where u(t) and y(t) are the input and output, re-

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1 Proceedings of the 35th FMil 2:lO Conference on Decision and Control Kobe, Japan December 1996 Robustness of Discrete Periodically Time Varying Control Under Different Model Perturbations * Jingxin Zhang Cishen Zhang* School of Electrical Engineering, University of South Australia, Whyalla Norrie, SA 5608 Cooperative Research Centre for Sensor Signal and Information Processing, Technology Park Adelaide, Australia Fax: $ j.zhang@unisa.edu.au Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, Vic. 3052, Australia Abstract: This paper presents a quantitative analysis on the robustness of discrete linear periodically time varying (LPTV) control under different perturbations in comparison with linear time invariant control. For unstructured perturbations, it is shown that the system stability margin is definitely deteriorated by LPTV control. For sturctured perturbations, the paper clarifies the mechanism of LPTV control in stability margin improvement and the side effect of this improvement - deterioration of the stability margin for unstructured perturbations. In a unified frequency domain framework, the paper exposes the relevance and conflictions of the previous results on the robustness of LPTV control, and therefore provides a deeper insight into the problem. 1. Introduction Robustness enhancement is the primary motivation for the investigation into linear periodically time varying (LPTV) control of linear time invariant (LTI) plants. It has been shown by many authors that LPTV control can improve significantly gainlphase margin and can provide strong and simultaneous stabilization. However, it has also been shown that nonlinear time varying (NLTV) control and linear time varying (LTV) control, which include LPTV control as a special case, offer no advantages over LTI control for robust stabilization of LTI plants subject to the norm bounded unstructured model perturbations, and that LPTV control systems may be sensitive to the high frequency band unstructured perturbations. For details of the above results, see e.g. [4, 10, 1, 7, 8, 9, 6, 111 and the references therein. These different results appear to be uncorrelated and conflict to certain extent, and the relevance and tradeoffs between the different results and cases are not clear. It is therefore necessary and important to gain further understanding of the intrinsic properties of LPTV control systems and the system robustness. In [12], the authors have analyzed the closed loop stability margin of discrete LPTV control system subject to NLTV unstructured model perturbations, and have shown that for this specific case, LPTV control deteriorates the stability margin. This result provides a deeper understanding for LPTV control. However, NLTV perturbation is a very large family, it contains many different practically important subsets, e.g. LPTV and LTI perturbations. It is known that for different perturbations, the robust stability conditions can be quite different. Thus [la] has not given a complete answer to the robustness of discrete LPTV control. Furthermore, [12] shows only the negative part of LPTV control. It gives no explanation on how can LPTV control improve closed loop robustness for some other classes of perturbations, e.g. gain variations, and what are the relationships among the different cases. This paper looks further into the problem. It investigates, respectively, the robustness of LPTV control system under LTI and a class of LPTV unstructured perturbations and structured real perturbations. It is shown that for unstructured perturbations, LPTV control definitely deteriorates the closed loop system stability margin. The paper also clarifies the mechanism of LPTV control in gain margin improvement and the side effect of this improvement - deterioration of the system robustness for unstructured perturbations. All the analyses are carried out in a unified frequency domain framework. The results exposes the relevance and the relationship of the previous results on this issue. Due to space limit, the proofs for all the lemmas and corollary in the sequel will be omitted. 