Analytic Comparison of Nonlinear H 1 -norm Bounding Techniques for Low Order Systems with Saturation

In this paper, we will compare di ering nonlinear H1 analysis techniques for linear systems with saturation nonlinearities, as applied to low order examples. The examples e ectively show the inter-relationship of stability and the behaviour of the induced norm and incremental gain, and the necessity of using bounded input sets. Finally, the state feedback synthesis problem receives preliminary attention, and two broad approaches are illustrated. The rst uses di erential inequality methods to generate a controller, the second method amounts to being a preliminary attempt at nonlinear H1 anti-windup synthesis. Keywords: nonlinear H1 control, systems with saturation, nite gain stability. 1 Introduction This paper consists of a series of examples which illustrate the inter-relationship of nonlinear H1 analysis techniques for feedback loops which contain saturation nonlinearities. As the examples themselves are all low order, it was possible to undertake closed form analysis (in general), and give analytic comparisons, which would not appear to be possible presently with higher order cases. There are two competing generalisations of the H1 norm available for nonlinear systems, being the incremental gain, and the induced L2+ norm. The incremental gain has many advantages over the induced norm in a theoretical sense, but it appears to be a quite conservative measure. For the class of systems studied, open loop unstable plants cannot be stabilised in an incremental gain sense, which appears to limit the incremental gain's usefulness as an analytical tool. The methods used to give bounds for the operator norms were developed elsewhere, although some have only appeared in the dissertation [8]. For continuous time systems, the Small Gain Theorem (Circle Criterion) estimate, the bound given in the paper by Liu, Chitour, Sontag [6], the bounds for the induced norm using dissipation functions as calculated by the method outlined in [7], and the bounds for the incremental gain using linear matrix inequalities developed in [8] are compared herein for the general rst order plant. The behaviour of the induced norm as the plant pole crosses from the closed left half plane to the open right half plane is pathological if one takes the input set to be L2+. The introduction of bounded input sets causes the induced norm to behave in a much more sensible fashion. It is believed that the use of such sets is necessary to give non-conservative analysis. This has been noted by other researchers, as in the paper by Lin, Saberi and Teel [4].Analysis techniques developed in [8] are used to compute the induced `2+ norm of a rst order discrete time example. Many of the results on continuous time induced norm computation in [7] have been extended to discrete time, but it is not possible to give the results in as clean a form. Finally, a preliminary examination of the state feedback synthesis problem has been made. Two di ering approaches to state feedback synthesis are illustrated for a rst order example. The rst method is an extension of that which appears in the paper [9] by A.J. van der Schaft, and which relies on the use of di erential inequalities. Even the rst order example shows some di culties with a naive extension of the methods in [9] to this class of systems. The second method relies on a parameterised piecewise linear controller, and the analysis undertaken to x suitable parameter values. The advantage of this approach is that it allows one to cleanly force the linearisation of the controller to be the linear controller derived by the much better understood linear design methodology, and the goal of the nonlinear synthesis is to best preserve the linear operating characteristics. In other words, this is viewed to be a preliminary attempt of using nonlinear H1 methods to synthesize anti-windup schemes. 1 2 Mathematical Preliminaries We will now provide the background notation and theory used in this paper. The dissertation [8] contains a fulller development of this problem framework and these surveyed results. The problems studied herein are the gain computations for a class of nonlinear operators. In order to measure the gain of an operator, we need to be able to measure signal size, which is done herein in terms of Lp+ spaces. The space norm of a vector in Rn used herein is the Euclidean (or 2-) Norm. It is de ned by kxk := px0x; and is the only norm used in this paper which is not distinguished by a subscript. The Lp+ norm (for p <1) of a (real) signal y is de ned by kxkp := Z 1 0 kx(t)kp dt 1 p ; and in the case p =1, kxk1 := ess sup t2R+fkx(t)kg: The space Lp+ is the set of all signals with nite Lp+ norm. For notational convenience, the dimension of the signal space is supressed, as it should be clear from the context. (We will implicitly use extended spaces (see the text by Willems [10] for an exposition using extended spaces), when we refer to test signals which give rise to unbounded outputs. From an operator theoretic point of view, if one does not use extended spaces, unbounded outputs do not formally exist, and the inputs correspond to a defect in the domain of the operator. It is felt that this distinction is not of supreme importance for the present work, and so the notational overhead of introducing extended spaces has been avoided. This distinction should be kept in mind when developing operator theoretic analysis, however.) In order to analyse systems with saturation nonlinearities, it is necessary to introduce bounded subsets of Lp+ spaces, for reasons which will be made apparent within this paper. We will now formally de ne the bounded sets used. De nition 2.1: The set Wp;q(N) is the intersection of the sets, Lp+ Tfw: kwkq Ng. Having de ned our signal sets, it is possible to de ne the gains to be bounded by the analysis undertaken in this paper. The induced Lp+ norm of an operator :Lp+ 7! Lp+ with domain Lp+ (resp. Wp;q(N)) is de ned (if it exists): k ki;p = sup w2Lp+(resp. Wp;q(N));w 6=0 k wkp kwkp : The Lp+ incremental gain of an operator :Lp+ 7! Lp+ with domain Lp+ (resp. Wp;q(N)) is de ned (if it exists): k k ;p = sup w; ~ w2Lp+(resp. Wp;q(N));w 6= ~ w k ~ w wkp k ~ w wkp : The motivation for the study of operator gains in robust control theory follows from the Small Gain Theorem (stated next). 