A Polynomial Chaos Approach to Robust \mathcal{H}_\infty Static Output-Feedback Control with Bounded Truncation Error
Yiming Wan, Dongying E. Shen, Sergio Lucia, Rolf Findeisen, Richard D. Braatz
AA Polynomial Chaos Approach to Robust H ∞ Static Output-FeedbackControl with Bounded Truncation Error
Yiming Wan, Dongying E. Shen, Sergio Lucia, Rolf Findeisen, and Richard D. Braatz
Abstract — This article considers the H ∞ static output-feedback control for linear time-invariant uncertain systemswith polynomial dependence on probabilistic time-invariantparametric uncertainties. By applying polynomial chaos the-ory, the control synthesis problem is solved using a high-dimensional expanded system which characterizes stochasticstate uncertainty propagation. A closed-loop polynomial chaostransformation is proposed to derive the closed-loop expandedsystem. The approach explicitly accounts for the closed-loopdynamics and preserves the L -induced gain, which resultsin smaller transformation errors compared to existing polyno-mial chaos transformations. The effect of using finite-degreepolynomial chaos expansions is first captured by a norm-bounded linear differential inclusion, and then addressed byformulating a robust polynomial chaos based control synthesisproblem. This proposed approach avoids the use of high-degreepolynomial chaos expansions to alleviate the destabilizing effectof truncation errors, which significantly reduces computationalcomplexity. In addition, some analysis is given for the conditionunder which the robustly stabilized expanded system impliesthe robust stability of the original system. A numerical exampleillustrates the effectiveness of the proposed approach. I. I
NTRODUCTION
Numerous studies have been reported on robust controldesign subject to model uncertainties, e.g., see [18] andreferences therein. The widely used worst-case strategy tendsto produce highly conservative performance because theworst-case scenario may have vanishingly low probability ofoccurrence. In contrast to a worst-case performance bound,practical interest in the performance variation or dispersionacross the uncertainty region has motivated recent researchon probabilistic robustness [18]. The design objective ei-ther adopts a probability-guaranteed worst-case performancebound [21], [25], or optimizes the averaged performance, atthe expense of an increased worst-case performance bound[2]. This line of research considers polytopic uncertainties[2], [25], or affine dependence on multiplicative white noises[8]. The randomized algorithm proposed in [21] can address
This work is supported by the National Natural Science Foundation ofChina under Grant No. 61803163, and the DARPA Make-It program undercontract ARO W911NF-16-2-0023.Yiming Wan is with School of Artificial Intelligence and Automation,Huazhong University of Science and Technology, and Key Laboratory ofImage Processing and Intelligent Control, Ministry of Education, Wuhan430074, China. Email address: [email protected]
Dongying E. Shen and Richard D. Braatz are with Massachusetts Instituteof Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA.Email address: { dongying, braatz } @mit.edu Sergio Lucia is with TU Dortmund University, 44227 Dortmund, Ger-many. Email address: [email protected]
Rolf Findeisen is with Otto-von-Guericke University Magdeburg, 39106Magdeburg, Germany. Email address: [email protected] nonlinear uncertainty structures, but can be computationallydemanding since a large number of samples is often required.As mentioned above, the probabilistic description of para-metric uncertainties allows optimizing the averaged per-formance. A common assumption is multiplicative whitenoise as in [8], with arbitrarily fast and unbounded rateof parameter changes, which is unrealistic for many ap-plications. Often, uncertain parameters change significantlyslower than the underlying system dynamics. For this reason,they can be regarded as time invariant, and their probabilis-tic information can be obtained from a priori knowledgeor parameter identification. When accounting for the timeinvariance of uncertain parameters, uncertainty propagationof the system state becomes non-Markovian [16], thus itscomputation usually resorts to computationally expensivesampling-based approaches [21]. Recently, some progressin robust control synthesis has been made by exploitingpolynomial chaos (PC) theory as a non-sampling approachfor uncertainty propagation, e.g., see [6], [10], [15], [17],[19]. The PC based stochastic Galerkin method characterizesthe evolution of probability distributions of the underlyingstochastic system states by a high-dimensional transformeddeterministic system that describes the evolution dynamics ofthe polynomial chaos expansion (PCE) coefficients. Then thecontrol synthesis problem can be solved by using the PCE-transformed system. An important consideration in PCE-based control is that due to using finite-degree PCEs, thePCE-transformed system has an inevitable approximationerror, which may destabilize the closed-loop dynamics andreduce closed-loop performance [1], [12], [14]. Althoughincreasing the PCE degree can reduce the effect of PCEtruncation errors, computational complexity increases signifi-cantly, as the state dimension of the PCE-transformed systemfactorially grows with the PCE degree.In this paper, a PCE-based H ∞ static output-feedback(SOF) control approach is proposed for uncertain lineartime-invariant (LTI) systems with polynomial dependenceon probabilistic time-invariant parametric uncertainties. Theproposed PCE-based transformation explicitly copes with theclosed-loop dynamics, and preserves the L -induced gain,which results in a smaller approximation error comparedto existing PCE-transformed systems reported in [19], [23].To account for the perturbations from the PCE truncationerrors, a norm-bounded linear differential inclusion (LDI) isconstructed for the PCE-transformed system. Then a robustPCE-based synthesis problem is formulated to minimize anupper bound of the L -induced gain from disturbances tothe controlled outputs. This formulation enforces closed-loop a r X i v : . [ ee ss . S Y ] F e b tability and robustifies the control performance against PCEtruncation errors. These benefits are achieved without thehigh computational complexity resulted from using a high-degree PCE.This paper is organized as follows. Section II statesthe robust H ∞ SOF control problem. Section III reviewspreliminaries on PC theory. Section IV derives the PCE-transformed closed-loop system. In Section V, a robust PCE-based SOF control synthesis method is proposed by using thenorm-bounded LDI to account for the PCE truncation errors.A simulation study and some concluding remarks are givenin Sections VI and VII, respectively.Notations: For a continuous-time vector-valued sig-nal x ( t ) , (cid:107) x ( t ) (cid:107) L denotes the L norm defined as (cid:0)(cid:82) ∞ x (cid:62) ( t ) x ( t ) d t (cid:1) / . The 2-norm of a vector x is defined as (cid:107) x (cid:107) = √ x (cid:62) x . For a square matrix X , let He { X } represent X + X (cid:62) . For a symmetric matrix X , its positive and negativedefiniteness is denoted by X > and X < , respectively.The vectorization of a matrix X is denoted by vec ( X ) . TheKronecker product is represented by ⊗ .II. P ROBLEM S TATEMENT