2 Problem statement Throughout the paper, II.II denotes the Lz norm and 12 norm on continuous and discrete signals; 11. 1, denotes the co-norm of a discrete system, or the H, norm if the system is stable; *(.) and p(.) denote the maxi- mum singular value and the spectral radius of a matrix, respectively; I. 1 denotes the magnitude of a complex variable; dim(.) denotes the dimension of a square matrix; X* denotes the complex conjugate of a variable X; 72. and C are the fields of real and complex numbers, respectively. In time domain, suppose that G is a causal, N- periodically time varying system, then it admits the following minimum state space representation *Work supported by Australian Research Council and Research Development Grant of University of South Australia /96 $ IEEE 3984 where u(t) and y(t) are the input and output, re-

2 spectively, r(t) is the n-dimensional state vector, and A(t), B(t), C(t) and D(t) are the N-periodically time varying matrices with appropriate dimensions and satisfyinga(t+in) = A(t), B(t+lN) = B(t), C(t+lN) = C(t), D(t + 1N) = D(t). Note that if N = 1, G reduces to an LIT system. Throughout the paper, this case is distinguished from the cases N # 1, i.e. an LPTV sys- tem is defined as not LTI. The system G in the form of (l), is a finite order system if n < 00. The system G is a real system if A(t), B(t), C(t) and D(t) are real matrices. The system G zs a stable system if Xi, i = 1,..., n, the eigenvalues of A := A(N - 1)A(N - 2).A(O), satisfy IXil<l, i=l,..., n (2) Denote Y(t) and F(z) the t-transforms of a discrete signal y(t) and an LTI discrete system F, respectively. F(z) zs defined as finite order if its pole polynomial 2s finite order. Where the pole polynomial is defined as the polynomial containing all the modes of F which is not necessarily minimum order. Write explicitly z = e'". The frequency responses of y(t) is defined as Y(ej"). The frequency transfer functzon of F zs defined as F(ej"'). Then Y(ej") and F(ej") are continuous functions in w E EO,%). The problems to be considered in the paper are the stability margin of 3 under different perturbations given in Al-A3, namely, the maximum tolerable r for which the system F controlled by LPTV Ii remains stable, and the comparison of the stability margin attainable by LPTV Ii with that attainable by LTI I<. For simplicity, only multiplicative perturbation case is considered. However, all the analyses in the sequel carry over to additive perturbation case. To make the analysis meaningful, the LPTV control and the resulting closed loop system are distinguished from the LTI control and the resulting closed loop system. It is defined that an LPTV system is not LTI. 3 Frequency domain lifting for systems For a discrete signal y(t), the norm of y(t) is llyll = d m. Let Y(ej") be the frequency reof Y(ej") is llyll = WN = Q. Then... Y(ej("+(N-l)"N) 11 is defined as the lifted frequency response of the signal w E [O,WN) Apparently, the norm is preserved under the lifting in frequency domain, i.e. IbII = IlYll = 1 1 ~ 1 1. Consider a discrete LPTV system G in the form of (1). Let ri(ej") and Y(ej") be the lifted frequency responses of the input U and output y, respectively. It has been shown in [13] that as i.