2 Theorem 2.2: Fix a nonlinear operator :Lp+ 7! Lp+, with domain Lp+. Then if k k ;p < 1, (I+ ) 1 exists, with (I+ ) 1 ;p < 1 1 k k ;p : Additionally, if (I+ ) 1 exists, and k ki;p < 1, then, (I+ ) 1 i;p < 1 1 k ki;p : Proof: See [10]. 2 Remark 2.3: It is an important caveat that the inverse of (I+ ) must exist if the induced norm version of the Small Gain Theorem is to be used. If the system is to be generated by a state space realisation, it would be necessary to verify the existence and uniqueness of the solution using standard theory on di erential equations. Analysis of generic nonlinear operators is quite di cult, hence it is usual to restrict ones attention to subclasses of systems. The class to be studied herein are those which arise when memoryless saturation nonlinearities (de ned next) are inserted into linear state space equations. The standard saturation function, denoted 0( ), is de ned: 0(x) := 8><>: 1 x < 1; x jxj 1; 1 x > 1: The nonlinear operators to be studied are of the form :w 7! z, generated by the state space equations: _ x(t) = Ax(t) B 0(w(t) K(x(t))); (1) x(0) = 0; (2) z(t) = Cx(t): (3) Restrictions on the terms appearing in (1) (3) has been imposed, listed in the following Assumption. Assumption 2.4: We assume that A 2 Rn n, B 2 Rn 1, (i.e., single input), C 2 Rp n, [A;B] controllable, [C;A] observable, and that dK(x) dx x=0 exists. We de ne K 2 R1 n as: K := dK(x) dx x=0 ; and the local linearisation of is the linear operator L, which has the transfer function L(s) = " A BK B C 0 # := C(sI (A BK)) 1B: (4) As is well known, k Lk ;2 = k L(s)k1. The Algebraic Riccati Equation can be used to compute this quantity, as a consequence of the following result. 3 Theorem 2.5: The linear operator L with the transfer function L(s) de ned in (4) is such that k Lk ;2 if and only if there exists a symmetric, positive de nite matrix P 2 Rn n which solves the Algebraic Riccati equation: (A BK)0P + P (A BK) + 1 2PBB 0P + C 0C = 0: (5) Proof: This is standard, see [11]. 2 It has been found to be possible to bound above the incremental gain using the solution of a coupled pair of linear matrix inequalities (see the text [1] for an exposition on linear matrix inequalities). This has been developed in [8], and will be stated in an appendix as an aid to readers. The two inequalities amount to being the simultaneous solution of the open loop Lyapunov equation, and the closed loop Riccati equation. There has been further assumptions on the form of systems added, as the present result is in a preliminary form, and some technical issues still appear to be open with regards to the extension to more general con gurations. Theorem 2.6: Fix an operator :w 7! z, generated by (1) (3), with the assumption that K(x) = Cx added to those listed in Assumption 2.4. Then, k k ;2 (over the set L2+) if there exists a (symmetric) positive de nite matrix P 2 Rn n which solves the pair of linear matrix inequalities: (A BC)0P + P (A BC) + 1 2PBB 0P + C 0C 0; (6) A0P + PA+ C 0C 0: (7) Proof: Given in the appendix. Originally in [8]. 2 It is conjectured that the \if" statement in Theorem 2.6 may be strengthened to \if and only if", but this is still an open problem. Examination of the inequalities (6) (7) indicates that no solution P will exist if either A or A BC is unstable, and this has been shown to be true for the corresponding discrete time problem. More importantly, bounding the signal sizes allowed is not su cient to allow less conservative analysis, as the construction uses signals of relatively small 1 norm, and with arbitrarily small di erences between the test pairs. (This is discussed in the appendix.) Finally, we will give the dissipation theory based results which are used to bound above the induced L2+ norm. We will need the following extra de nitions and results. De nition 2.7: A function V :Rn 7! R+ is a candidate dissipation function if it is uniformly Lipschitz over compact sets, V (0) = 0, and V ( ) > 0 if 6= 0. Theorem 2.8 (Rademacher's): A candidate dissipation function V is di erentiable almost everywhere. Proof: See [12, Theorem 2.2.1] for details. 2 De nition 2.9: The operator :w 7! z generated by (1) (3) is L2-detectable if w; z 2 L2+ implies that x 2 L2+. 4 Lemma 2.10: If the form of K:Rn 7! R is restricted to be of the form K(x) = KCx, then is L2-detectable if [C;A] is detectable. Proof: This preliminary result is taken from [8, Theorem 3.6.11]. Theorem 2.11: Fix the operator :w 7! z. Assume that is L2-detectable, and that for w 2 W2;q(N), the reachable set lies within 0 Rn. Let V (x) be a candidate dissipation function de ned on 0, and d 0 be de ned d = 8<: : 2 0; dV dx exists9=; : Then, k ki;2 (over W2;q(N)) if sup 2 d fmaxf A( ); B( )gg 0; with A( ); B( ) de ned in (8),(9). A:1 is de ned by A( ) := ( A:1( ) j aj 1; A:2( ) j aj > 1; (8) with a; A:1; A:2 de ned: a( ) := K( ) + 1 2B0 dTV dx ; A:1( ) := 2 dV dx (A +B a( )) 2 kK( ) + a( )k2 + 0C 0C ; A:2( ) := 2 dV dx (A +Bsign( a( ))) 2 kK( ) + sign( a( ))k2 + 0C 0C : The function B( ) is de ned: B( ) := 2 dV dx (A B 0(K( ))) + 0C 0C : (9) Proof: This is taken from [7]. 