Consider an uncertain LTI system described by ˙ x ( t, ξ ) = A ( ξ ) x ( t, ξ ) + B w ( ξ ) w ( t ) + B ( ξ ) u ( t, ξ ) (1a) z ( t, ξ ) = C z x ( t, ξ ) + D zw w ( t ) + D z u ( t, ξ ) (1b) y ( t, ξ ) = C ( ξ ) x ( t, ξ ) + D w ( ξ ) w ( t ) (1c)where x ∈ R n x is the state, u ∈ R n u is the control input, w ∈ R n w is the external disturbance, y ∈ R n y is themeasured output, z ∈ R n z is the controlled output, and ξ ∈ R n ξ is the uncertain parameter vector that lies within abounded set Ξ . In the closed-loop system, the system state x , control input u , measured output y , and controlled output z all depend on ξ . The matrices C z , D zw , and D z in (1b)are independent of ξ , because they are determined by theperformance specifications in the controlled output z . Assumption 1:
The system matrices A , B w , B , C , and D w in (1) are polynomial functions of ξ .The above assumption of polynomial dependence on ξ isnot restrictive since non-polynomial nonlinear dependencecan be accurately approximated by polynomials or piecewisepolynomials [20].The objective of this paper is to design an SOF controller u ( t, ξ ) = Ky ( t, ξ ) (2)that solves the H ∞ -optimal control problem min K ,γ> γ s.t. E ξ (cid:110) (cid:107) z ( t, ξ ) (cid:107) L (cid:111) ≤ γ (cid:107) w ( t ) (cid:107) L , ∀ (cid:107) w ( t ) (cid:107) L < ∞ (3)for the closed-loop system ˙x ( t, ξ ) = A cl ( ξ ) x ( t, ξ ) + B cl ( ξ ) w ( t ) (4a) z ( t, ξ ) = C cl ( ξ ) x ( t, ξ ) + D cl ( ξ ) w ( t ) (4b) with A cl ( ξ ) = A ( ξ ) + B ( ξ ) KC ( ξ ) , B cl ( ξ ) = B w ( ξ ) + B ( ξ ) KD w ( ξ ) , C cl ( ξ ) = C z + D z KC ( ξ ) , D cl ( ξ ) = D zw + D z KD w ( ξ ) . (5)The L -induced gain γ from the disturbance w ( t ) to thecontrolled output z ( ξ , t ) generalizes the H ∞ norm of a de-terministic LTI system. The mathematical expectation E ξ {·} in (3) accounts for the time-invariant probabilistic parametricuncertainties ξ .To explicitly account for the polynomial dependence onprobabilistic uncertain parameters ξ , the basic idea of solvingthe problem (3) is: (i) first apply the PCE-based transforma-tion to derive a high-dimensional transformed closed-loopsystem that describes the dynamics of PCE coefficients;and (ii) then construct a H ∞ -control synthesis problemfor the PCE-transformed closed-loop system. However, dueto using finite-degree PCEs, PCE truncation errors leadto discrepancy between the PCE-transformed closed-loopsystem and the original closed-loop system. Even whenthe PCE-transformed closed-loop system is stabilized, thepresence of PCE truncation errors can destabilize the originalclosed-loop system or reduce its closed-loop performance[12]. Most PCE-based control literature have not addressedthis challenge, which motivates this investigation in howto (i) perform the PCE-based transformation to achievesmall transformation errors; (ii) take into account the PCEtruncation errors in solving the problem (3); and (iii) ensurethe stability of the original closed-loop system by stabilizingthe PCE-transformed system.III. P RELIMINARIES ON POLYNOMIAL CHAOS THEORY
This section is a brief overview on polynomial chaostheory to facilitate the PCE-based control synthesis proposedin the next sections.For a random vector ξ , a function ψ ( ξ ) : R n ξ → R witha finite second-order moment admits a PCE [24] ψ ( ξ ) = ∞ (cid:88) i =0 ψ i φ i ( ξ ) , where { ψ i } denotes the expansion coefficients and { φ i ( ξ ) } denotes the multivariate PC bases. By using the Askeyscheme of orthogonal polynomial bases, this expansion expo-nentially converges in the L sense, which results in accurateapproximations even with a relatively small number of terms[24]. For the sake of notation simplicity, these basis functionsare normalized in the rest of this paper such that they becomeorthonormal with respect to the probabilistic distribution µ ( ξ ) of the random vector ξ , i.e., (cid:104) φ i ( ξ ) , φ j ( ξ ) (cid:105) = (cid:90) Ξ φ i ( ξ ) φ j ( ξ ) µ ( ξ ) d ξ = E ξ { φ i ( ξ ) φ j ( ξ ) } = (cid:40) if i = j otherwise , (6)here Ξ is the support of µ ( ξ ) . By exploiting the orthonor-mality in (6), each PCE coefficient ψ i is computed by ψ i = (cid:104) ψ ( ξ ) , φ i ( ξ ) (cid:105) , (7)which can be calculated via numerical integration [24]. Inparticular, the mean and variance of ψ ( ξ ) can be computedfrom the PCE coefficients as below: E ξ { ψ ( ξ ) } = (cid:104) ψ ( ξ ) , φ ( ξ ) (cid:105) = ψ , var { ψ ( ξ ) } = ∞ (cid:88) i =1 ( ψ i φ i ( ξ )) = ∞ (cid:88) i =1 ψ i . (8)In practical computations, a PCE with an infinite numberof terms is truncated to a finite degree p : ψ ( ξ ) ≈ ˆ ψ ( ξ ) = N p (cid:88) i =0 ψ i φ i ( ξ ) . (9)The total number of terms in (9) is N p + 1 = ( n ξ + p )! n ξ ! p ! ,depending on the dimension n ξ of ξ and the highest degree p of the retained polynomials { φ i ( ξ ) } N p i =0 .Let ξ [ S i ] denote a monomial ξ s i, ξ s i, · · · ξ s i,nξ n ξ with S i =( s i, , s i, , · · · , s i,n ξ ) . The monomial has an exact PCE ξ [ S i ] = N pi (cid:88) k =0 β ik φ k ( ξ ) (10)with a degree p i = (cid:80) n ξ j =1 s i,j without any truncation errors,since any monomial can be exactly expressed as a linearcombination of the corresponding orthonormal polynomialbases, as illustrated by the below example. Example 1:
The normalized Legendre polynomials formorthonormal bases with respect to a random scalar ξ uni-formly distributed over [ − , . The first four normalizedLegendre polynomials are given by φ ( ξ ) φ ( ξ ) φ ( ξ ) φ ( ξ ) = √ − √ √ − √ √ ξξ ξ , such that { φ i ( ξ ) , i = 0 , , , } are orthonormal, see Ap-pendix B of [13]. By inverting the coefficient matrix, themonomials ξ , ξ , and ξ can be represented as ξξ ξ = √ √ √ √ φ ( ξ ) φ ( ξ ) φ ( ξ ) φ ( ξ ) , where each row of the above coefficient matrix is the PCEcoefficients for the corresponding monomial.With the monomial bases { ξ [ S i ] } , any multivariate poly-nomial function ψ ( ξ ) is expressed as ψ ( ξ ) = (cid:88) S i ∈S α [ S i ] ξ [ S i ] , (11)where { α [ S i ] } are the monomial coefficients, and the set S is determined by all the monomials presented in ψ ( ξ ) . Then the exact PCE for ψ ( ξ ) in (11) can be directly computedfrom (10) as ψ ( ξ ) = N p (cid:88) k =0 (cid:32) (cid:88) S i ∈S α [ S i ] β ik (cid:33) φ k ( ξ ) (12)with p = max i { p i } denoting the degree of ψ ( ξ ) in (11).IV. PCE- TRANSFORMED CLOSED - LOOP SYSTEM