(ejw) Fig. 1 LPTV feedback control system F Fig. 1 gives the feedback control system, denoted F, to be consider in the paper. It is assumed that all the blocks of F are SISO discrete systems. In the figure, P denotes a real, causal, finite order and stabilizable LTI nominal plant, K is a real, causal and finite order LPTV controller with period N, the dashed-line enclosed subsystem S is the nominal closed loop system resulting from the interconnection of P, I< and weighting function W, and A is a multiplicative perturbation to P. The following different cases of A will be considered. Al. A is an LPTV, causal, possibly unstable operator with period N, llal)m 5 r. A2. A is an LTI, causal, possibly unstable operator, llalloo 5 r, and P + AWP possesses the same number of unstable poles as P. A3. A = 6, ER, 0 5 b 5 r. In Al-A3, r E R, 0 5 r < 03. W in AI and A2 is a stable LTI weighting function with stable inverse, and in A3 W = 1. The As in A1 and A2 represent two different classes of unstructured perturbations. While A in A3 represents a class of structured uncertainties - gain variations. where G(ej") is defined on w E [ 0,w~) and is in the following form G(ejw) = [Gk(e3(w+'"N))]k,~=~,l,..,,~-l = G(z) = [ G~(z~~'"~)]~,I=o,I,_.., N-1 (3) G(ej") or G(z) as given above is defined as the lifted frequency transfer function of the LPTV system G, which transfers the lifted input U(ej") to the lifted output?(elw). Lemma 3.1: Suppose that G is a real, causal, n-th order LPTV system in the form of (1) and G(z) in the form of (3) is the lifted transfer function of G. Then a) G(t) is a proper, finite order transfer function matrix an z with an N x (n x N)-th order pole polynomial D(rN). b) The zeros of D(zN) are given by XiejlWN, i=1,2,..., n,1=0,1,...) N-1. cj G is stable zf and only if all the zeros of D(zN) belong to the open unit disk. d) If G is stable, then llgllm = llgllm. Remark 3.1: Lemma 3.3 is in fact the theoretical basis of the paper. It reveals an important fact that G(t) 3985

3 carries the same stability information of G as its time domain counterpart (l), and G(z) is a proper, finite order transfer function matrix of z with finite number of poles. Thus the frequency domain techniques for LTI MIMO system analysis can be readily applied to G(z). Actually, this is the basic thought of the paper. Lemma 3.2: Suppose that G is a real, causal, and finite order LPTV system with the lifted transfer function G(z) in the form of (3). Then Go(.) on the diagonal of (3) is a real-rational function of z and it gives rise to a real, causal and finite order LTI system Go with Go(%) as its transfer function. Moreover, Go is a stable system if G is a stable system. Lemma 3.3: A system F is LTI if and only if the lifted frequency transfer function F(ejw) has a diagonal structure written as P(ejw) = c ~iag[~o(ej(~+~~~))], i = 0,1,..., N-1. 4 Robust stability conditions of closed loop LPTV systems Consider the nominal closed loop subsystem S in Fig. 1. It is assumed that S is internally stabilized by IC. Apparently, S is LPTV if I< is LPTV. Denote S(&'), i)(ejw), r/z.(ejw), k(ej"), and A(ej") the lifted transfer function matrices of S, P, W, I< and A, respectively. Then F can be represented equivalently by an LTI MIMO system with a nominal closed loop,$(ejw> = r/ir(ej")p(ej")li.(ejw)[i - p(ejw)li.(ejw)]-' and a perturbation block A(.&"). According to Lemmas and the assumptions on P, W, K and A, S(ejw), P(ej"), r/z.(ej"),.k(ej"), and A(ejw) are all N x N dimensional, proper, finite order transfer function matrices, and S(ej") is in the form of (3) with So(ej(w) = diag[so(ej(w+'"n))],=o,~,...,~-l on the diagonal and some nonzero Sk(ej("+lWN)), k = l,..., N - 1, 1 = 0,1,..., N - 1 on the off-diagonals. Hence, the frequency domain techniques for LTI MIMO system analysis can be readily applied to the equivalent system. Lemma 4.1: Let I< be LPTV. Suppose that A satisfies A1 and that p(ej") + A(ej")r/ir(ej")P(ej") possesses the same number of unstable poles as P(ej"). Then the closed loop system 3 is stable if and only if llsll, < 1/r. Lemma 4.2: Let Ir' be LPTV. Suppose that A satisfies A2. Then the closed loop system F is stable if and only if supwe[o,wn) p[s(ejw)] < 1/r, where p[s(ej")] is the structured singular value of S(ej") [3]. Lemma 4.3: Let K be LPTV. Denote yi(w), i = 1,2,..., N, the N eigenvalues of S(ejw) and &(U) the phase of 7i(w). Suppose that A satisfies A3. Then the closed loop system 3 is stable if and only if for w E [0,w~) and i = 1,2,..., N, IYi(0)l # 1/r, h(w) = (4) 5 Stability margin for different