2 Remark 2.12: There has been a minor re-de nition of B to take advantage of the simpler form of the system given here than that appears in [7], the equivalence of the test condition given here to that in [7] follows from the observations in [7, Remark 3.3]. The problem with characterising the induced norm in terms of dissipation functions is the computation of the functions themselves. There are generic existence theorems available, which rely on viscosity solution theory (see the paper by James [3]). The dissipation functions used herein were computed via the use of a (sub-optimal) heuristic algorithm developed in [7]. Preliminary de nitions required will now be stated. 5 We assume that K(x) is linear, denoted K(x) = Kx. Let P be the positive de nite (symmetric) solution to the algebraic Ricatti equation (A BK)0P + P (A BK) + 1 2 l PBB 0P + C 0C +Q = 0; for some arbitrary Q > 0. Let :Rn 7! R be de ned: ( ) := minf0; 2 0P (A B 0(K )) + 0C 0C g : Let the induced trajectory from ( 2 Rn) (denoted x(t; )) be the state space trajectory generated by: _ x(t) = Ax(t) B 0(Kx(t)); x(0) = : Assumption 2.13: It is assumed that for any xed disturbance set W2;q(N), the reachable set (denoted ) has compact closure, and for all points 2 @ , x(t; ) enters in nite time. (The latter essentially entails that there neither be equilibria nor limit cycles on the boundary of the reachable set.) Theorem 2.14: Fix the operator :w 7! z generated by (1) (3), with K(x) = Kx. The following statements are then equivalent: 1. for all w 2 W2;q(N), w is an element of L2+; 2. for all 2 , x(t; ) (i.e., with w = 0) is an element of L2+; 3. the function V : 7! R+ de ned: V ( ) = 1 2 Z 1 0 (x(t; )0C 0Cx(t; ) (x(t; ))) dt; is a valid dissipation function for some nite l. (Hence k ki;2 overW2;q(N)). Proof: In [7]. 2 3 The First Order Continuous Time Plant We will now apply the analysis techniques surveyed in Section 2 to the simplest possible non-trivial case, being the general rst order plant with proportional feedback. Although it may appear simple, this system appears to display most of the important properties which distinguish feedback loops with saturation nonlinearities from purely linear systems. Additional, as the analysis can be done in closed form, it is easier to compare di ering analysis techniques. The nonlinear system operator :w 7! z has the state space realisation: _ x = ax(t) + 0( kx(t) + w(t)); z(t) = x(t); x(0) = 0; with a; k 2 R. The remainder of this section will consist of the application of the analysis techniques to the cases of interest which are determined by the values of a; k. 6 f 6 w 1 s+a z k Figure 1: Example System 3.1 Case a > 0, k < a This is a somewhat unusual case, as the choice of feedback destabilises the stable linear plant. We can easily verify that the induced Lp+ norm does not exist for all p (take as a test input w(t) = for 0 t , w(t) = 0 otherwise). However, it is also simple to verify that any trajectory is con ned to = ( 1 a; 1a), and so the state is con ned to a compact set for all choices of input, and could even be considered to be stable under a loose enough de nition of state space stability. The importance (albeit limited) of this example is that it is possibly one of the simplest which shows that global state space properties are not necessarily determined by the behaviour at the origin, as is the case for linear systems. Additionally, it underlines the fact that the behaviour of the (closed loop) local linearisation provides a lower bound for operator gain properties. 3.2 Case a > 0, k = 1 We will now x k = 1, and then examine the stability properties as the plant pole is shifted along the real axis. Examination of this case will provide a basic motivation as to the importance of the use of bounded input sets. Additionally, we can compare the bounds generated by di erent analysis techniques. Once again, as a > 0, we have that the state is con ned to = ( 1 a ; 1 a). The system is locally stable, and the local linearisation has the transfer function L(s) = 1 s+a+1 , which is in RH1, with k L(s)k1 = 1 a+1. We will now examine the di erent bounds for the operator gain given by the use of the Small Gain Theorem (Circle Criterion), incremental gain analysis, and the result provided by the computational heuristic. 3.2.1 Small Gain Estimate The rst basic approach to bound the operator gain is to use the estimate provided by the small gain theorem, equivalent to the Circle Criterion. The means by which the estimate is made is via the use of loop shifting, a tactic long used in this situation (for example, see [10, 7 f f 6 ? -w u 12 1 s+a 1 s+a z Figure 2: The System after Loopshifting Section 6.2]). Loop transformations are the usual means of proving the Circle Criterion, and so the details are of enough interest to be given here. The rst step is to realise that we can express the function 0 as 0(x) = 12x+ (x); with (x) given by: (x) := ( 1 2x jxj 1; sign(x) 1 2x jxj > 1: The importance of this is that 1 2x is the centre of the sector [0; 1], and has the minimal norm to cover the sector. Now examine Figure 2. For ease of notation, let the linear operator with transfer function 1 s+a be denoted as G, and let H = G I+ 12G . We then can see that u = G u 1 2Gu+ w: Then,u = (I+H ) 1(I+ 1 2G) 1w; and so, z =G (I+H ) 1(I+ 12G) 1w: The incremental gain can thus be bounded by the use of the Small Gain Theorem, k k ;p kGk ;p k k ;p 1 1 kHk ;p k k ;p! (I+ 1 2G) 1 ;p : (10) For the signal space L2+, we arrive at k k ;2 1 2a 1 + 1 2a : (11) 8 (The most di cult part of the transition from from the inequality (10) to (11) is to verify that 1 + 1 2G(s) 1 1 = s+ a s+ a+ 12 1 = 1; which is true for all a > 1 4. This can be veri ed by the use of the Riccati equation (5), with 2 = 1 + , for > 0.) This analysis agrees with the expected result from the graphical analysis used in the Circle Criterion that the gain is nite if the system is stable, but becomes unbounded as a! 0. 