In this section, a PCE-transformed system is constructedfor the closed-loop system (4), which will be used in PCE-based control synthesis.Let x i denote the i th component of the state vector x . Thescalar x i ( t, ξ ) is expressed as x i ( t, ξ ) = ˆ x i ( t, ξ ) + ˜ x i ( t, ξ ) , ˆ x i ( t, ξ ) = N p (cid:88) j =0 x i,j ( t ) φ j ( ξ ) = x (cid:62) i ( t ) φ ( ξ ) , (13)where ˆ x i ( t, ξ ) is the truncated PCE with a degree p , x (cid:62) i ( t ) and φ ( ξ ) denote x (cid:62) i ( t ) = (cid:2) x i, ( t ) x i, ( t ) · · · x i,N p ( t ) (cid:3) , φ ( ξ ) = (cid:2) φ ( ξ ) φ ( ξ ) · · · φ N p ( ξ ) (cid:3) (cid:62) , (14)and ˜ x i ( t, ξ ) represents the truncation error. Define ˆx ( t, ξ ) = (cid:2) ˆ x ( t, ξ ) ˆ x ( t, ξ ) · · · ˆ x n x ( t, ξ ) (cid:3) (cid:62) , ˜x ( t, ξ ) = (cid:2) ˜ x ( t, ξ ) ˜ x ( t, ξ ) · · · ˜ x n x ( t, ξ ) (cid:3) (cid:62) , x PCE ( t ) = (cid:2) x ( t ) · · · x n x ( t ) (cid:3) , then the PCE of the vector x ( t, ξ ) can be written as x ( t, ξ ) = ˆx ( t, ξ ) + ˜x ( t, ξ ) = x (cid:62) PCE ( t ) φ ( ξ ) + ˜x ( t, ξ )= (cid:16) φ (cid:62) ( ξ ) ⊗ I n x (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) Φ (cid:62) x ( ξ ) vec (cid:0) x (cid:62) PCE ( t ) (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) X ( t ) + ˜x ( t, ξ ) . (15)The last equation of (15) applies the property of the Kro-necker product vec ( EF G ) = ( G (cid:62) ⊗ E ) vec ( F ) .With the PCE of x ( t, ξ ) in the form of (15), the closed-loop system (4) can be equivalently rewritten as Φ (cid:62) x ( ξ ) ˙X ( t ) = A cl ( ξ )Φ (cid:62) x ( ξ ) X ( t ) + B cl ( ξ ) w ( t ) + r x ( t, ξ ) , (16a) r x ( t, ξ ) = − ˙˜x ( t, ξ ) + A cl ( ξ ) ˜x ( t, ξ ) , (16b) z ( t, ξ ) = z nom ( t, ξ ) + r z ( t, ξ ) , (16c) z nom ( t, ξ ) = C cl ( ξ )Φ (cid:62) x ( ξ ) X ( t ) + D cl ( ξ ) w ( t ) , (16d) r z ( t, ξ ) = C cl ( ξ ) ˜x ( t, ξ ) . (16e)The PCE-based transformation of of the closed-loop dynamicequation (4a) follows the typical stochastic Galerkin method(16) as follows. Left-multiplying (16a) by Φ x ( ξ ) leads to Φ x ( ξ )Φ (cid:62) x ( ξ ) ˙X ( t ) = Φ x ( ξ ) A cl ( ξ )Φ (cid:62) x ( ξ ) X ( t )+ Φ x ( ξ ) B cl ( ξ ) w ( t ) + Φ x ( ξ ) r x ( t, ξ ) . (17)aking the mathematical expectation with respect to ξ onboth sides of (17), the PCE-transformed closed-loop dynamicequation is obtained as ˙ X ( t ) = ¯ A cl X ( t ) + ¯ B cl w ( t ) + ¯R x ( t ) , (18a) ¯R x ( t ) = E ξ { Φ x ( ξ ) r x ( t, ξ ) } = E ξ { Φ x ( ξ ) A cl ( ξ ) ˜x ( t, ξ ) } , (18b) ¯ A cl = E ξ { Φ x ( ξ ) A cl ( ξ )Φ (cid:62) x ( ξ ) } , (18c) ¯ B cl = E ξ { Φ x ( ξ ) B cl ( ξ ) } , (18d)which describes the dynamics of the PCE coefficient vector X ( t ) . The above derivations exploit E ξ { Φ x ( ξ )Φ (cid:62) x ( ξ ) } = I n x ( N p +1) and E ξ { Φ x ( ξ ) ˙˜x ( t, ξ ) } = , according to (6) andthe definition of Φ x ( ξ ) in (14) and (15).To compute ¯ A cl and ¯ B cl defined in (18c) and (18d), thepolynomial dependence of Φ x ( ξ ) B ( ξ ) , C ( ξ )Φ (cid:62) x ( ξ ) , and D w ( ξ ) on ξ should be exploited according to Assumption1. Let q denote the maximal degree of these polynomialmatrices above. Then according to (10)–(12), their exactPCEs are expressed as Φ x ( ξ ) B ( ξ ) = N q (cid:88) i =0 ˆB i φ i ( ξ ) , (19a) C ( ξ )Φ (cid:62) x ( ξ ) = N q (cid:88) i =0 ˆC i φ i ( ξ ) , (19b) D w ( ξ ) = N q (cid:88) i =0 ˆD w ,i φ i ( ξ ) , (19c)where N q + 1 is the number of terms in the PCE of degree q . Proposition 1:
With the PCE coefficients { ˆB i } , { ˆC i } ,and { ˆD w ,i } in (19a)–(19c), ¯ A cl in (18c) and ¯ B cl in (18d)are computed as ¯ A cl = A + ¯ B ¯ K ¯ C , ¯ B cl = B w + ¯ B ¯ K ¯ D w , (20a) A = E ξ { Φ x ( ξ ) A ( ξ )Φ (cid:62) x ( ξ ) } , (20b) B w = E ξ { Φ x ( ξ ) B w ( ξ ) } , (20c) ¯ B = (cid:2) ˆB ˆB · · · ˆB N q (cid:3) , (20d) ¯ C = (cid:104) ˆC (cid:62) ˆC (cid:62) · · · ˆC (cid:62) N q (cid:105) (cid:62) , (20e) ¯ D w = (cid:104) ˆD (cid:62) w , ˆD (cid:62) w , · · · ˆD (cid:62) w ,N q (cid:105) (cid:62) , (20f) ¯ K = I N q +1 ⊗ K . (20g)The proof of Proposition 1 is in Appendix A.The matrices A in (20b), B w in (20c), { ˆB i } in (19a), { ˆC i } in (19b), and { ˆD w ,i } in (19c) can be computed byeither projections as in (7)–(8) via numerical integration orthe use of exact PCEs of monomials as in (10)–(12).The PCE-transformed approximation of the output equa-tion (16c) is described in the proposition below, which isdifferent from the standard Galerkin projection. Proposition 2:
The equation (cid:107) Z nom ( t ) (cid:107) L = E ξ {(cid:107) z nom ( t, ξ ) (cid:107) L } (21) holds, where z nom ( t, ξ ) is defined in (16d), and Z nom ( t ) isdefined by Z nom ( t ) = ¯ C cl X ( t ) + ¯ D cl w ( t ) , (22a) ¯ C cl = ¯ C Z + ¯ D Z ¯ K ¯ C , ¯ D cl = ¯ D Zw + ¯ D Z ¯ K ¯ D w , (22b) ¯ C Z = (cid:20) C Z (cid:21) ∈ R n z ( N q +1) × n x ( N p +1) , (22c) ¯ D Zw = (cid:20) D Zw (cid:21) ∈ R n z ( N q +1) × n w , (22d) C Z = I N p +1 ⊗ C z , D Zw = E ξ { Φ z ( ξ ) D zw } , (22e) ¯ D Z = I N q +1 ⊗ D z , (22f)with Φ z ( ξ ) defined similarly to Φ x ( ξ ) in (15).The proof of Proposition 2 is in Appendix B.The above derivations (16)–(18) are exact, including theeffect of unknown truncation errors ˜x ( t, ξ ) captured by r x ( t, ξ ) , r z ( t, ξ ) , and ¯R x ( t ) . After neglecting these unknownterms, a PCE-transformed approximation ˙ X a ( t ) = ¯ A cl X a ( t ) + ¯ B cl w ( t ) Z a ( t ) = ¯ C cl X a ( t ) + ¯ D cl w ( t ) , (23)is obtained according to Propositions 1 and 2. Note that X a ( t ) and Z a ( t ) in (23) are approximates of X ( t ) in (18)and Z nom ( t ) in (22a). This PCE-transformed system (23) isdifferent from the existing ones in two aspects:(i) The existing approach in [6], [19], [23] separatelytransforms the four system equations in (1) and (2)by applying standard Galerkin projection, to constructa PCE-transformed closed-loop system. In this case, ifthe open-loop system is unstable due to its parametricuncertainties, the divergent open-loop system state as afunction of ξ would not admit a PCE. In contrast, ourproposed transformation directly considers the originalclosed-loop dynamics, which allows PCEs of systemstates by stabilizing the closed-loop dynamics.(ii) With the error term ¯R x ( t ) removed from (18a), thesquared L -induced gain from w ( t ) to z nom ( t, ξ ) in(18a) and (16d) is equal to the squared H ∞ -norm ofthe PCE-transformed approximation (23), i.e., sup (cid:107) w (cid:107) L ≤ E ξ (cid:110) (cid:107) z nom ( t, ξ ) (cid:107) L (cid:111) = sup (cid:107) w (cid:107) L ≤ (cid:107) Z nom ( t ) (cid:107) L , (24)according to (21). This property does not hold for thePCE-transformed approximation given in [6], [19], [23].Moreover, it is proved by (49)–(52) in Appendix C that theworst-case upper bound for the approximation error x ( t, ξ ) − Φ (cid:62) x ( ξ ) X a ( t ) from (23) is smaller than the one obtained fromthe PCE-transformed system in [6], [19], [23].V. R OBUST
PCE-
BASED S TATIC O UTPUT -F EEDBACK C ONTROL
