998.

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J. P. Hespanha. Logic-Based Switching Algorithms in Control. PhD thesis, Dept. of Electrical Engineering, Yale University, New Haven, CT, 1998.

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J.P. Hespanha. Logic-based switching algorithms in control. PhD thesis, Dept. of Electrical Engineering, Yale University, 1998.

....9] does not have this desirable property. However, it is not difficult to modify the logic by introducing a multiplicative hysteresis constant instead of an additive one. The resulting scale independent hysteresis switching logic was studied and applied to control of uncertain nonlinear systems in [1] and elsewhere. These results still relied on the termination of switching in finite time, and were thus limited to situations where the parametric uncertainty range is described by a finite set and there are no noise, disturbances, or unmodeled dynamics. The scale independent hysteresis ....

....issues regarding minimization over P. The understanding here is that minimization over Di s is computationally tractable if these sets are sufficiently small. The above procedure yields a piecewise constant signal a which is continuous from the right everywhere. By the same argument as in [1], one can show that chattering is avoided if all p, p 6 P are bounded below by some positive number. In fact, there exists a maximal interval [0, Tm= on which a is defined, and there can only be a finite number of switches on each proper subinterval of [0, Tm= In the supervisory control ....

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J. P. Hespanha, Logic-based switching algorithms in control, Ph.D. Thesis, Dept. of Electrical Engineering, Yale University, 1998.

....stability of hysteresis based adaptive control systems in the presence of measurement noise. 1 Introduction Adaptive control algorithms that employ a logic based supervisor to orchestrate the switching between a family of candidate controllers have been quite successful in numerous applications [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. The need for switching usually arises from the fact that no single candidate controller would be capable, by itself, of guaranteeing stability and good performance when connected with a poorly modeled process. This type of supervisory control results in a switched closedloop system of the form ....

....Unfortunately, with nonlinear systems this may lead to finite escape of the closed loop. Adaptive switching algorithms for nonlinear systems have therefore avoided a fixed dwelltime, and have been mostly based on hysteresis switch ing [13, 14] or on its more recent scale independent version [8, 15]. To date, the analysis of algorithms based on hysteresis switching relied heavily on showing that switching stops in finite time [7, 8, 9, 10, 11, 12] However, in the presence of noise and disturbance inputs, this is hardly the case. In fact, the only known algorithms for which switching can be ....

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J.P. Hespanha, Logic-Based Switching Algorithms in Control. PhD thesis, Yale University, New Haven, CT, 1998.

....is given next. 4 Scale independent hysteresis switching logic So far we have closely followed the set up of [16, 17] However, the switching logic that we will consider here is different from the one employed there. Namely, we will work with the scale independent hysteresis switching logic [6, 12], whose functioning is as follows (see Figure 2) Assumed given are continuous monitoring signals p, p G 7 (to define the switching logic, we do not need to require that they be defined by the above formulas, although this is the case to which we will specialize later) Let us pick a number h 0 ....

....state. If all p, p 7 are bounded below by some positive number, then infinitely fast switching (chattering) is avoided. In fact, there must be an interval [0, Tnax) of maximal length on which exists, and there can only be a finite number of switches on each proper subinterval of [0, Tna) see [6, 12] for details. In all the cases treated below, it will be clear that Tnax c. Hybrid systems having the above property are known as non Zeno. One can easily see from the definition of the above switching logic that replacing the signals p 7 by their scaled versions p(t) O(t)tp(t) p P ....

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J.P. Hespanha, Logic-based switching algorithms in control, Ph.D. Thesis, Dept. of Electrical Engineering, Yale University, 1998.

....the smallest monitoring signal. In this way, supervisory control selects the candidate controller that corresponds to an estimator error that has been small for some time (in an integral sense) To prevent chattering, one approximates this mechanism by introducing a dwell time [27] or hysteresis [9, 13, 12, 19]. As discussed in these references, different versions of the latter approach can be applied to nonlinear systems with 7 finite or infinite and Q being equal to 7 or a finite subset of 7; an overview of various switching logics is given in [19] Two properties need to be satisfied by the switching ....