3.2.2 Incremental Gain Analysis We will next bound above the incremental gain of the system, by using the result given in Theorem 2.6. The continuous time incremental gain algebraic inequalities are particularly easy to solve as they are given in terms of a scalar variable. The inequalities which result are: 2ap + 1 0; (12) 2(a+ 1)p + 1 2p2 + 1 0; (13) p > 0: (14) It is an obvious consequence of linear system analysis that the minimal value of for which there exists a solution to the inequalities (13) (14) is l = 1 a+ 1 : The solution p which determines this value of is given by: p = 1 a+ 1 : However, from (12), it is necessary that p > 1 2a . Hence, if a 1 we can use this value of p for solution to (12) (13). This implies that the nonlinear system has the same incremental gain as the local linearisation. If a < 1, the minimising choice of p is to take p = 1 2a. In this case, substitution into (13) yields: 2 1 4a; (15) which gives an upper bound of 1 2pa for . Once again, the bound for the incremental gain becomes unbounded as a! 0, but the bound given for a > 0 is tighter than that provided by Small Gain analysis. Remark 3.1: It is interesting that no deterioration from the lower bound provided by the local linearisation for a > 1. This may in fact point to a general principle that if one is \stable enough", the introduction of the saturation does not a ect the performance (in the present system con guration, at least). How this conjecture could be proven is not readily apparent, however. 9 3.2.3 Induced L2+ Norm Bounding We will now apply the heuristic computational method given in Theorem 2.14. The analysis here gives non-surprisingly the least conservative results for bounding the induced L2+ norm. When the analysis is extended to the case where the plant is unstable, the importance of doing such a direct analysis will be underlined. We will rst need the solution for the local linearisation. The algebraic Riccati equation which results from (5) is: 2p( a 1) + 1 2 l p2 + 1 = 0: We have that l = 1 a+1 , with a corresponding p = 1 a+1 . The dissipation function Vl associated with the linear problem is given by: Vl( ) = 12 1 a+ 1 2: It is then easy to verify that the state is con ned to the set = ( 1 a ; 1a) for all nite time. We now need to see over what subset of the solution V = Vl is a valid candidate function. Referring to Theorem 2.11 for the de nitions of A, etc., it is obvious that Vl satis es A( ) 0 for ; we need only verify the region where B( ) 0. We have that B( ) = 2 dVl dx ( a sign( )) + 2; = 2 a+ 1(a 2 + j j) + 2: And so, B( ) 0 if (1 a)j j 2 0, or j j 2 1 a. We then have veri ed that V = Vl is su cient if ( 2 1 a ; 2 1 a), which occurs if a 1 3 . For such cases then, = l = 1 a+1. Considering the result of the incremental gain analysis, it is obvious that the norm would not be a ected for a > 1. For a < 13, we take V ( ) to be such that B( ) = 0 for j j > 2 1 a . Then, dV dx = 2 2(aj j+ 1) : (16) As an aside, the anti-derivative of (16) is V ( ) = 1 2a3 1 2(aj j+ 1)2 2(aj j+ 1) + ln(aj j+ 1) + ; (although the actual analytic form of V ( ) is somewhat unimportant its derivative is the important quantity, and is of a simple form).10 As B( ) 0 is automatically satis ed by the construction of V , the A expression needs to be examined to determine the value of . The functions of interest are given by: a( ) = + 1 2 dV dx ; = + 1 2 2 2(aj j+ 1) ; A:1( ) = 2 dV dx ( a + a( )) 2 1 2 dV dx 2 + 2 = 2 aj j+ 1 (a+ 1)j j+ 1 4 2 4 (aj j+ 1)2 + 2; A:2( ) = 2 dV dx ( a + sign( a( ))) 2 k + sign( a( ))k2 + 2: As A:1( ) A:2( ), we will examine it rst. For the expression to be negative (semi)de nite, we need that ( ) :=vuut14 2 a 2 + (1 a)j j 1 : If we express the bounding expression as a function of , basic calculus indicates that the minimum choice occurs at = 1 1 a , and then monotonically increases. When matched at the beginning of the interval of interest (i.e., = 2 1 a), the non-surprising result is that a choice of = 1 1+a is indicated. An interesting case is where we solve the equations for all of R1, we see that lim !1 ( ) = 1 2pa . This is precisely the bound determined by the inequality (15)! This is not a mere coincidence; this is the behaviour that should be expected in this case. The A:1 equation corresponds directly to the algebraic Riccati equation, and the B expression to the algebraic Lyapunov equation. Their simultaneous solution corresponds to the continuous incremental gain inequalities. Re ection upon this example gives the reasons why the more detailed analysis undertaken in [7] is necessary to give less conservative results. Approaches which rely on quadratic properties will correspond to the A:1 expression, and neglect the fact that we can use the less strict A:2 expression if a is larger than 1, which is extremely likely to occur in the large. The other cause for conservatism which can be eliminated is that the limiting properties are used to determine the norm, which neglects the fact that the state will be con ned to a bounded set for any xed initial condition. When the full analysis is undertaken, we nd that is determined by the A:2 expression, and by the fact that = ( 1 a; 1 a). The full analysis is rather tedious, and in fact will not be given here explicitly. The important point of interest is in fact that lima!0 (a) = p2, which is obviously much better behaved than the prediction given by the small gain or incremental gain analysis, or relying upon the limiting behaviour of A:1. The bound determined as a function of a was numerically computed, and is plotted in Figure 3, along with the bounds given by the small gain and incremental gain analysis. 