perturbations Theorem 5.1: Given a real, causal, finite order, LPTV controller I< which internally stabilizes the closed loop 3986 system S, a real, causal, finite order LTI controller KO can always be constructed such that when applied to the plant P, KO yields a stable LTI closed loop system So and the lifted transfer function of So is just So(ej"), the diagonal of S(ejw). For the proof of the theorem and the details of construc- tion of KO, readers are referred to [13]. From Lemmas 4.1 to 4.3 it is obvious that the stability margin maximization problem for case A1 amounts to the H,-optimization problem of finding a IC which minimizes llsll, = 11S11,; for case A2 it amounts to the p-optimization problem of finding a K which minimizes p(s); while for case A3 it amounts to the characteristic locus shaping problem of finding a IC which shapes the characteristic loci of S so as to maximize r for which the condition (4) holds. The rest of this section will analyze 11S11, p(s) and the characteristic loci of S attainable by the LPTV controller K in comparison with those attainable by the LTI controller KO and the optimal LTI controller I<:. In the following, it is assumed that I<,+ is an optimal LTI controller that yields the nominal LTI closed loop system Sg and the maximumstability margin attainable by LTI controllers. Following the same convention, the lifted frequency transfer function of S,* is denoted as S,+(ej"). Note that for different cases of perturbations, IC: may be different, and that IC: does not necessarily belong to the set KO which, as shown in Theorem 5.1, is constructed from a given LPTV IC set. Accordingly, Sg is not necessarily given by the diagonal of S, the lifted transfer function of an LPTV S. Theorem 5.2: The H, norm of the LPTV system S satisfies IISII, > IISgllm. Before embarking on the proof of the theorem, two lemmas required in the proof are introduced first. Lemma 5.1: The H, norm o the LPTV s stem s satisfies IlSllm L d----k-+ s~p,e[o,2?r)ck=o S,(ej") 2 I IS0 (ei 1 I loo Lemma 5.2: Subject to the assumptions on the nominal plant P, there exists an optimal LTI controller IC: such that the nominal closed loop S$(ej") resulting from IC,+ is all-pass, i.e. IS,+(ejw)l = a G cont, Vw. Proof of Theorem 5.2: Suppose So # S:. It follows from Lemma 5.1 and the optimality of 5': that llslloo 2 IlS0lloo > IISGllm. Conversely, suppose So = Sg. Since S is LPTV, there must be k # 0 and w E [0,2s] such that Sk(eJw) # 0. It then follows from Lemma 5.1 and N--l 2 jw 5.2 that IISIIL 2 SuP,E[0,27r) Ck=O = N-1 2 jw CY2 + SUPwE[0,zrr)Ck=1 &(e > CY2 = 11%112 This proves the theorem. 0 Theorem 5.3: For the LPTV system S with period N 5 3, and for the LPTV system S having period N > 3 and possessing a scaling matrix F(w) which gives ~[F(w)S(ej~)F-'(w)l = p[s(ejw)l, w E [o,wn) (5) the following holds: "P[O,WN) p[s('jw)1 > supio,wn) p[sg

4 Proof: In [3] it is shown that for a square complex matrix M associated with a block diagonal matrix A from which p(m) is defined, p(m) 5 inffa(fmf-'), and the equality always holds if the number of the blocks of A is less than or equal to 3. Where F is a block diagonal scaling matrix with the same number of blocks as that of A. In the special case considered in the theorem, A and hence F are diagonal. Thus the equality holds for dim(a) = N 5 3 and some S with N > 3 and satisfying (5). Suppose F(w) is a scaling matrix which gives the equality (5). According to [3] F(w) = diag[fk(w)], fk(w) E 72, fk(w) >0,.k = 0,1,..., N - 1, w E [0, ON). Then SUP[^,^,) p[s(ejw)] can be written as where S(ejw) := F(w)S(ej")F-'(w) = (ej(w+'wn))}k 1=o,1,..., N- 1, 3 k (e.f(w+lwn) = %sk(ej(w+iwn)!i 1 From the last equations and the structure of S(ej"> it can be observed that S[ej") has the same structure as S(ejw) and contains So(ej") as its diagonal. Since So(eJw) is diagonal, sup p[so(eiw)] = sup S[SO(S")] [OWN) [OWN) = IISo(.iW)IIm = 11soIIw (7) Thus following the same line as that in the proof of Theorem 5.2, it can be readily shown that llsllm > llsg[lm. The theorem then follows from (6) and (7). 