The original synthesis problem (3) can be transformed intoa standard H ∞ -control problem min K ,γ nom > γ nom s.t. (cid:107) Z nom ( t ) (cid:107) L ≤ γ nom (cid:107) w ( t ) (cid:107) L , or the PCE-transformed closed-loop system (23). By ap-plying the bounded real lemma [5], the above problem isreformulated as the following synthesis problem [23]: min P , K ,γ nom γ nom s.t. P > , γ nom > , He { P ¯ A cl } P ¯ B cl ¯ C (cid:62) cl ¯ B (cid:62) cl P − γ nom I n w ¯ D (cid:62) cl ¯ C cl ¯ D cl − γ nom I n x ( N p +1) < . (25)The third constraint in (25) is a bilinear matrix inequality(BMI) due to the multiplication between P and ¯ K in P ¯ A cl and P ¯ B cl . These bilinear terms cannot be converted to linearterms by using the conventional change of variables, due tothe block-diagonal structure of ¯ K = I N q +1 ⊗ K .It should be noted that the PCE truncation error is ne-glected in (23), thus is not addressed by the above synthesisproblem (25). Due to this reason, even if the nominal PCE-transformed closed-loop system (23) is stabilized, the trueclosed-loop system (4) is not necessarily stable. A com-monly adopted remedy in literature is to use higher-degreePCEs with smaller truncation errors, which still providesno theoretical guarantee. In addition, as the PCE degree p becomes higher, the number of PCE terms grows factorially,which significantly increases the computational complexityinvolved in solving the PCE-based synthesis problem.To address the above stability issue, a robust PCE-basedcontrol synthesis method is proposed in this section byconstructing a norm-bounded LDI to describe the effect ofPCE truncation errors. This approach robustifies stability andperformance without increasing the PCE degree. A. Linear differential inclusion for PCE-transformed closed-loop system
Before deriving a LDI for the PCE-transformed closed-loop system, the following proposition is given for the PCEtruncation error ˜x ( t, ξ ) . Proposition 3:
For the PCE truncation error ˜x ( t, ξ ) , thereexists a non-unique matrix M ( t, ξ ) ∈ R n x × n x such that ˜x ( t, ξ ) = M ( t, ξ )Φ (cid:62) x ( ξ ) X ( t ) = Φ (cid:62) x ( ξ ) N ( t, ξ ) X ( t ) , (26)with N ( t, ξ ) defined as N ( t, ξ ) = I N p +1 ⊗ M ( t, ξ ) .The proof of Proposition 3 is given in Appendix D.It is assumed in the following derivations that the uncer-tainty matrix N ( t, ξ ) in (26) is norm-bounded, i.e., N ( t, ξ ) ∈ F x = { ∆ | ∆ (cid:62) ∆ ≤ ρ I n x ( N p +1) } . (27)Then, the norm-bounded LDI system [5] Φ (cid:62) x ( ξ ) ˙X ( t ) = A cl ( ξ )Φ (cid:62) x ( ξ )( I + ∆ x ( t )) X ( t )+ B cl ( ξ ) w ( t ) − ˙˜x ( t, ξ ) (28a) z rob ( t, ξ ) = C cl ( ξ )Φ (cid:62) x ( ξ )( I + ∆ x ( t )) X ( t )+ D cl ( ξ ) w ( t ) (28b)is constructed for the closed-loop system (16) by substituting(26) into (16b) and (16e) and replacing N ( t, ξ ) with ∆ x ( t ) ∈F x , where I represents an identity matrix of appropriate dimensions. Under the condition (26), the system trajectoryset of the LDI (27)–(28) includes the system trajectory of(16). Hence, given X (0) = X (0) = 0 , the followinginequality holds: sup w ( t ) , ∆ x ( t ) ∈F x E ξ {(cid:107) z rob ( t, ξ ) (cid:107) L } ≥ E ξ {(cid:107) z ( t, ξ ) (cid:107) L } . (29)With similar procedures as in Section IV, the PCE-transformed LDI, i.e., the PCE transformation of (28), isobtained as ˙ X ( t ) = ¯ A cl ( I + ∆ x ( t )) X ( t ) + ¯ B cl w ( t ) , (30a) Z rob ( t ) = ¯ C cl ( I + ∆ x ( t )) X ( t ) + ¯ D cl w ( t ) , (30b)where ¯ A cl , ¯ B cl , ¯ C cl , and ¯ D cl are computed according toPropositions 1 and 2. Similarly to (21), the above PCEtransformation ensures (cid:107) Z rob ( t ) (cid:107) L = E ξ {(cid:107) z rob ( t, ξ ) (cid:107) L } ,which further implies sup w ( t ) , ∆ x ( t ) ∈F x (cid:107) Z rob ( t ) (cid:107) L ≥ E ξ {(cid:107) z ( t, ξ ) (cid:107) L } (31)for any X (0) = X (0) = 0 according to (29). B. Robust PCE-based control synthesis
Based on the PCE-transformed LDI (30), a H ∞ controlsynthesis problem can be formulated as min K ,γ rob > γ rob s.t. (cid:107) Z rob ( t ) (cid:107) L ≤ γ rob (cid:107) w ( t ) (cid:107) L , ∀ (cid:107) w ( t ) (cid:107) L < ∞ , ∆ x ( t ) ∈ F x . (32)According to (31), this problem minimizes an upper boundof γ in the original problem (3). Theorem 1:
The PCE-transformed LDI (30) is quadrati-cally stable, and its H ∞ norm is upper bounded by γ rob , ifthere exist a positive definite matrix P and a scalar τ > such that He { P ¯ A cl } + τ ρ I P ¯ B cl P ¯ A cl ¯ C (cid:62) cl ¯ B (cid:62) cl P − γ rob I 0 ¯ D (cid:62) cl ¯ A (cid:62) cl P 0 − τ I ¯ C (cid:62) cl ¯ C cl ¯ D cl ¯ C cl − γ rob I < (33)where ρ is defined in (27) to quantify the effect of PCEtruncation errors.The proof of Theorem 1 is given in Appendix E.The robust PCE-based control synthesis problem is thenformulated as min P , K ,γ rob ,τ γ rob s.t. (33) , P > , γ rob > , τ > . (34)The constraint (33) is a bilinear matrix inequality (BMI) withrespect to P and K . It should be noted that the PCE-basedsynthesis solution to (34) might fail to stabilize the originalclosed-loop system (4) if the bound ρ in (27) is inadequateto address the underlying uncertainty. This issue brings upthe following discussions.irst of all, the corollary below presents a sufficientcondition under which the control gain derived from (34)stabilizes the original closed-loop system (4). Corollary 1.1:
If the PCE-transformed LDI (30) is ro-bustly stabilized and the uncertainty matrix N ( t, ξ ) in (26)remains bounded with probability 1 at any time t , the originalclosed-loop system (4) is mean-square stable.The proof of the above corollary is given in Appendix F.However, it is a challenging task to determine the uncer-tainty bound ρ in (27) for N ( t, ξ ) . On one hand, N ( t, ξ ) isrelated to the PCE truncation error of the closed-loop state x ( t, ξ ) , hence it is bounded if the closed-loop state has aconvergent PCE for all t > . Since this requirement forthe closed-loop state can be check only after constructingthe closed-loop system (4), ρ in (27) cannot be verifiedbefore solving the synthesis problem (34). On the other hand,it is still challenging to determine the uncertainty bound ρ after having a given control gain. Generally, computing ρ involves deriving the PCE of x ( t, ξ ) whose dependenceon ξ is usually non-polynomial. This requires numericalintegration at each time instant (see Section IV in [12]), thusis infeasible to be performed for all t > .As a remedy to the above difficulty of directly deriving theuncertainty bound ρ in (27), we propose to perform a post-analysis of robust stability and re-tune ρ as a robustifyingparameter. For stability analysis of polynomially uncertainsystems considered in this paper, various methods such aspolynomially parameter-dependent Lyapunov function [4]and sum-of-squares [3] are available. By combining suchstability analysis and an iterative bisection search strategy, alower bound ρ can be determined such that any ρ ≥ ρ results in a stabilizing gain for the closed-loop system (4).The basic idea is as follows. We start with a value of ρ that ensures the feasibility of the synthesis problem (34).Then, in each iteration, the following successive steps areperformed: i) solving the synthesis problem (34); ii) doingthe post-analysis of robust stability with the obtained controlgain; and iii) tuning ρ with a bisection search strategy,i.e., decreasing ρ if the resulting closed-loop system isstabilized, or increasing ρ if it is unstable. Re-tuning ρ ≥ ρ involves tradeoffs between stability and performance.The use of a more conservative bound ρ enhances stability,but sacrifices control performance. Remark 1:
In [10], [22], an alternative approach is pro-posed to ensure the stability of the original closed-loopsystem (4). It includes the conventional worst-case stabilitycondition as a complementary constraint for the nominalPCE-based synthesis problem (25).