11 −0.1 0 0.1 0.2 0.3 0.4 0.5 0 2 4 6 8 10 12 14 16 18 20 Value of a B ou nd Figure 3: Bounds for as a function of a determined by (from top): Small Gain, incremental gain, induced L2+ norm analysis. 12 3.3 Case a = 0, k = 1 The case where a = 0 is of particular interest, being the case of an integrator following the standard saturation with unity negative feedback. 3.3.1 Incremental Gain In the paper by Chitour, Liu, and Sontag, there is a proof that the incremental gain does not exist for L2+ spaces for this system ([2, Example 5]). However, the construction therein relies on a parameterised pair of signals which become arbitrarily large in in nity norm as the di erential gain becomes in nite. It is possible to construct signal pairs which have much smaller in nity norms which display unbounded incremental gain behaviour. The following construction is a continuous time generalisation of the generic construction developed for discrete time systems in [8]. Let M > 1 be a free parameter, and de ne the signals w(t); ~ w(t) for t 2 [0;M ] as follows: w(t) = ( 1 + t 2T 2T t < 2T + 1; :1 t+ 2T 2T + 1 t < 2T + 2; ~ w(t) = 8><>: 0 0 t < :1; t :1 :1 t < 1; w(t) otherwise; with T 2 f0; 1; 2; : : :g, and with w(t) = ~ w(t) = 0 for t > M . We have that kwk1 = k ~ wk1 = 2. To prove that k k ;2 does not exist, assume the converse, i.e., that there exists some nite such that k w ~ wk2 kw ~ wk2 < . It can be veri ed that kw ~ wk22 = 358 3000, and that k w ~ wk22 > M 1 100 . Thus, by choosing M such that M 358 30 2 + 1; we contradict the assumption that incremental gain is bounded by . The importance of verifying that small signals give unbounded incremental gain behaviour is to show that the tactic of using bounded input spaces is not likely to be helpful to give less conservative bounds. Additionally, when one attempts to use Small Gain arguments to bound the operator norm, in the induced norm case one can re-de ne the standard saturation at large value so as to keep the lower sector bound away from 0. This cannot be done for the incremental gain, as it is the \incremental sector" which is important. This example (at least) shows that this is not a conservative assumption, as the small signal sizes show that re-de nition at large values could not help in any event. 3.3.2 Induced L2+ Norm Despite the fact that the incremental gain is unbounded, the induced L2+ norm is nite. If the input set is taken to be L2+, we see that the reachable set = R1, and that = p2. A candidate dissipation function which determines this value is given by: V ( ) = ( 12 2 j j 2; 1 6 3 + 23 otherwise. 13 Once again, the full analysis will be omitted, but the norm bound results by taking the limiting value of A:2. It is interesting to note that = p2 is a tight bound; this can be shown by examining the limiting properties of the input signal wT given by wT (t) = ( t+ 1 0 t T; 0 t > T; as T !1. (The response is given by: x(t) = 8><>: t 0 t T; 2T t T t 2T 1; e t+2T 1 t 2T 1:) In this case, we are able to compare this norm bound with that given in the paper by Liu, Chitour and Sontag [6]. The bound given by their analysis is that k ki;2 2(1+1) k Lk1 = 4. The analysis in [6] holds of course for general nonlinearities, and so it is not surprising that some deterioration would occur in the bound. 3.4 Case a < 0, k = 1 When the plant is open loop unstable, all signal bounds become unde ned, at least over unbounded sets. Once again, the incremental gain displays explosive behaviour, even within bounded input sets. The test signal w given by: w(t) = ( 1 + (a+ 1)t 0 t 1 a + 1; 0 t > 1 a + 1; lies in all Lp+ spaces, yet gives rise to an output which does not lie in any Lp+ space. Therefore, the failure of all analysis techniques is not conservative in this case. 3.5 Analysis with Bounded Inputs The conclusions in Section 3.4 may seem somewhat unusual from point of view of robust control theory. When operator theoretic analysis is applied to feedback loops, one expects results to conform to the so-called Meta-Theorem of robust control (stated next). Theorem 3.2 (Meta-Theorem of Robust Control): If a (closed loop) control system is stable (in some input/output sense), then the system will remain stable (in the same sense) for all small (in some sense) perturbations to the system model. For linear systems, the Meta-Theorem underlies H1 control (for example), as there is a characterisation of classes of perturbations. Such reasoning is exempli ed in the Gap Metric approaches to robust control. Much of the underlying belief in the validity of the MetaTheorem comes from the stability implied by the Small Gain Theorem. This belief gave the author the initial justi cation for the examination of the gain properties of these system operators. 