Robustness Analysis of Linear Parameter Varying Systems Using Integral Quadratic Constraints Harald Pfifer and Peter Seiler 1 Abstract A general approach is presented to analyze the orst case input/output gain for an interconnection of a linear parameter arying (LPV) system and an uncertain or nonlinear element. The input/output behaior of the nonlinear/uncertain block is described by an integral quadratic constraint (IQC). A dissipation inequality is proposed to compute an upper bound for this gain. This orst-case gain condition can be formulated as a semidefinite program and the result can be interpreted as a Bounded Real Lemma for uncertain LPV systems. The paper shos that this ne condition is a generalization of the ell knon Bounded Real Lemma for LPV systems. The effectieness of the proposed method is demonstrated on a simple numerical example. I. INTRODUCTION This paper presents a method to analyze the robustness of a linear parameter arying (LPV) system ith respect to nonlinearities and/or uncertainties. LPV systems are a class of linear systems here the state matrices depend on (measurable) time-arying parameters. The existing analysis and synthesis results for LPV systems proide a rigorous frameork for design of gain-scheduled controllers. These results can roughly be categorized based on ho the state matrices depend on the scheduling parameters. One approach is to assume the state matrices of the LPV system hae a rational dependence on the parameters. In this case finite dimensional semidefinite programs (SDPs) can be formulated to synthesize LPV controllers [1], [2], [3]. An alternatie approach is to assume the state matrices hae an arbitrary dependence on the parameters. The controller synthesis problem leads to an infinite collection of parameter-dependent linear matrix inequalities (LMIs) [4], [5]. A brief reie of this technical result is proided in Section II. The computational solution of such parameter-dependent LMIs requires some finite-dimensional approximation and is typically more inoled. The benefit is that arbitrary parameter dependence can be considered hich appears in many applications, e.g. aeroelastic ehicles [6] and ind turbines [7], [8], by linearization of nonlinear models. Integral quadratic constraints (IQCs) are used in this paper to model the uncertain and/or nonlinear components. IQCs, introduced in [9], proide a general frameork for robustness analysis. In [9] the system is separated into a feedback connection of a knon linear time-inariant (LTI) system and a perturbation hose input-output behaior is described by an IQC. An IQC stability theorem as formulated in [9] 1 H. Pfifer and P. Seiler are ith the Aerospace Engineering and Mechanics Department, Uniersity of Minnesota, emails: hpfifer@umn.edu, seiler@aem.umn.edu ith frequency domain conditions and as proed using a homotopy method. The contribution of this paper is to proide robust performance conditions for a feedback interconnection of an LPV system and a perturbation hose input-output behaior is described by an IQC. The frequency-domain stability condition in [9] does not apply for this case because the LPV system is time-arying. Instead, this paper uses a timedomain interpretation for IQCs, described in Section II, that builds upon the ork in [1], [11]. The time-domain iepoint is used in Section III to derie dissipation-inequality conditions that bound the orst-case induced gain of the system. The conditions are deried for LPV systems hose state matrices hae arbitrary dependence on the scheduling parameters. The robust performance analysis conditions can thus be ieed as generalizations of those gien for nominal (not-uncertain) LPV systems in [4], [5]. The results in this paper also complement the recent robust performance results obtained for LPV systems hose state matrices hae rational dependence on the scheduling parameters [12], [13], [14], [15]. In contrast to [12], [13], [14], [15], an arbitrary dependence on the parameters is assumed in this ork and a finite-dimensional approximation based approach is taken. As noted aboe, this ill enable applications to systems, e.g. aeroelastic ehicles or ind turbines, for hich arbitrary dependence on scheduling ariables is a natural modeling frameork. A. Notation II. BACKGROUND Standard notation is used except for a fe cases hich are specifically mentioned in this section. S n denotes the set of n n symmetric matrices. R + describes the set of nonnegatie real numbers. B. Analysis of LPV Systems Linear parameter arying (LPV) systems are a class of systems hose state space matrices depend on a timearying parameter ector ρ : R + R nρ. The parameter is assumed to be a continuously differentiable function of time and admissible trajectories are restricted, based on physical considerations, to a knon compact subset P R nρ. In addition, the parameter rates of ariation ρ : R + Ṗ are assumed to lie ithin a hyperrectangle Ṗ defined by Ṗ := {q R nρ ν i q i ν i, i = 1,..., n ρ }. (1)