Remark 2:
Compared to our previous paper [23], theproposed robust PCE-based control synthesis here has thefollowing advantages. Firstly, two terms of norm-boundeduncertainties are used in [23], while only one term of norm-bounded uncertainty ∆ x ( t ) is introduced in (28). Hence onlyone robustifying parameter ρ in (27) needs to be determined,which is much simpler than determining two robustifyingparameters in [23]. Secondly, the stability issue of theoriginal closed-loop system (4) was just briefly mentioned in [23], whilst detailed discussions are given here includingCorollary 1.1 and post-analysis with the above bisectionsearch strategy to determine ρ .VI. S IMULATION EXAMPLE
Consider the system (1): A ( ξ ) = (cid:20) . ξ − . . . (cid:21) , B w = (cid:20) (cid:21) , B ( ξ ) = (cid:20) . ξ . (cid:21) , C z = (cid:20) (cid:21) (cid:62) , C ( ξ ) = (cid:20) ξ (cid:21) , D w ( ξ ) = (cid:20) ξ
00 0 0 1 (cid:21) , D z = (cid:2) . (cid:3) (cid:62) , D zw = , with the uncertain scalar ξ uniformly distributed over theinterval [ − , . Three H ∞ SOF control synthesis methodsare implemented for comparison: (i) worst-case robust con-trol synthesis; (ii) the nominal PCE-based control synthesisin (25); and (iii) the proposed robust PCE-based controlsynthesis in Section V-B.The worst-case robust controller accounts for the polytopicuncertainty ξ ∈ [ − , by solving [7] min P , K ,γ γ s.t. He { PA cl ( ξ i ) } PB cl ( ξ i ) C (cid:62) cl ( ξ i ) B (cid:62) cl ( ξ i ) P − γ I D (cid:62) cl ( ξ i ) C cl ( ξ ) D cl ( ξ i ) − γ I < ,i = 1 , , with A cl , B cl , C cl , and D cl defined in (5). The parameters { ξ i } are set to be ξ = − and ξ = 1 such that { A cl ( ξ i ) , B cl ( ξ i ) , C cl ( ξ i ) , D cl ( ξ i ) } defines a polytope as anoverbounding uncertainty set, i.e., (cid:20) A cl ( ξ ) B cl ( ξ ) C cl ( ξ ) D cl ( ξ ) (cid:21) = (cid:88) i =1 β i (cid:20) A cl ( ξ i ) B cl ( ξ i ) C cl ( ξ i ) D cl ( ξ i ) (cid:21) with (cid:80) i =1 β i = 1 and β i ≥ . The TOMLAB/PENBMIsolver [9] is used to solve the BMI optimization whichproduces the controller K wc = (cid:2) − . − . (cid:3) .To explicitly address the probabilistic uncertain parameter ξ , the nominal PCE-based control synthesis (25) is appliedwith different PCE degrees from 1 to 10. Post-analysisshows that the obtained control gain does not robustlystabilize the closed-loop system if the adopted PCE degreeis less than 2. Fig. 1 depicts the distributions of H ∞ normsobtained from the worst-case robust control and the PCE-based control for 1000 values for ξ uniformly distributedwithin [ − , . The nominal PCE-based control synthesisusing PCEs with degrees 2, 3, and 10 gives K PCE = (cid:2) . − . (cid:3) , K PCE = (cid:2) . − . (cid:3) , and K PCE = (cid:2) . − . (cid:3) , respectively. The nominalPCE-based controls achieve much better averaged perfor-mance than worst-case robust control, at the cost of largerworst-case H ∞ norms when ξ approaches − . As theadopted PCE degree increases to 10, the PCE truncation error Fig. 1. Distributions of H ∞ norms over ξ ∈ [ − , generated by worst-case robust control and nominal PCE-based controls. becomes smaller, and the worst-case H ∞ norm decreasesaccordingly.Without increasing the PCE degree, the robust PCE-basedcontrol synthesis (34) introduces a robustifying parameter ρ to address the PCE truncation errors. To illustrate thispoint, we consider low PCE degrees 1, 2, and 3. Firstly,the bisection algorithm described at the end of Section V-B is applied to determine the lower bound ρ for ρ thatensures closed-loop stability. The obtained lower bound ρ is 0.0027, 0, and 0 for the synthesis problem (34) withPCE degrees 1, 2, and 3, respectively. In the following,the robust PCE-based control synthesis (34) with the nd -degree PCE is solved with different values of ρ . In thiscase, the synthesis problem (34) has 25 decision variables,which is significantly fewer than 256 decision variables in(25) using a th -degree PCE. The resulting distributions of H ∞ norms are compared to the worst-case robust control andthe nominal PCE-based control in Fig. 1. Compared to thenominal PCE-based control with nd -degree PCEs, the robustPCE-based control with nd -degree PCEs and ρ = 0 . achieves both a much smaller worst-case H ∞ norm and asmaller averaged H ∞ norm. By just increasing ρ to 0.0225,the worst-case H ∞ norm is further reduced to be about 26%smaller than for worst-case robust synthesis. However, as ρ increases from 0.0036 to 0.0225, a more conservative boundis used to quantify the effect of the PCE truncation errors,which leads to an increase in the averaged H ∞ norm from14.6731 to 16.4820. For a detailed comparison, Table I liststhe worst-case and averaged H ∞ norms obtained by differentmethods and tuning parameters.The results of nominal PCE-based synthesis using thePCE-transformed system (45) as in [23] are also includedin Table I, in order to illustrate the benefit of the proposedclosed-loop PCE transformation in Section IV. To comparethe approximation errors, Fig. 3 depicts the mean andvariance of state trajectories produced by the true systemand the two different PCE-transformed systems (23) and(45), with zero disturbance w , nd -degree PCEs, and a SOF -1 -0.95 -0.9 -0.85 -0.80102030405060 Worst-case robust SOFNominal PCE-based SOFPCE degree = 10 -0.8 -0.5 0 0.5 10102030405060 Robust PCE-based SOF PCE degree=2, =0.0 Robust PCE-based SOF
PCE degree=2, =0. Fig. 2. Distributions of H ∞ norms over ξ ∈ [ − , generated by worst-case robust control, nominal PCE-based control, and robust PCE-basedcontrols. TABLE IW ORST - CASE AND AVERAGED H ∞ NORMS ACHIEVED BY DIFFERENTSYNTHESIS METHODS ( p REPRESENTS THE
PCE
DEGREE , AND ρ IS THEROBUSTIFYING PARAMETER IN THE ROBUST
PCE-
BASED SYNTHESIS )Synthesis method Worst-case Averaged H ∞ norm H ∞ normWorst-case robust SOF . . Nominal PCE-based SOF p = 2 80 . . p = 3 57 . . p = 10 55 . . Robust PCE-based SOF p = 2 , ρ = 0 . . . p = 2 , ρ = 0 . . . p = 2 , ρ = 0 . . . Nominal PCE-based SOFusing the PCE-transformed system (45) p = 2 168 . . p = 3 67 . . p = 10 55 . . control gain K = (cid:2) . − . (cid:3) . The true statestatistics are obtained from Monte Carlo simulations, whilethe state means and variances from the PCE-transformedsystems are computed using (8)–(9). The proposed PCE-transformed closed-loop dynamics (23) have higher accuracythan the PCE-transformed system (45), although both havegood approximation to the mean of the true system states.This observation is consistent with the reasons explained inAppendix C, The proposed closed-loop PCE transformationresults in significantly improved control performance whenthe PCE degree is 2 or 3, as shown in Table I. As the PCEdegree increases to 10, the difference between the nominalPCE-transformed systems proposed in this paper and thatreported in [23] become negligible, which result in almostidentical worst-case and averaged H ∞ norms.