14 If we con ne ourselves to the use of incremental gain, problems do not occur. The incremental gain increases in a continuous fashion, becoming unbounded as a approaches zero. In fact, we can use the stronger version of the Small Gain Theorem, and so we could develop quite a strong theory. The only major source of conservatism introduced may be due to that associated with the Small Gain Theorem. The restriction of theory to the study of the incremental gain does not seem a completely satisfactory option, as this restriction would a priori exclude plants with unstable modes which are to be controlled by bounded inputs. From an engineering standpoint, this might seem to largely con ne its usefulness to be an academic exercise. This is not to say that there could be no useful results which could be developed, as in areas such as process control, plants may be more likely to be stable. In such a case, the fact that the computation may be done as the solution of linear matrix inequalities may allow easily understood and implemented algorithms to be developed. Although conservative, incremental gain analysis may still be su cient for many purposes. However, the preference is to use the induced norm of the systems, as it will give rise to less conservative bounds. The uniqueness/existence di culties associated with the induced norm version of the Small Gain theorem are likely to be unimportant in non-pathological classes of nonlinear systems, allowing the development of the corresponding robust control theory. The fact that the induced norm bound behaves in a sensible fashion for a > 0 seems encouraging. But the the model seems to display remarkable lack of robustness with a = 0. This could be predicted by the explosion in the incremental gain as a goes to zero, but this seems to be a very unsatisfactory situation. The Meta-Theorem seems to have a a counterexample to it, as it seems unreasonable that replacing 1s by 1 s is a dramatic perturbation. We therefore need to re-examine the situation. It would appear something would have to be eliminated: the Meta-Theorem, the class of systems or perturbations, or the theoretical framework. The rst two options are unsatisfying, so a need to re-examine the operator theoretic based de nitions of stability used. Examination of the destabilising signals indicates that they may need to be pathologically large in order to destabilise the system. Extremely large excursions in the state space are necessary if the plant is \nearly stable". As it is expected that our models/systems would fail long before such excursions became possible, such test inputs seem suspect in terms of an engineering intuition. The problem is eliminated if we con ne the state to bounded subsets of the state space. In order for this to make sense from an operator theoretic standpoint (and to put the means of restriction into a realisation independent form),Wp;q(N) sets are introduced. And, in fact the pathological behaviour exhibited for the induced norm disappears when we examine the bound given in terms of a. If we take our input set to be W2;1(N1), it is readily veri ed that the reachable set = ( N1 a+1 ; N1 a+1), if N1 a+1 < 1 jaj. The basic existence result given by Theorem 2.14 then indicates a nite norm bound exists. The analysis was made on a digital computer, and the bound dependence vs. a for various values of N1 are given in Figure 4. The behaviour of the bound is now sensible, as although there is a value of a at which it becomes unbounded, the failure occurs in a continuous fashion. 15 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Value of a N or m B ou nd Figure 4: Bound for as a function of a for values of (from top) N1 = 15; 10; 6; 5. 16 4 The Double Integrator We will now brie y comment on a result given the paper by Liu, Chitour and Sontag [5]. In that paper, they prove the following Proposition. Proposition 4.1: Consider the system operator generated by: _ x1(t) = x2; _ x2(t) = (ax1(t) + bx2(t) + w(t)); x(0) = 0; with a; b > 0. Then the mapping w 7! x is not Lp+ stable for all p 2 [1;1]. Proof: This requires a fairly detailed analysis of test signals, and is done in [2, Section 8]. 2 As pointed out in [2], the system in the proposition is globally state space stable. This system provides a very good understanding as to the di erences between linear and nonlinear systems when it comes to global properties. By the use of Theorem 2.14, the fact that the system is globally state space stable means that for any compact subset of R2, trajectories con ned to that set will exhibit a nite gain property. The result is that the norm bound will deteriorate as the set size increases in an unbounded fashion. This dependence upon the set size may be of extreme importance for numerical computations. 5 Discrete Time Case The induced norm computation problem for discrete time systems was examined in [8], but the results were less satisfactory than in the continuous time case. The inability to assume a quadratic form for the dissipation function makes the straight extension of the discrete time completion of squares impossible. The following test theorem on dissipation functions will now be stated without proof. Theorem 5.1: Take the (discrete time) system operator :w 7! z, which is generated by the di erence equations: x[k + 1] = Ax[k] +B 0( K(x[k]) + w[k]) =: f(x[k]; w[k]); z[k] = Cx[k]; x[0] = 0: Here, A 2 Rn n, B 2 Rn 1, C 2 Rp n, and K:Rn 7! R continuous. Then, let V :Rn 7! R+ be continuous, with V (0) = 0, V ( ) > 0 for all 6= 0. If for a xed choice of , we have that for all 2 Rn, max !2S( )nH( ; !) = V (f( ; !)) V ( ) 2 k!k2 + kh( ; !)k2o 0; (17) whereS( ) = [K( ) 1;K( ) + 1][ f0g ; 17 then k ki;2 . Proof: This is a simpli cation of [8, Theorem 6.4.1]. 