Controllability and Observability Imply Exponential Decay of Sensitivity in Dynamic Optimization
EExponential Decay of Sensitivity inDynamic Optimization:A Graph-Theoretic Approach
Sungho Shin ∗ Victor M. Zavala ∗ , ∗∗∗ Department of Chemical and Biological Engineering,University of Wisconsin-Madison, Madison, WI 53706 USA(e-mail: { sungho.shin,victor.zavala } @wisc.edu). ∗∗ Mathematics and Computer Science Division,Argonne National Laboratory, Argonne, IL 60439 USA
Abstract:
We study exponential decay of sensitivity (EDS) in dynamic optimization (DO)problems, which include model predictive control (MPC) and moving horizon estimation(MHE). The property indicates that the sensitivity of the solution at time i against a dataperturbation at time j decays exponentially with | i − j | . We use a graph-theoretic analysis ofthe optimality conditions of DO problems to prove that EDS holds under uniform boundednessof the Lagrangian Hessian, a uniform second order sufficiency condition (uSOSC), and a uniformlinear independence constraint qualification (uLICQ). Furthermore, we prove that uSOSC anduLICQ can be obtained under uniform controllability and observability. These results provideinsights into how perturbations propagate along the horizon and enable the development ofapproximation and solution schemes. We illustrate the developments with numerical examples. Keywords: sensitivity analysis, nonlinear, model predictive control, graph-theory1. INTRODUCTIONWe study the DO problem:min x N u N − N − (cid:88) i =0 (cid:96) i ( x i , u i ; d i ) + (cid:96) N ( x N ; d N ) (1a)s.t. T x = d − | λ − (1b) x i +1 = f i ( x i , u i ; d i ) , i ∈ I [0 ,N − | λ i . (1c)Here, N ∈ I > is the horizon length; for each stage (time) i , x i ∈ R n x are the states, u i ∈ R n u are the controls, d i ∈ R n d are the data (parameters), λ i ∈ R n x are thedual variables, (cid:96) i : R n x × R n u × R n d → R are the stagecost functions, f i : R n x × R n u × R n d → R n x are thedynamic mapping functions, (cid:96) N : R n x × R n d → R is thefinal cost function. The initial state constraint is enforcedwith the initial state mapping T ∈ R n × n x and parameter d − ∈ R n . We let x − , u − , u N , λ N be empty vectors(for convenience), and define z i := [ x i , u i ], w i := [ z i ; λ i ],and ξ i := [ w i ; d i ] for i ∈ I [ − ,N ] , and we use the syntax v a : b := [ v a ; v a +1 ; · · · ; v b ] for v = x, u, λ, d, z, w, ξ . Problem(1) is a parametric nonlinear program that we denote as P N ( d − N ). We assume that all functions are twice con-tinuously differentiable and potentially nonconvex. Typi-cal MPC problems are formulated with T = I and typicalMHE problems are formulated with an empty matrix T ∈ R × n x (i.e., initial constraint is not enforced). State-outputmappings encountered in such problem formulations areassumed to be embedded within the stage costs. (cid:63) We acknowledge support from the Grainger Wisconsin Distin-guished Graduate Fellowship.
In this paper we show that, under uniform boundedness ofthe Lagrangian Hessian (uBLH), a uniform second ordersufficiency condition (uSOSC), and a uniform linear in-dependence constraint qualification (uLICQ), the primal-dual solution of the DO problem at a given stage decaysexponentially with the distance to the stage at which adata perturbation is introduced. In particular, around agiven base data d (cid:63) − N and associated solution w (cid:63) − N atwhich uBLH, uSOSC, and uLICQ are satisfied, there existuniform constants Υ > ρ ∈ (0 ,
1) and neighborhoods D (cid:63) − N of d (cid:63) − N and W (cid:63) − N of w (cid:63) − N such that : (cid:107) w † i ( d − N ) − w † i ( d (cid:48)− N ) (cid:107) ≤ (cid:88) j ∈ V Υ ρ | i − j | (cid:107) d j − d (cid:48) j (cid:107) (2)holds for i ∈ I [ − ,N ] . Here, w †− N : D (cid:63) − N → W (cid:63) − N is theprimal-dual solution mapping of P N ( · ) and w † i ( · ) is suchmapping at stage i . In other words, the sensitivity Υ ρ | i − j | of the solution at stage i against a data perturbation atstage j decays exponentially with respect to the distance | i − j | . We call this property exponential decay of sensitivity (EDS). Here, it is important that (Υ , ρ ) are uniformconstants (independent of horizon length N ). This allowsus to maintain (Υ , ρ ) unchanged even if