10 20Time t M ean o f x ( t, ) t M ean o f x ( t, ) True statePCE- transformed system ( ) PCE- transformed system ( ) t V a r i an c e o f x ( t, ) t V a r i an c e o f x ( t, ) -3 Fig. 3. State means and variances obtained from PCE-transformed systems(18a) and (45) compared to the true system state, with zero disturbances w , nd -degree PCEs, and a SOF control gain K = [1 . − . . VII. C
ONCLUSION
A robust PCE-based H ∞ SOF control synthesis method ispresented to address probabilistic time-invariant parametricuncertainties. A closed-loop PCE-based transformation isproposed to achieve smaller transformation errors comparedto reported PCE-based transformations in the literature. Theeffect of PCE truncation errors is captured by a norm-bounded LDI to formulate a robust PCE-based synthesismethod. The proposed approach allows the use of relativelylow-degree PCEs by tuning a robustifying parameter. Incontrast, most existing PCE-based control design methodsrely on using high-degree PCEs to alleviate the effect of trun-cation errors, which results in much higher computationalcost. APPENDIX
A. Proof of Proposition 1
According to (5), (6), (20b), (20d), (20e), and (20g), itfollows that ¯ A cl = E ξ { Φ( ξ ) A cl ( ξ )Φ (cid:62) ( ξ ) } = A + E ξ { Φ( ξ ) B ( ξ ) KC ( ξ )Φ (cid:62) ( ξ ) } = A + E ξ N q (cid:88) i =0 ˆB i φ i ( ξ ) K N q (cid:88) j =0 ˆC j φ j ( ξ ) = A + E ξ N q (cid:88) i =0 N q (cid:88) j =0 ˆB i K ˆC j φ i ( ξ ) φ j ( ξ ) = A + N q (cid:88) i =0 ˆB i K ˆC i = A + ¯ B ¯ K ¯ C . The proof of ¯ B cl = B w + ¯ B ¯ K ¯ D w follows similar proceduresas above. B. Proof of Proposition 2
Since E ξ {(cid:107) z nom ( t, ξ ) (cid:107) L } can be expressed as E ξ {(cid:107) z nom ( t, ξ ) (cid:107) L } = (cid:90) ∞ (cid:20) X ( t ) w ( t ) (cid:21) (cid:62) (cid:20) Γ Γ (cid:62) Γ Γ (cid:21) (cid:20) X ( t ) w ( t ) (cid:21) d t, the proof of Proposition 2 is equivalent to the proof of Γ = E ξ { Φ x ( ξ ) C (cid:62) cl ( ξ ) C cl ( ξ )Φ (cid:62) x ( ξ ) } = ¯ C (cid:62) cl ¯ C cl , (35) Γ = E ξ { D (cid:62) cl ( ξ ) C cl ( ξ )Φ (cid:62) x ( ξ ) } = ¯ D (cid:62) cl ¯ C cl , (36) Γ = E ξ { D (cid:62) cl ( ξ ) D cl ( ξ ) } = ¯ D (cid:62) cl ¯ D cl . (37)According to the definition of Φ (cid:62) x ( ξ ) in (15), the followingequations hold: C z Φ (cid:62) x ( ξ ) = Φ (cid:62) z ( ξ ) C Z , (38) K (cid:62) D (cid:62) z C z Φ (cid:62) x ( ξ ) = Φ (cid:62) y ( ξ ) K (cid:62) D (cid:62) Z C Z , (39)with Φ y ( ξ ) and Φ z ( ξ ) defined similarly to Φ x ( ξ ) , K = I N p +1 ⊗ K , D Z = I N p +1 ⊗ D z . (40)From (19b), define C as C = E ξ { Φ y ( ξ ) C ( ξ )Φ (cid:62) x ( ξ ) } = E ξ Φ y ( ξ ) N q (cid:88) i =0 ˆC i φ i ( ξ ) = (cid:104) ˆC (cid:62) ˆC (cid:62) · · · ˆC (cid:62) N p (cid:105) (cid:62) . (41)Since the degree of C ( ξ )Φ (cid:62) x ( ξ ) is higher than for Φ (cid:62) x ( ξ ) ,i.e., q > p and N q > N p , it follows from (41) and (40) that ¯ C in (20e), ¯ K in (20g) and ¯ D Z in (22f) can be partitioned as ¯ C = (cid:20) CC (cid:21) , ¯ K = (cid:20) K K (cid:21) , ¯ D Z = (cid:20) D Z D Z , (cid:21) . (42)With the above equations, (5), (22c)–(22f), (38), and (39), itcan be derived that Γ = E ξ { Φ x ( ξ ) C (cid:62) cl ( ξ ) C cl ( ξ )Φ (cid:62) x ( ξ ) } = Γ + Γ + Γ with Γ = E ξ { Φ x ( ξ ) C (cid:62) z C z Φ (cid:62) x ( ξ ) } = E ξ {C (cid:62) Z Φ z ( ξ )Φ (cid:62) z ( ξ ) C Z } = C (cid:62) Z E ξ { Φ z ( ξ )Φ (cid:62) z ( ξ ) }C Z = C (cid:62) Z C Z = ¯ C (cid:62) Z ¯ C Z , Γ = He (cid:8) E ξ { Φ x ( ξ ) C (cid:62) ( ξ ) K (cid:62) D (cid:62) z C z Φ (cid:62) x ( ξ ) } (cid:9) = He (cid:8) E ξ { Φ x ( ξ ) C (cid:62) ( ξ )Φ (cid:62) y ( ξ ) }K (cid:62) D (cid:62) Z C Z (cid:9) = He {C (cid:62) K (cid:62) D (cid:62) Z C Z } = He { ¯ C (cid:62) ¯ K (cid:62) ¯ D (cid:62) Z ¯ C Z } , Γ = E ξ { Φ x ( ξ ) C (cid:62) ( ξ ) K (cid:62) D (cid:62) z D z KC ( ξ )Φ (cid:62) x ( ξ ) } = E ξ N q (cid:88) i =0 N q (cid:88) j =0 ˆC (cid:62) i K (cid:62) D (cid:62) z D z K ˆC i φ i ( ξ ) φ j ( ξ ) = N q (cid:88) i =0 ˆC (cid:62) i K (cid:62) D (cid:62) z D z K ˆC i = ¯ C (cid:62) ¯ K (cid:62) ¯ D (cid:62) Z ¯ D Z ¯ K ¯ C . The above expressions of Γ , Γ , and Γ prove that Γ =¯ C (cid:62) cl ¯ C cl with ¯ C cl defined in (22b). The proofs of (36) and (37)follow similar procedures and so are omitted. . Comparison with existing PCE-based approximations Firstly, the two PCE-transformed closed-loop systems forcomparison are summarized. The first one is (23), while thesecond one is given in [6], [19], [23], and briefly reviewedbelow without detailed derivations. By applying standardGalerkin projection, the four equations in (1) and (2) areseparately transformed into ˙X b ( t ) = A X b ( t ) + B U b ( t ) + B w w ( t ) , (43a) Z b ( t ) = C Z X b ( t ) + D Zw w ( t ) + D Z U b ( t ) , (43b) Y b ( t ) = C X b ( t ) + D w w ( t ) , (43c) U b ( t ) = K Y b ( t ) , (43d)where the matrices B and D w are defined as B = E ξ { Φ x ( ξ ) B ( ξ )Φ (cid:62) u ( ξ ) } and D w = E ξ { Φ y ( ξ ) D w ( ξ ) } , (44)with Φ u ( ξ ) and Φ y ( ξ ) defined similarly to Φ x ( ξ ) in (15), C Z , D Zw , D Z , K , and C are defined in (22e), (40), (41),respectively. Similarly to (23), the error terms introduced inGalerkin projection are not included in (43), hence U b , Y b ,and Z b are approximates of the PCE coefficient vectors ofthe control input, measured system output, and controlledsystem output, respectively. Combining (43a)–(43d) leads tothe PCE-transformed closed-loop system given in [19], [23]: ˙ X b ( t ) = A cl X b ( t ) + B cl w ( t ) , Z b ( t ) = C cl X b ( t ) + D cl w ( t ) , (45)with