The set of admissible trajectories is defined as A := {ρ : R + R nρ : ρ(t) P, ρ(t) Ṗ t }. The parameter trajectory is said to be rate unbounded if Ṗ = R nρ. The state-space matrices of an LPV system are continuous functions of the parameter: A : P R nx nx, B : P R nx n d, C : P R ne nx and D : P R ne n d. An n th x order LPV system, G ρ, is defined by ẋ(t) = A(ρ(t))x(t) + B(ρ(t))d(t) e(t) = C(ρ(t))x(t) + D(ρ(t))d(t) The state matrices at time t depend on the parameter ector at time t. Hence, LPV systems represent a special class of time-arying systems. Throughout the remainder of the paper the explicit dependence on t is occasionally suppressed to shorten the notation. The performance of an LPV system G ρ can be specified in terms of its induced L 2 gain from input d to output e. The induced L 2 -norm is defined by G ρ = (2) e sup d,d L 2,ρ A d, (3) here. represents the signal L 2 -norm, i.e. e = ( e(t) T e(t) dt) 1 2. The initial condition is assumed to be x() =. The notation ρ A refers to the entire (admissible) trajectory as a function of time. The analysis belo leads to conditions that inole the parameter and rate at a single point in time, i.e. (ρ(t), ρ(t)). The parametric description (p, q) P Ṗ is introduced to emphasize that such conditions only depend on the (finite-dimensional) sets P and Ṗ. In [5] a generalization of the LTI Bounded Real Lemma is stated, hich proides a sufficient condition to bound the induced L 2 gain of an LPV system. The sufficient condition uses a quadratic storage function that is defined using a parameter-dependent matrix P : P S nx. It is assumed that P is a continuously differentiable function of the parameter ρ. In order to shorten the notation, a differential operator P : P Ṗ is introduced as in [16]. P is defined Snx as: n ρ P (p) P (p, q) = q i (4) p i i=1 The next theorem states the condition proided in [5] to bound the L 2 gain of an LPV system. Theorem 1. ([5]): An LPV system G ρ is exponentially stable and G ρ γ if there exists a continuously differentiable P : P S nx, such that (p, q) P Ṗ P (p) >, (5) [ ] P (p)a(p) + A(p) T P (p) + P (p, q) P (p)b(p) B T (p)p (p) I [ ] + 1 C(p) T [ ] γ 2 D(p) T C(p) D(p) <. (6) The conditions (5) and (6) are parameter-dependent LMIs that must be satisfied for all possible (p, q) P Ṗ. Thus (5) and (6) represent an infinite collection of LMI constraints. Since q enters only affinely into the LMI and the set Ṗ is a polytope, it is sufficient to check the LMI on the ertices of Ṗ. On the other hand, p can enter (6) nonlinearly and the set P does not hae to be conex. A remedy to this problem, hich orks in many practical examples, is to approximate the set P by a finite set P grid P that represents a gridding oer P. In order to aoid the functional dependence of the decision ariable, P (p) has to be restricted to a finite dimensional subspace. A common practice [5], [17] is to restrict the storage function ariable P (p) to be a linear combination of basis functions, N b P (p) = g i (p)p i (7) i=1 here g i : R nρ R are basis functions (i = 1,..., N b ) and the matrix coefficients P i S nx are the decision ariables in the optimization. C. Integral Quadratic Constraints An IQC is defined by a symmetric matrix M = M T R nz nz and a stable linear system Ψ RH nz (m1+m2). Ψ is denoted as Ψ(jω) := C ψ (jωi A ψ ) 1 [B ψ1 B ψ2 ] + [D ψ1 D ψ2 ] (8) A bounded, causal operator : L m1 2e L m2 2e satisfies an IQC defined by (Ψ, M) if the folloing inequality holds for all L m1 2 [, ), = () and T : T z(t) T Mz(t) dt (9) here z is the output of the linear system Ψ: ẋ ψ (t) = A ψ x ψ (t) + B ψ1 (t) + B ψ2 (t), x ψ () = (1) z(t) = C ψ x ψ (t) + D ψ1 (t) + D ψ2 (t) (11) The notation IQC(Ψ, M) is used if satisfies the IQC defined by (Ψ, M). Fig. 1 proides a graphic interpretation of the IQC. The input and output signals of are filtered through Ψ. If IQC(Ψ, M) then the output signal z satisfies the (time-domain) constraint in (9) for any finitehorizon T. To simple examples are proided belo to connect this terminology to standard results used in robust control. z Fig. 1. Ψ Graphic interpretation of the IQC Example 1. Consider a causal (SISO) operator that satisfies the bound b. The norm bound on implies