....Property if it preserves the detectability, i.e. if the switched system (2) 3) is detectable with respect to the output e. The Non Destabilization Property trivially holds if the switching stops in finite time (which is the case if the scale independent hysteresis switching logic of [9] or its variants proposed in [12, 19] are applied in the absence of noise, disturbances, and unmodeled dynamics) In the linear case, a standard output injection argument shows that detectability is not destroyed by switching 4 if the switching is sufficiently slow (so as not to destabilize the ....

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J.P. Hespanha. Logic-Based Switching Algorithms in Control. PhD thesis, Dept. of Electrical Engineering, Yale University, New Haven, CT, 1998.

....to (2) with x(0) 0. When (2) is uniformly exponentially stable over S, the induced norm gS is known to be finite. Since both slow dwell time switching and slow switching on theaverage result in exponential stability, both these forms of slow switching result in a finite induced norm gS . In [16, 11, 17] one can find upper bounds on L 2 induced norms over appropriately defined sets of switching signals that satisfy the slow switching properties mentioned above. However, these bounds are typically very conservative and prompted us to compute the precise value of the induced norm under ....

J. P. Hespanha, Logic-Based Switching Algorithms in Control. PhD thesis, Yale University, New Haven, CT, 1998.

.... ] with arbitrary N 0 , and not only for the switching signals in S[ D ] Systems like (1) arise in an adaptive context when a high level, logic based supervisor orchestrates the switching between a family of candidate controllers so as to achieve some desired behavior for the closed loop system [3, 4, 5, 2, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1]. The need for switching usually arises from the fact that no single candidate controller would be capable, by itself, of guaranteeing stability and good performance when connected with a poorly modeled process. In several of these algorithms the supervisor guarantees, by construction, that there ....

.... D . Unfortunately, with nonlinear system this may lead to nite escape of the closed loop. Adaptive switching algorithms for nonlinear system have therefore avoided a xed dwell time, and have been mostly 3 based on hysteresis switching [17, 18] or on its more recent scale independent version [11, 19]. To date, the analysis of algorithms based on hysteresis switching relied heavily on showing that switching stops in nite time [10, 11, 12, 13, 14, 15] However, in the presence of noise and disturbance inputs, this is hardly the case. In fact, the only known switching algorithms for which ....

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J. P. Hespanha, Logic-Based Switching Algorithms in Control. PhD thesis, Yale University, New Haven, CT, 1998.

....Section 5 contains some concluding remarks and directions for future research. A preliminary version of the results in this paper was presented at the 12th Int. Symposium on the Mathematical Theory of Networks and Syst. St. Louis, MO, June 1996. These were subsequently improved in the PhD thesis [16]. 2 Stable controller switching The feedback con guration used in this paper is shown in Figure 1. In this gure u denotes the PSfrag replacements controller process y u eT u C d n r Figure 1: Feedback con guration control input, y the process output, r a bounded reference signal, d ....

J. P. Hespanha, Logic-Based Switching Algorithms in Control. PhD thesis, Yale University, New Haven, CT, 1998.

....these sets are. In particular, we were able to constructively define the set of point coincidence tasks, and as a result were also able to construct an encoding for any task in the class. Is this also possible for the sets T uncal and T PInv The structure of the set T PInv has been studied in [20] with some generality. Here, we take from [21] a characterization for T PInv for the special cases when n 2 f2; 3; 4g. For a given set of tasks T , let us say that the task T (f) 0 can be generated by T if it can be obtained by applying any number of the three task construction primitives ....

J. Hespanha. Logic-Based Switching Algorithms in Control. PhD thesis, Yale University, New Haven, CT, 1998.

.... ] with arbitrary N 0 , and not only for the switching signals in S[ D ] Systems like (1) arise in an adaptive context when a high level, logic based supervisor orchestrates the switching between a family of candidate controllers so as to achieve some desired behavior for the closed loop system [2, 3, 1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. The need for switching usually arises from the fact that no single candidate controller would be capable, by itself, of guaranteeing stability and good performance when connected with a poorly modeled process. In several of these algorithms the supervisor guarantees, by construction, that there ....