2 We will now give an example of the use of Theorem 5.1 for a simple discrete time system. The analysis is complicated by the fact that it is not su cient to evaluate the derivative, and so it is likely that pen and paper calculations will be of more limited value. The system is a \summer" following the standard saturation; the di erence equations generating the system operator are given by: x[k + 1] = x[k] + 0(w[k] x[k]) = f(x[k]; w[k]); x[0] = 0; with x[k] 2 R. The induced L2+ norm for the equivalent linear system is 1. When the saturation is included, the norm deteriorates to p2, as will be shown below. The fact that these match the values for the continuous time equivalent is somewhat suggestive; this may point to a more general principle. The nature of the saturation would discourage the use of high amplitude high frequency test signals, and so there may be some form of sampling property which could be exploited. The dissipation function to be used is: V ( ) = ( 2 j j 1; j j(j j+1)(2j j+1) 6 j j > 1 (this was generated by the dicrete time version of Theorem 2.14 [8, Theorem 6.5.6]). The function 2 is the solution to the corresponding (discrete) algebraic Riccati equation, and so for test values of the Hamiltonian which correspond to di erences between two points lying in [ 1; 1], the test holds with a test value of 1. If we examine some such that j j 1, but jf( ; !)j > 1, we rst note that for all values of ! in the test interval, f( ; !) = !. Then, H( ; !) = V (f( ; !)) V ( ) 2!2 + 2 = V (!) 2!2 = 1 3 j!j3 3 2!2 + 16 j!j: As j!j 2 over the test interval, we nd that H( ; !) 0. For j j > 1, it is easily tested that H( ; 0) = 0. Otherwise, we need to test H( ; !) for 1 ! +1. Without loss of generality, assume that > 1. If we write ! = + , we have that: H( ; !) = " + ( + + 1)(2 + + 1) 6 ( + 1)(2 + 1) 6 # 2( + )2 + 2; = 16 (6 6) 2 + (6 2 + 6 24) + 2 3 9 2 + : The maximising choice of over the interval [ 1; 1] is = 1, and then we see that H( ; !) = 2j j 1: 18 If one veri es the algebra without setting = p2, one will see that the choice of = p2 is the minimal one such that the leading coe cient in the H( ; !) expression is non-positive. 6 A First Order State Feedback Synthesis We will now examine a simple application which illustrates the potential power of the analysis technique developed in [7]. The rst order integrator with input saturation will once again be examined, but we will now synthesize a general nonlinear (state) feedback, using two di erent approaches. The rst technique uses partial di erential inequalities, following the sort of technique developed by A.J. van der Schaft in [9]. The second technique is similar in principle to the most basic anti-windup schemes, as the controller is xed to be piecewise linear, and then algebraic conditions solved to x the value of free parameters. We take the plant to be controlled as: _ x(t) = 0( K(x(t)) + w(t)); x(0) = 0: 6.1 Controller Synthesis Using Partial Di erential Inequalities The analysis here follows closely that given in [9], and is of a preliminary nature. It may be possible to solve the state feedback problem using dynamic programming methods, however the preference is once again is to see whether more class speci c algorithmsmay be developed. The somewhat unusual nature of the test condition in Theorem 2.11 seems to indicate that the rather naive extension attempted here will not be su cient. The controller is a state feedback, and is to be generated by the equation K( ) = dV dx ; (18) with V ( ) a smooth function mapping R 7! R. We would like that the induced L2+ norm for the mapping w 7! z to be less than or equal one. As is typical in this situation, the function V in (18) is to be a valid dissipation function for = 1. The rst unusual point to note is that the \optimal" input function a( ) is identically zero, as a( ) = K( ) + 1 2 dV dx : Therefore, V needs to satisfy A:1( ); B( ) 0 for all . The inequalities of interest become: dV dx 2 + 2 0; 2 dV dx 0@ 00@ dV dx 1A1A+ 2 0: 19 The constraining equations then become: x2 dV dx 2 ; 1 2x2 8><>: dV dx 2 if dV dx < 1, dV dx otherwise. One possible smooth solution is to take dV dx = + 3 2 . The intuition behind the solution is that the controller has unit gain near the origin, giving the correct local properties. The gain is then increased as the output get larger, and reduces the \system sensitivity". The controller is thus able to preserve the local norm bound. An interesting issue arises as a consequence of the behaviour of a with = 1. The synthesis does not take advantage of the fact that the A:2( ) A:1( ), and hence is a weaker condition. It that if we are extend the methods in [9], this issue will have to be seriously examined if the controller performance is to be satisfactory. 6.2 Synthesis of a Piecewise Linear Controller The control synthesised by the straight use of partial di erential inequalities may be unsatisfactory, however. The additional nonlinear terms may complicate linear system analysis, and lead to too complex controllers. One of the important inplication of the analysis in [7] is that linear controllers will preserve their norm properties over a non-negligible region around the origin. As the nonlinear analytic techniques and design intuition are nowhere near as developed as that for linear control theory, the preference is that the nonlinear techniques do not a ect linear design processes unnecessarily. As such, an alternative state feedback will be given which preserves the induced norm, but is in fact piecewise linear. The control is xed to be of the form K( ) = k1 + k2M 0 M ! : (19) There are three free parameters (k1; k2;M), but as we assume that the