0 Corollary 5.1: Suppose that the LPTV system S satisfies the condiiions of Theorem 5.2 and Theorem 5.3. Then llsllm > llsollm and su~~o,~~)~[s(ej")l > s'p[o,wn) p[ S 0 (ej")] if 3k, k # 0, such that Sk(z) does not possess a zero at ejw-. Where wm is the w at which So(ej") takes its Ha norm. Remark 5.1: Theorem 5.2 and Theorem 5.3 reveal an important intrinsic property of LPTV control - for unstructured perturbations, LPTV control can neither improve stability margin nor attain the same level of stability margin as LTI control, instead, it deteriorates stability margin. The cause of this deterioration is the presence of the nonzero off-diagonals of S arising from the LPTV dynamics of IC. Corollary 5.1 further shows that if the channels arising from LPTV dynamics do not all possess a zero at e'"-, the stability margin of LPTV closed loop system S is simply poor than that of the LTI closed loop system So given by the diagonal of 3. A demonstrative example is given in section 6. Note that the condition N 5 3 in Theorem 5.3 is not restrictive. It has been shown in [7] that N = 2 is sufficient for the stabilization of LTI plants subject to structured perturbations. The theorem is in fact valid for the N suggested (implicitly) in the previous literature Now consider cases A.3. Let &i, i = 1,2,3, be the stability margin for cases A1 - A3, respectively, i.e. R,i = the maximum r in case Ai, i = 1,2,3, for which 3 remains stable. Further, let &i(kz) and Rmi(K) be the stability margins attained by K; and the maximum stability margins attainable by the LPTV K set, respectively. Define r,i := the maximum r for which yj(w) satisfies condition (4). Theorem 5.4: a) R,,,3(K) > &3(KO+) provided P is biproper. b) Rm3(I() = minlsig rmi. c) a) and b) are achieved only if S(ejw) possesses nonzero off-diagonals resulting from the periodically tame varying dynamics of LPTV Ii. d) Provided that S(ejw) satisfies the conditions of Theorem 5.2 and 5.3, Rm3(K) > Rm3(K;) impltes Rm1(K) < Rml(K;) and & 2(K) < &2(K;). Proof: a) This is the well known result of [7]. b) Directly from condition (4) and the definition of Rm3(K). c) Suppose $(ejw) does not have nonzero off-diagonals, S(ejw) reduces to a diagonal matrix which, as shown in Theorem 5.1, corresponds to an LTI closed loop system resulting from an LTI controller Iih. Denote &a(ka) the stability margin given by this K;. From the optimality of Kz, &3(KA) 5 Rm3(Kz). This proves c). d) From c), if a) holds, S(eJw) must have nonzero offdiagonals. d) then follows from Theorem 5.2 and Theorem Remark 5.2: Theorem 5.4 reveals how can LPTV control improve gain margin - shaping n(w) loci with nonzero off-diagonals. In case S is LTI, yi(w) loci are determined solely by the diagonals of S, which are the frequency responses of an LTI nominal closed loop system at different frequency bands. It is known that for LTI controllers, if P is nonminimum phase, gain margin is always limited [7]. Notice that the gain margin defined in [7] equals &3 - l. This means that with only diagonals, 7i(ejw) is confined to certain region which gives the limited Rm3. Whereas, if S is LPTV, S has nonzero off-diagonals. These off-diagonals provide extra freedom for further tuning of yi(ej") so that a better shape of 7i(ej") which gives larger &3 can be obtained. This is the mechanism of LPTV K in gain margin improvement. The example in next section will demonstrate this graphically. Apparently, any benefit that LPTV IC can bring is from tuning the magnitudes and/or phases of yi(ej") with nonzero off-diagonals. The nonzore off-diagonals are essentially indispensable for this tuning. However, as shown in the theorem, these nonzero off-diagonals definitely deteriorate the stability margin for unstructured perturbations. Hence the LPTV Ii' designed for gain margin maximization can be very sensitive to other classes of perturbations. The example in next section will show how sensitive it could be.