....D . Unfortunately, with nonlinear system this may lead to finite escape of the closed loop. Adaptive switching algorithms for nonlinear system have therefore avoided a fixed dwell time, and have been mostly 3 based on hysteresis switching [15, 16] or on its more recent scaleindependent version [9, 17]. To date, the analysis of algorithms based on hysteresis switching relied heavily on showing that switching stops in finite time [8, 9, 10, 11, 12, 13] However, in the presence of noise and disturbance inputs, this is hardly the case. In fact, the only known switching algorithms for which ....

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J. P. Hespanha. Logic-Based Switching Algorithms in Control. PhD thesis, Yale University, New Haven, CT, 1998.

....weighted norm of the corresponding estimation errors. The reason for defining the performance signals in this manner will become clear later. 3. 4 Switching logic The switching logic we consider here is a modified version of the scale independent hysteresis switching logic proposed in [7, 8]. Let us pick a number h 0 called the hysteresis constant. The functioning of the switching logic is as follows (see Figure 3) First, we set oe(0) arg min p2P f p (0)g. Suppose that at a certain time t i the value of oe has just switched to some q 2 P. If r q (t i ) 6= 0, we set s = 1. The ....

....is monotonically nondecreasing. This, the finiteness of P, and the fact that p (0) 0 for each p 2 P guarantee the existence of a positive number ffl such that p (t) ffl 8t 0; 8p 2 P. It is not hard to conclude now from the way the switching logic was defined that chattering cannot occur [7, 8]. Observe that p is bounded by virtue of (10) It follows that the signals p satisfy the assumptions of the Hysteresis Switching Lemma [7, 8, 15] which enables us to conclude that the switching stops in finite time 1 . More precisely, if [0; T ) is the largest interval on which the ....

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J. P. Hespanha, Logic-based switching algorithms in control, Ph.D. Thesis, Dept. of Electrical Engineering, Yale University, 1998.

....The use of integral input to state stability and integral detectability also seems quite natural when the perfor mance signals are defined as integral norms of output estimation errors. In fact, with integral detectability we were able to avoid many of the technical difficulties that arose in [1, 6]. Working with time domain definitions of integral input to state stability and integral detectability instead of definitions based on dissipationlike inequalities, as in [1, 6] also helped simplifying the analysis. This paper is organized as follows. In Section 2 the notion of ....

....errors. In fact, with integral detectability we were able to avoid many of the technical difficulties that arose in [1, 6] Working with time domain definitions of integral input to state stability and integral detectability instead of definitions based on dissipationlike inequalities, as in [1, 6] also helped simplifying the analysis. This paper is organized as follows. In Section 2 the notion of integral input to state stability is reviewed and a dual definition of integral detectability is introduced. Section 3 describes the overall control problem addressed in this paper namely the ....

[Article contains additional citation context not shown here]

J. P. Hespanha. Logic-Based Switching Algorithms in Control. PhD thesis, Yale University, New Haven, CT, 1998.

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Hespanha, J.P. (1998). Logic-Based Switching Algorithms in Control. Ph.D. Thesis. Dept. of Electrical Engineering, Yale University, New Haven, CT.

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J. P. Hespanha, Logic-based switching algorithms in control, Ph.D. Thesis, Dept. of Electrical Engineering, Yale University, 1998.

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J. P. Hespanha, Logic-based switching algorithms in control, Ph.D. Thesis, Dept. of Electrical Engineering, Yale University, 1998.

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Hespanha, J.P. (1998). Logic-Based Switching Algorithms in Control. Ph.D. Thesis. Dept. of Electrical Engineering, Yale University, New Haven, CT.

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Hespanha, J.: Logic-Based Switching Algorithms in Control. PhD thesis, Yale University (1998)

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