that b for any input/output pair L 2 and = (). This constraint on (, ) can be expressed as the folloing infinite-horizon inequality: [ ] T [ ] [ ] (t) b 2 (t) dt (12) (t) 1 (t) Next, the causality of is used to demonstrate that, in fact, the inequality inoling (, ) holds oer all finite horizons. Gie any T, define a ne input ṽ by ṽ(t) = (t) for t T and ṽ(t) = otherise. This truncated signal ṽ generates an output = (ṽ). This ne input/output pair (ṽ, ) also satisfies b ṽ. This implies [ ] T [ ] [ ] ṽ(t) b 2 ṽ(t) dt (a) T (b) T (t) [ ṽ(t) (t) [ (t) (t) 1 (t) ] [ ] ṽ(t) dt 1 (t) ] [ ] (t) dt 1 (t) ] T [ b 2 ] T [ b 2 Inequality (a) follos because ṽ(t) = for t T. Inequality (b) follos from the causality of. Specifically, ṽ(t) = (t) for t T and hence (t) = (t) for t T. The final conclusion is that satisfies the IQC defined by (Ψ, M) ith Ψ = I 2 and M = [ ] b 2 1. In this example Ψ contains no dynamics and hence z = [; ]. Example 2. Next consider an LTI (SISO) system that satisfies the bound b. Since is LTI it commutes ith any stable, minimum phase system D(s), i.e. D = D. Thus the frequency-scaled system := DD 1 is also norm-bounded by b. Let (, ) be any input-output pair D 1 Fig. 2. D Scaling of the LTI system for the scaled system, see Fig. 2. The first example implies that [ ] T [ ] [ ] T (t) b 2 (t) dt (13) (t) 1 (t) The associated input/output pair for the original system = () is related to the input/output pair for the scaled system by = D = D. Thus the inequality in (13) can be equialently ritten as T z(t) T Mz(t) (14) here M = [ ] b 2 1 and z is the output of Ψ := [ D D ] generated by the signals [ ], see Fig. 1. Hence satisfies the IQC defined by (Ψ, M). Note that the use of D(s) directly corresponds to the multipliers used in classical robustness analysis, e.g. the structured singular alue µ [18], [19], [2], [21]. The to examples aboe are simple instances of IQCs. The proposed approach applies to the more general IQC frameork introduced in [9] but ith some technical restrictions. In particular, [9] proides a library of IQC multipliers that are satisfied by many important system components, e.g. saturation, time delay, and norm bounded uncertainty. The IQCs in [9] are expressed in the frequency domain as an integral constraint defined using a multiplier Π. The multiplier Π can be factorized as Π = Ψ MΨ and this connects the frequency domain formulation to the timedomain formulation used in this paper. One technical point is that, in general, the time domain IQC constraint only holds oer infinite horizons (T = ). The ork in [9], [1] dras a distinction beteen hard/complete IQCs for hich the integral constraint is alid oer all finite time interals and soft/conditional IQCs for hich the integral constraint need not hold oer finite time interals. The formulation of an IQC in this paper as a finite-horizon (time-domain) inequality is thus alid for any frequency-domain IQC that admits a hard/complete factorization (Ψ, M). While this is somehat restrictie, it has recently been shon in [1] and [11] that a ide class of IQCs hae a hard factorization. The remainder of the paper ill simply treat, ithout further comment, (Ψ, M) as the starting point for the definition of the finite-horizon IQC. Integral quadratic constraints ere introduced in [9] to proide a general frameork for robustness analysis. In this frameork the system is separated into a feedback connection of a knon linear time-inariant (LTI) system and a perturbation hose input-output behaior is described by an IQC. The IQC stability theorem in [9] as formulated ith frequency domain conditions and as proed using a homotopy method. A contribution of this paper is to use the time-domain ie of IQCs in order to derie stability conditions for the interconnection of a knon linear parameterarying (LPV) system and a perturbation hose input-output behaior is described by an IQC. The LPV system is time arying and hence a frequency-domain stability condition is not suitable for such interconnections. A dissipation inequality condition is deried to determine stability of the LPV system in the presences of perturbations. III. LPV ROBUSTNESS ANALYSIS An uncertain LPV system is described by the interconnection of an LPV system G ρ and an uncertainty, as depicted in Fig. 3. This interconnection represents an upper linear fractional transformation (LFT), hich is denoted F u (G ρ, ). The uncertainty is assumed to satisfy an IQC described by (Ψ, M). Note that the perturbation can include hard nonlinearities (e.g. saturations) and infinite dimensional operators (e.g. time delays) in addition to true system uncertainties. The term uncertainty is used for simplicity hen referring to the the perturbation.