A cl = A + BKC , B cl = B w + BKD w , C cl = C Z + D Z KC , D cl = D Zw + D Z KD w . (46)Next, a relationship between the system matrices in (23)and (46) is established. Similarly to the partition of ¯ C in (42), ¯ B in (20d) and ¯ D w in (20f) are partitioned as ¯ B = (cid:2) B B (cid:3) , ¯ D w = (cid:20) D w D w , (cid:21) (47)according to the definition of B and D w in (44). By exploit-ing (42) and (47), ¯ A cl , ¯ B cl in (20a) and ¯ C cl , ¯ D cl in (22b) canbe rewritten as ¯ A cl = A cl + B K C , ¯ B cl = B cl + B K D w , , ¯ C cl = (cid:20) C cl D Z , K C (cid:21) , ¯ D cl = (cid:20) D cl D Z , K D w , (cid:21) . (48)From (20d)–(20f) and the matrix partitions in (42) and (47),it can be seen that B , C , and D w , are determined by thehigh-degree PCE coefficients { ˆB i , ˆC i , ˆD w ,i } N q i = N p +1 in (19),but they are missing in the system matrices in (45). Note thatif the system matrix B in (1) is independent of ξ , B definedin (47) and (42) becomes a zero matrix, which results in ¯ A cl = A cl and ¯ B cl = B cl . Similar consequences occur if C or D w is independent of ξ .Since X a ( t ) and X b ( t ) in the PCE-transformed systems(23) and (45) are approximates of PCE coefficients X ( t ) ofthe truce system state x ( t, ξ ) , the state reconstruction error ˜x (cid:63) ( t, ξ ) = x ( t, ξ ) − Φ (cid:62) x ( ξ ) X (cid:63) ( t ) (49) is used to compare the approximation errors of these PCE-transformed systems, with (cid:63) being a or b . Such a stateapproximation error can be decomposed as ˜x (cid:63) ( t, ξ ) = ˜x ( t, ξ ) + Φ (cid:62) x ( ξ ) ( X ( t ) − X (cid:63) ( t )) . (50)with ˜x ( t, ξ ) being the PCE truncation error defined in(15). The first term ˜x ( t, ξ ) in (50) is the orthogonalprojection error determined by the true closed-loop sys-tem state and its PCE, thus is independent of the PCE-transformed system (23) or (45). In contrast, the second term Φ (cid:62) x ( ξ ) ( X ( t ) − X (cid:63) ( t )) in (50) depends on the approximationadopted in deriving the PCE-transformed systems. Accordingto (50), the bound of ˜x (cid:63) ( t, ξ ) can be described as E ξ {(cid:107) ˜x (cid:63) ( t, ξ ) (cid:107) } ≤ E ξ {(cid:107) ˜x ( t, ξ ) (cid:107) } + E ξ { (cid:13)(cid:13) Φ (cid:62) x ( ξ ) ( X ( t ) − X (cid:63) ( t )) (cid:13)(cid:13) } = E ξ {(cid:107) ˜x ( t, ξ ) (cid:107) } + (cid:107) X ( t ) − X (cid:63) ( t ) (cid:107) . (51)To further derive an upper bound for (51), the dynamics of ˜X (cid:63) ( t ) = X ( t ) − X (cid:63) ( t ) is derived as ˙˜X a ( t ) = ¯ A cl ˜X a ( t ) + ¯R x ( t ) , ˙˜X b ( t ) = ¯ A cl ˜X b ( t ) + B K C X b ( t ) + B K D w , w ( t )+ ¯R x ( t ) according to (18a), (23), (45), and (46). For the sake ofbrevity, assume X (0) = X a (0) = X b (0) , then the upperbounds for the approximation errors E ξ {(cid:107) ˜x (cid:63) ( t, ξ ) (cid:107) } in (51)are expressed as E ξ {(cid:107) ˜x a ( t, ξ ) (cid:107) } ≤ π ( t ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) t e ¯ A cl ( t − τ ) ¯R x ( τ ) dτ (cid:13)(cid:13)(cid:13)(cid:13) , (52a) E ξ {(cid:107) ˜x b ( t, ξ ) (cid:107) } ≤ π ( t ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) t e ¯ A cl ( t − τ ) ¯R x ( τ ) dτ (cid:13)(cid:13)(cid:13)(cid:13) (52b) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) t e ¯ A cl ( t − τ ) B K ( C X b ( τ ) + D w , w ( τ )) dτ (cid:13)(cid:13)(cid:13)(cid:13) for the PCE-transformed systems (23) and (45), respectively.Note that π ( t ) in (52) represents the upper bound for the PCEtruncation error E ξ {(cid:107) ˜x ( t, ξ ) (cid:107) } . The detailed expression of π ( t ) has been given in literature such as [1], [14], hence isnot detailed here.From (51) and (52), it can be concluded that E ξ {(cid:107) ˜x a ( t, ξ ) (cid:107) } from our proposed PCE-transformed sys-tem (23) has a smaller worst-case upper bound than E ξ {(cid:107) ˜x b ( t, ξ ) (cid:107) } from the PCE-transformed system in [6],[19], [23]. D. Proof of Proposition 3
With Kronecker product, the first equation in (26) can beequivalently expressed as ˜x ( t, ξ ) = (cid:0)(cid:0) X (cid:62) ( t )Φ x ( ξ ) (cid:1) ⊗ I n x (cid:1) vec ( M ( t, ξ )) . (53)For any given value of t and ξ , the above equation (53) musthave a non-unique solution for vec ( M ( t, ξ )) since the matrix X (cid:62) ( t )Φ x ( ξ ) (cid:1) ⊗ I n x has full row-rank. The second equalityof (26) exploits M ( t, ξ )Φ (cid:62) x ( ξ ) = Φ (cid:62) x ( ξ ) N ( t, ξ ) accordingto the definition of Φ (cid:62) x ( ξ ) in (15). E. Proof of Theorem 1
By applying the Schur complement lemma, (33) is equiv-alent to He { P ¯ A cl } P ¯ B cl P ¯ A cl ¯ B (cid:62) cl P − γ rob I 0 ¯ A (cid:62) cl P 0 0 + diag ( τ ρ I , , − τ I )+ γ − rob (cid:2) ¯ C cl ¯ D cl ¯ C cl (cid:3) (cid:62) (cid:2) ¯ C cl ¯ D cl ¯ C cl (cid:3) < . (54)Define ϕ x ( t ) = ∆ x ( t ) X ( t ) and χ ( t ) = (cid:2) X (cid:62) ( t ) w (cid:62) ( t ) ϕ (cid:62) x ( t ) (cid:3) (cid:62) . Let V ( t ) = X (cid:62) ( t ) PX ( t ) represent the Lyapunov function. Left-multiplying (54) by χ (cid:62) ( t ) and right-multiplying (54) by χ ( t ) results ind V d t + γ − rob Z (cid:62) rob ( t ) Z rob ( t ) − γ rob w (cid:62) ( t ) w ( t )+ τ ( ρ X (cid:62) ( t ) X ( t ) − ϕ (cid:62) x ( t ) ϕ x ( t )) < . Note that τ > , and ρ X (cid:62) ( t ) X ( t ) − ϕ (cid:62) x ( t ) ϕ x ( t ) ≥ for ∆ x ( t ) ∈ F x . According to the S -procedure, the aboveinequality is equivalent tod V d t + γ − rob Z (cid:62) rob ( t ) Z rob ( t ) − γ rob w (cid:62) ( t ) w ( t ) < . (55)With ϕ x ( t ) = ∆ x ( t ) X ( t ) , (55) can be rewritten as χ (cid:62) ( t ) P χ ( t ) < with P = He { P ¯ A cl ( I + ∆ x ( t )) } P ¯ B cl ( I + ∆ x ( t )) (cid:62) ¯ C (cid:62) cl ¯ B (cid:62) cl P − γ rob I ¯ D (cid:62) cl ¯ C cl ( I + ∆ x ( t )) ¯ D cl − γ rob I . Since χ (cid:62) ( t ) P χ ( t ) < holds for arbitrary χ ( t ) and ∆ x ( t ) ∈F x , the quadratic stability and the H ∞ norm of the system(30) is proved according to the bounded real Lemma [5]. F. Proof of Corollary 1.1