linearisation of the controller is xed by a priori linear analysis, there are in fact only two. In order to motivate setting the problem in this framework, it may be noted that the most basic anti-windup schemes are also piecewise linear, although in that case the controller has dynamics. If this method is generalised, it would then amount to providing a synthesis method for anti-windup schemes using nonlinear H1 methods. We assume that the linearisation of the controller is xed to be Kl = k1 + k1 = 1, as a result of linear analysis (for example, this controller gives the optimal (unweighted) robustness measure with respect to the Gap Metric). The resulting (closed loop) local linearisation has an in nity norm of l = 1. We now wish to nd a controller K( ) of the form determined by (19), with k1 + k2 = 1, and which ensures that k ki;2 = 1. 20 We choose M = 1 2 , by re ection upon the fact that Vl( ) = 2 2 is a valid dissipation function for the unity feedback controller for j j 2. We then see that the controller has one free parameter, and the control form may be expressed as: K( ) = ( j j 2; k + 2(1 k) j j > 2: It is readily apparent that in order to preserve the operator norm of 1, k 1. Take a candidate dissipation function V to be: V ( ) = ( 1 2 2 j j 2; 1 6 3 + 23 j j > 2: Examination of the asymptotic value of A:2( ) (with = 1) gives: A:2( ) = 2 2 (k + 3 2k)2; for large enough ( a( ) 1 for 2). Choosing k = p2 is su cient for the candidate function to be valid for = 1. In order to generalise this synthesis technique, many computational issues would need to be explored. However, by the tactic of xing the form of the controller to be piecewise linear with free parameters, it is conjectured that the resulting computations should be algorithmically tractable. 7 Conclusions The examples in this paper are felt to provide a very good test bed of illustrations of the properties of feedback loops with saturation nonlinearities. Most of the observed theoretical peculiarities of these systems, and how they di er from linear systems, are given concrete manifestation. The induced norm analysis for the plant as the plant pole shifts from the left half plane to the right half plane gives a very good motivation for the use of bounded input sets if one is to undertake less conservative analysis. If one is to con ne one's attention to purely stable systems, it may be preferable to use the incremental gain, which is far more easily computed, and has nice properties for analysis. Finally, the preliminary state feedback synthesis example illustrates some of the problems with a straightforward extension of di erential inequality based methods to controller synthesis, and how it may be possible to synthesize anti-windup schemes based on nonlinear H1 methods in the near future. Appendix We will now state the proof to Theorem 2.6. Theorem 2.6: Fix an operator :w 7! z, generated by (1) (3), with the assumption that K(x) = Cx added to those listed in Assumption 2.4. Then, k k ;2 (over the set 21 L2+) if there exists a (symmetric) positive de nite matrix P 2 Rn n which solves the pair oflinear matrix inequalities:(A BC)0P + P (A BC) + 12PBB 0P + C 0C 0;(6)A0P + PA+ C 0C 0:(7)Proof: Take any two signals w; ~w 2 L2+, with kw ~wk2 6= 0. Let x; ~x be the statetrajectories associated with w; ~w, respectively. Let = x ~x. Fix V ( ) = 12 0P . It is thena su cient condition to prove that k k ;2ifsupt2R+(H(t) := 2dVdx (t) _(t)2 k(w ~w)(t)k2 + kC (t)k2) 0;by the use of standard dissipation arguments.Note that_(t) = A (t) +B( 0( Cx(t) + w(t))0( C~x(t) + ~w(t))):Then, by using the fact that we can write 0(a) 0(b) = (a b), for some with 01,we can re-write H(t) as (suppressing t dependence):H2dVdx (A BC)2 kw ~wk2 + 0C 0C !+(1 ) 2dVdx A + 0C 0C !=:1 + (1 ) 2:It is a standard consequence of linear dissipation theory that 1 0 is implied by P beingthe solution to (6), and that 2 0 is implied by P also being the solution to (7). Hence,H(t) 0, and so Theorem 2.6 has been veri ed. 2Remark 7.1: Although similar linear matrix inequalities have been used to verify upperbounds for induced norms, the author is unaware of a reference which explicitly pointsto the fact that the linear matrix inequality is being used to compute the incremental gain.Additionally, in [8], discrete time constructions have been made to show that if either theopen or closed loop dynamics matrices are unstable, the incremental gain does not exist.These constructions use signals of small in nity norm, so bounding signal sizes appears tobe unhelpful, unlike the induced norm problem. The result will be now stated without proof.(There is a further test condition for the non-existence of the incremental gain in [8], but theresult will not be stated due to its added notational complexity, and the fact that the resultis somewhat less satisfactory in that there is no a priori characterisation of the size of the\destabilising" (incremental gain sense) test signal pairs.) See also the paper by Chitour,Liu and Sontag [2] for further characterisations of the incremental gain properties of systemswith saturation nonlinearities.22 Theorem 7.2: Fix a discrete time operator :w 7! z generated by the di erence equations:x[k + 1] = Ax[k] +B 0(w[k] Cx[k]);x[0] = 0;z[k] = Cx[k];with [A;B] controllable, [C;A] observable, A 2 Rn n, B 2 Rn 1,C 2 R1 n. Then k k ;pdoes not exist if (A) 1 (if p < 1), or if (A) > 1 (if p = 1). Additionally, the testsignals constructed to show that the incremental gain is unbounded have an `1+ norm of1 + , for any > 0.Proof: See [8]. 2References[1] S. 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