d Fig. 3. G ρ e Uncertain LPV System The robust performance of F u (G ρ, ) is measured in terms of the orst case induced L 2 gain from the input d to the output e. The orst-case gain is defined as sup F u (G ρ, ). (15) IQC(Ψ,M) This gain is orst-case oer all uncertainties that satisfy the IQC defined by (Ψ, M) and admissible trajectories ρ. A. Bounded Real Lemma including IQCs As proposed in [11], the basic LFT interconnection in Fig. 3 is considered here satisfies the IQC defined by (Ψ, M). In this basic interconnection the filter Ψ is included as shon in Fig. 4. z Fig. 4. Ψ d G ρ Analysis Interconnection Structure The dynamics of the interconnection in Fig. 4 are described by = () and ẋ = A(ρ)x + B 1 (ρ) + B 2 (ρ)d z = C 1 (ρ)x + D 11 (ρ) + D 12 (ρ)d e = C 2 (ρ)x + D 21 (ρ) + D 22 (ρ)d, e (16) here the state ector is x = [x G ; x ψ ] ith x G and x ψ being the state ectors of the LPV system G ρ and the filter Ψ respectiely. The uncertainty is shon in the dashed box in Fig. 4 to signify that it is remoed for the analysis. The signal is treated as an external signal subject to the constraint T z(t) T Mz(t) dt. (17) This effectiely replaces the precise relation = () ith the quadratic constraint on z. A dissipation inequality can be formulated to upper bound the orst-case L 2 gain of F u (G ρ, ) using the system (16) and the time domain IQC (17). The folloing theorem is based on the dissipation inequality frameork gien in [11]. Theorem 2. Assume F u (G ρ, ) is ell posed for all IQC(Ψ, M). Then the orst-case gain is γ if there exists a continuously differentiable P : P S nx and a scalar λ > such that (p, q) P Ṗ P (p) >, (18) P (p)a + A T P (p) + P (p, q) P (p)b 1 P (p)b 2 B1 T P (p) + B2 T P (p) I C1 T ] +λ M [C 1 D 11 D 12 + 1 γ 2 D T 11 D T 12 C T 2 D T 21 D T 22 ] [C 2 D 21 D 22 < (19) In (19) the dependency of the state space matrices on p has been omitted to shorten the notation. The same considerations in regard to gridding and basis functions for P (p) as in Section II-B hae to be taken into account, in order to turn Theorem 2 into a computational tractable optimization problem. In many practical examples, the IQCs include frequencydependent eightings hich do not necessarily hae to be rational functions. For instance, the IQC factorization for an LTI norm-bounded operator uses a scaling D(s). In the classical µ-frameork, a frequency gridding is applied, so that D(jω) can be soled for at each frequency indiidually. This freedom is lost in the presented approach, as a single D RH has to be specified. To oercome this shortcoming a basis function approach is used, i.e. N D D(s) = α i D i (s). (2) i=1 α i can be treated as free decision ariable by using an independent factorization (Ψ i, M i ) for each D i (s) and alloing different λ i in condition (19) for each factorization. This is analogous to the use of many multipliers in the original IQC analysis [9]. IV. NUMERICAL EXAMPLE A simple example is used to demonstrate the applicability of the proposed method. The example is a feedback interconnection of a first-order LPV system ith a gain-scheduled proportional-integral controller as shon in Fig. 5. d Fig. 5. e C ρ u T del G ρ Closed Loop Interconnection ith Dynamic Uncertainty The system G ρ, taken from [22], is a first order system ith dependence on a single parameter ρ. It can be ritten y
as ẋ G = 1 τ(ρ) x G + 1 τ(ρ) u G y = K(ρ)x G (21) ith the time constant τ(ρ) and output gain K(ρ) depending on the scheduling parameter as follos: τ(ρ) = 133.6 16.8ρ K(ρ) = 4.8ρ 8.6. (22) The scheduling parameter ρ is restricted to the interal [2, 7]. For all the folloing analysis scenarios a grid of six points is used that span the parameter space equidistantly. A timedelay of.5 seconds is included at the control input. The time delay in the system is represented by a second order Pade approximation: T del (s) = (T d s) 2 12 T ds (T d s) 2 12 + T ds 2 2 + 1 (23) + 1, here T d =.5. For G ρ a gain-scheduled PI-controller C ρ is designed that guarantees a closed loop damping ζ cl =.7 and a closed loop frequency ω cl =.25 at each frozen alue of ρ. The controller gains that satisfy these requirements are gien by K p (ρ) = 2ζ clω cl τ(ρ) 1 K(ρ) K i (ρ) = ω2 cl τ(ρ) K(ρ). (24) The controller is realized in the folloing state space form: ẋ c = K