The additive noise w ( t ) is set to zero for the internalstability analysis. The SOF controller obtained from solving(34) robustly stabilizes the PCE-transformed LDI (30), hence X ( t ) converges to zero with time. Since the trajectory of X ( t ) in the system (16) belongs the trajectory set of X ( t ) in the LDI (27)–(28), then we have X ( ∞ ) = 0 implied by X ( ∞ ) = 0 . As N ( t, ξ ) in (26) is bounded with probability1 for all t , the truncation error ˜x ( t, ξ ) approaches zero atalmost every ξ as t goes to infinity. Therefore, we have lim t →∞ E ξ {(cid:107) x ( t, ξ ) (cid:107) } = lim t →∞ E ξ { (cid:13)(cid:13) ˜x ( t, ξ ) + Φ (cid:62) x ( ξ ) X ( t ) (cid:13)(cid:13) }≤ lim t →∞ E ξ {(cid:107) ˜x ( t, ξ ) (cid:107) } + lim t →∞ (cid:107) X ( t ) (cid:107) = 0 (56)which proves the mean-square stability of the original closed-loop system (4). R EFERENCES[1] C. Audouze and P. B. Nair. A priori error analysis of stochasticGalerkin projection schemes for randomly parameterized ordinarydifferential equations.
International Journal for Uncertainty Quan-tification , 6(4):287–312, 2016.[2] S. Boyarski and U. Shaked. Robust H ∞ control design for best meanperformance over an uncertain-parameters box. Systems & ControlLetters , 54(6):585–595, 2005.[3] G. Chesi. Exact robust stability analysis of uncertain systems with ascalar parameter via LMIs.
Automatica , 49:1083–1086, 2013.[4] G. Chesi, A. Garulli, A. Tesi, and A. Vicino.
Homogeneous PolynomialForms for Robustness Analysis of Uncertain Systems . Springer-Verlag,2009.[5] G. E. Dullerud and F. Paganini.
A Course in Robust Control Theory:a Convex Approach . Springer Science & Business Media, 2013.[6] J. Fisher and R. Bhattacharya. Linear quadratic regulation of systemswith stochastic parameter uncertainties.
Automatica , 45:2831–2841,2009.[7] J. C. Geromel, R. H. Korogui, and J. Bernussou. H and H ∞ robustoutput feedback control for continuous time polytopic systems. IETControl Theory & Applications , 1(5):1541–1549, 2007.[8] D. Hinrichsen and A. J. Pritchard. Stochastic H ∞ . SIAM Journal onControl and Optimization , 36(5):1504–1538, 1998.[9] K. Holmstr¨om, A. O. G¨oran, and M. M. Edvall.
User’s Guide forTOMLAB/PENOPT . Tomlab Optimization Inc.[10] S. Hsu and R. Bhattacharya. Design of linear parameter varyingquadratic regulator in polynomial chaos framework.
InternationalJournal of Robust and Nonlinear Control , 30(16):6661–6682, 2020.[11] J. Lavaei and A. G. Aghdam. Robust stability of LTI systems oversemialgebraic sets using sum-of-squares matrix polynomials.
IEEETransactions on Automatic Control , 53(1):417–423, 2008.[12] S. Lucia, J. A. Paulson, R. Findeison, and R. D. Braatz. On stabilityof stochastic linear systems via polynomial chaos expansions. In
Proceedings of the 2017 American Control Conference , pages 5089–5094, Seattle, WA, 2017.[13] O. P. Le Maˆıtre and O. M. Knio.
Spectral Methods for UncertaintyQuantification With Applications to Computational Fluid Dynamics .Springer, 2010.[14] T. M¨uhlpfordt, R. Findeisen, V. Hagenmeyer, and T. Faulwasser.Comments on truncation errors for polynomial chaos expansions.
IEEE Control Systems Letters , 2(1):169–174, 2018.[15] S. Nandi, V. Migeon, T. Singh, and P. Singla. Polynomial chaos basedcontroller design for uncertain linear systems with state and controlconstraints.
Journal of Dynamic Systems, Measurement, and Control ,140(7):071009, 2017.[16] J. A. Paulson, E. Harinath, L. C. Foguth, and R. D. Braatz. Nonlinearmodel predictive control of systems with probabilistic time-invariantuncertainties. In
Proceedings of 5th IFAC Conference on NonlinearModel Predictive Control , pages 16–25, Seville, 2015.[17] J. A. Paulson and A. Mesbah. An efficient method for stochas-tic optimal control with joint chance constraints for nonlinear sys-tems.
International Journal of Robust and Nonlinear Control , 2018.DOI:10.1002/rnc.3999.[18] I. R. Petersen and R. Tempo. Robust control of uncertain systems:classical results and recent developments.
Automatica , 50(5):1315–1335, 2014.[19] D. Shen, S. Lucia, Y. Wan, R. Findeisen, and R. D. Braatz. Polynomialchaos-based H -optimal static output feedback control of systemswith probabilistic parameter uncertainties. In Proceedings of 20thIFAC World Congress , pages 3595–3600, Toulouse, France, 2017.[20] E. S¨uli and D. F. Mayers.
An Introduction to Numerical Analysis .Cambridge University Press, 2003.[21] R. Tempo, G. Calafiore, and F. Dabbene.
Randomized Algorithmsfor Analysis and Control of Uncertain Systems with Applications .Springer-Verlag, London, 2013.[22] Y. Wan and R. D. Braatz. Mixed polynomial chaos and worst-casesynthesis approach to robust observer based linear quadratic regulation.In
Proceedings of the 2018 American Control Conference , pages 6798–6803, Milwaukee, USA, 2018.[23] Y. Wan, D. E. Shen, S. Lucia, R. Findeisen, and R. D. Braatz.Robust static H ∞ output-feedback control using polynomial chaos. In Proceedings of the 2018 American Control Conference , pages 6804–6809, Milwaukee, USA, 2018.24] D. Xiu and G. E. Karniadakis. The Wiener-Askey polynomial chaosfor stochastic differential equations.