5 6 Example It can be readily shown using bilinear transform and the Consider the LTI plant P(z) = k ~ This is ~ an ex- ~ well known ~ procedure ~ ~ given in.[5] that the optimal conample studied by many authors, e.g. [7, 21. It is shown troller I<$ for the plant is I{$ = $& and the resulting in [7] that with LTI controller R-3 = 1.25 (R-3 = gain all-pass LTI closed loop system is S: = -2.5,-, Bz-f margin -1, where the gain margin is as defined in [7]). with ls811m = lls;,.llw = p[s,*] = 14db. Thus Whereas in [2] it is shown that with the LPTV con- troller K(q-') = l+(-l)'+(6+6(-l)t)q-1, the closed loop system is stable for IC < -5 and k > 4. Note that for the given I<, N = 2, WN = 7r, ej'"n = (-1)'. Take W = 1 (i@(z) = I> and assume k = ko, a known nominal value. The lifted transfer function of the nominal closed loop system S can be calculated as S( z) = P(,)I? (2)[I-P( z )I? (z)]- 1 = [ $?) )I : :?: Sl(2) = ko(1-2~-')(1+ 3.~-~)(1-6z-l) 1 + 2k0-9r-2 Following the line of [13], the LTI controller I?o(r) as described in Theorem 5.1 can be calculated as It can be easily checked that when applied to the nominal plant P(z), I<o(z) gives the closed loop system So(z). This verifies the result of Theorem 5.1. Since S1(z) does not have zero on unit disk, from Corollary 5.1 it is known that llsll > lls0ll and ' suplde[o,wn) p[s(ejw 11 > supcde[o,wn) piso Because N = 2, according to [3], (5) always holds and p[$ can be calculated precisely (e.g. using MATLAB p-toolbox). Hence IIS(ej")llm = 6[S(ejw)] and p[s(ej")] can be evaluated from the peak values of e[s(ej")] and p[s(ejw)] plots, and can be compared with ISo(ej")l = 5[S?(ej,")] = p[so(ej")] which is given by the diagonals of S(eJW). All the plots given are plotted for ko = -6 and w E [0,7r) since WN = r. Fig. 2 gives the 5[S(ej")], p[s(ejw)] and 5[S0(ejw)] plots. From the peak values of the three plots, it can be seen clearly that lls(ejw)llm = s~p,~[~-,,~),p[s(ej")] = 48db > IISo(ej")llw = sup,e[o,wn) p[so(ej")] M 43db. This coincides with the analysis of Corollary 5.1. Fig. 3 gives the plots for n (w), the minimumeigenvalue of S(ej"). It can be observed that Iyl(w)l < -250db and ld1(w)l 5 7r. Because I-yl(w)I is virtually zero, r-1 = 00. Fig. 4 gives the plots for yz(w), the maximum eigenvalue of S(ej"). As shown in the figure, -5db < yz(w) 5 16db and 1~#2(w)l 5 ~/2. Since &?(U) is always smaller than T, r-2 = 03. Thus, for -cm < ko 5-6 the closed loop system is always stable, i.e. Rm3 = m(azl<i<2~~i = gain margin -1 = 00. The results coincide with the analysis of Theorem 5.4 and explain how the gain margin is improved llsll00 > IlS,'llm, SUPW [0,2?r) > SUPWE[0,27r) P[S;l. This coincides with the analyses of Theorem 5.2 and Theorem 5.3. It is obvious that although the LPTV control K provides a very significant gain margin improvement, the resultant IIS(eJ")llm and p[s(ej")] are very large. Hence it is very sensitive to the unstructured perturbations. When ko = -6, a perturbation A of form AI or A2 with r > will destabilize the system, while a destabilizing A for the optimal LTI I(: must have r > 5, which is 1250 times of that for the LPTV K. Similar results have also been observed in the computation for ko = 5, which show that though the shapes of the plots are different, the conclusions of Theorem 5.2, Theorem 5.3 and Corollary 5.1 are still true. Conclusions The necessary and sufficient conditions on the robust stability of LPTV control systems have been presented in frequency domain for the different classes of perturbations. These conditions are the extension of the LTI system robust stability conditions to LPTV systems and can be further extended to the analysis and design of general LPTV systems with LPTV plants and controllers. The three classes of model perturbations considered in the paper are all the theoretically and practically important ones, and were most often analyzed separately by different techniques in the previous literature - resulting in somewhat uncorrelated and conflict results. The quantitative analysis and computational results have shown that for unstructured perturbations, LPTV control definitely deteriorates the system stability margin. The condition (4) have provided a frequency domain insight into the mechanism of LPTV control in gain margin improvement and have enable us to revealed the consequence of this improvement - deterioration of the stability margin for unstructured perturbations. References &I Chapellat, H. and M. Dahleh, (1992), Analysis of time varying control strategies for optimal disturbance rejection and robustness, IEEE Trans. on Automatzc Control, Vol. AC-37, pp Das, S. K. and Rajagopalan, P. K., (1992), Periodic discrete-time systems: stability analysis and robust control using zero placement, IEEE Trans. on Automatic Control, Vol. AC- 37, No. 3, pp

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