i (ρ)e u = x c + K p (ρ)e (25) As an extension to the original example in [22], a multiplicatie norm bounded uncertainty is inserted at the input of the plant. The norm of the uncertainty is assumed to be less than a positie scalar b, i.e. b. The uncertainty is described by the integral quadratic constraints (Ψ, M) ith Ψ = I and M = [ ] b 2 1. The gain of the channel from the reference signal d to the control error e is used as performance measurement and γ is an upper bound on the orst case gain as defined by (15). Established techniques are used to compute to gains for comparison ith the proposed method: (I) the orst-case LTI gain in the presence of uncertainty at each frozen alue of ρ, and (II) the bound on induced L 2 gain of the nominal LPV system ( = ). The orst-case LTI gain (I) at each frozen alue of ρ is computed using the Matlab command cgain [23]. The nominal LPV gain (II) is computed using the standard LMI condition in Theorem 1. The first analysis is performed assuming unbounded parameter ariation rates. Hence, a constant matrix P is assumed. In this case, the nominal gain for the system ithout uncertainty is 18.7. The results of the robust LPV performance analysis for different uncertainty sizes is shon in the top plot of Fig. 6. The conex optimization could not find feasible solutions for b >.1, meaning that for uncertainties larger than ten percent no finite gain can be guaranteed. For the case b =.5 this analysis yields a bound on the orst case gain of 29.1. The folloing approach is used to estimate the conseratieness of the upper bound. First, the frozen LTI analysis (I) is performed for b =.5 to obtain the orst alues of ρ and. The orst alue of returned by cgain is used to construct a (not-uncertain) LPV system F u (G ρ, ). The conditions in Theorem 1 are then used to compute the induced L 2 gain bound for this system. The result of this analysis is a gain of 27.5 hich his close to the computed orst-case gain of 29.1. This approach does not proide a true loer bound on the orst-case gain due to the conseratism in the Bounded Real Lemma for LPV systems. Hoeer, it does proide an indication of the conseratism in the proposed robustness bound. In the next step, the parameter ariation rate is bounded by ρ 2. Affine and quadratic parameter dependences are considered for P, i.e. P (ρ) = P + ρp 1 and P (ρ) = P + ρp 1 + ρ 2 P 2 respectiely. The results are gien in the middle plot of Fig. 6. Alloing a higher order P (ρ) reduces the orst case gains and also allos finding finite gains up to b =.3. In comparison, the affine storage function could only guarantee finite gains for b.25. Finally, to compare the proposed method ith the LTI gains the rate is bounded by ρ.1. Both affine and quadratic dependences of P (ρ) are presented in relation to the results of the Matlab function cgain in the loer plot of Fig. 6. The folloing to facts mostly contribute to the difference beteen the results from cgain and the proposed method: First, the parameter dependence of the storage function is restricted in the latter, hile the first can compute a indiidual P at each grid point. Second, cgain assumes that is an LTI operator, hereas it can be any norm-bounded operator in the latter case. V. CONCLUSIONS In this paper the analysis frameorks for LPV systems and uncertainties described by IQCs hae been combined. This leads to computationally efficient conditions to assess the robust performance of an LPV system interconnected ith uncertainties and nonlinearities. The proposed robust LPV analysis frameork is a generalization of the ell knon nominal LPV Bounded Real Lemma. A simple numerical example as presented to sho the potential of the proposed approach. Future ork ill consider the synthesis of robust controllers for uncertain LPV systems. ACKNOWLEDGMENTS This ork as partially supported by NASA under Grant No. NRA NNX12AM55A entitled Analytical Validation Tools for Safety Critical Systems Under Loss-of-Control Conditions, Dr. C. Belcastro technical monitor. This ork as also partially supported by the National Science Foundation under Grant No. NSF-CMMI-1254129 entitled CA- REER: Probabilistic Tools for High Reliability Monitoring and Control of Wind Farms. Any opinions, findings, and
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