the horizonlength becomes indefinitely long (e.g., when approachingan infinite horizon). Furthermore, we show that uLICQand uSOSC can be obtained from uniform controllabilityand observability.The main result of this paper is a specialization ofa recently-established EDS result for general graph-structured optimization problems (Shin et al., 2021).Specifically, the DO problem (1) is a special class of such a r X i v : . [ m a t h . O C ] M a r roblems (the graph is a line). Here, we also establish anexplicit connection between regularity conditions (uLICQand uSOSC) with controllability and observability. Re-cently, EDS has been established for MPC problems underuSOSC and uniform controllability (Na and Anitescu,2019); however, the proof uses a different technique (basedon a Riccati recursion). In this paper, we establish EDS byusing a more general graph-theoretic analysis of the opti-mality conditions of the DO problem. EDS for continuous-time, linear-quadratic MPC problems has been reportedin Gr¨une et al. (2019, 2020b,a). Recently, it has also beenshown that EDS is an important property that can beused to construct efficient time-coarsening or discretizationschemes (Shin and Zavala, 2020; Gr¨une et al., 2020b) andto establish convergence of decomposition algorithms (Naet al., 2020; Na and Anitescu, 2020). Notation:
The set of real numbers and the set of integersare denoted by R and I , respectively, and we define I A := I ∩ A , where A is a set; I > := I ∩ (0 , ∞ ); I ≥ := I ∩ [0 , ∞ ).We consider vectors always as column vectors. We usethe syntax: { M i } Ni =1 := [ M ; · · · ; M N ] := [ M (cid:62) · · · M (cid:62) n ] (cid:62) .Furthermore, v [ i ] denotes the i -th component of v . Fora function φ : R n x → R and variable vectors y ∈ R n u , z ∈ R n d , ∇ yz φ ( x ) := { ( { ∂ ∂y [ i ] ∂z [ j ] φ ( x ) } qj =1 ) (cid:62) } pi =1 . For avector function ϕ : R n x → R m and a variable vector w ∈ R s , ∇ w ϕ ( x ) := { ( { ∂∂w [ j ] ϕ ( x )[ i ] } sj =1 ) (cid:62) } mi =1 . Vector 2-norms and induced 2-norms of matrices are denoted by (cid:107) · (cid:107) . For matrices A and B with the same dimensions, A (cid:31) ( (cid:23) ) B indicates that A − B is positive (semi) definite.We use the convention: if m or n is zero, R m × n is asingleton only containaining the m × n null matrix.2. MAIN RESULTS In this section, we establish EDS (2) for P N ( · ). Wefirst formally define sufficient conditions for EDS to hold(uBLH, uSOSC, and uLICQ). We begin by defining thenotion of uniformly bounded quantities. Definition 1. (Uniform Bounds). A set { A i } i ∈A is called L -uniformly bounded above (below) if there exists a uni-form constant L ∈ R (independent of N ) such that (cid:107) A i (cid:107) ≤ ( ≥ ) L holds for any i ∈ A .We say that a set of a quantity is uniformly bounded above(below) if there exists a uniform constant L such that theset is L -uniformly bounded above (below). Also, we willwrite that some quantity a is uniformly bounded above(below) if { a } is uniformly bounded above (below).The Lagrangian function of P N ( d − N ) is defined as L N ( w − N ; d − N ) := N (cid:88) i =0 L i ( z i , λ i − i ; d i ) , where: L i ( z i , λ i − i ; d i ) := (cid:96) i ( z i ; d i ) − λ (cid:62) i − x i + λ (cid:62) i f i ( z i ; d i ) L N ( x N , λ N − ; d N ) := (cid:96) N ( x N ; d N ) − λ (cid:62) N − x N . Definition 2. (uBLH). Given d (cid:63) − N and the solution w (cid:63) − N of P N ( d (cid:63) − N ), L -uBLH holds if: (cid:107)∇ w − N ξ − N L N ( w (cid:63) − N ; d (cid:63) − N ) (cid:107) ≤ L, (3) with uniform constant L < ∞ .The primal Hessian H N of the Lagrangian and theconstraint Jacobian J N are: H N := ∇ z N ,z N L N ( w (cid:63) − N ; d (cid:63) − N ) (4a) J N := ∇ z N c − N − ( z (cid:63) N ; d (cid:63) − N ) , (4b)where c − N − ( · ) is the constraint function for P N ( · ). Definition 3. (uSOSC). Given d (cid:63) − N and the solution w (cid:63) − N of P N ( d (cid:63) − N ), γ -uSOSC holds if: ReH ( H N , J N ) (cid:23) γI, (5)with uniform constant γ > ReH ( H N , J N ) := Z (cid:62) H N Z is the reducedHessian and Z is a null-space matrix of J N . Definition 4. (uLICQ). Given d (cid:63) − N and the primal-dualsolution w (cid:63) − N of P N ( d (cid:63) − N ), β -uLICQ holds if: J N J (cid:62) N (cid:23) βI (6)with uniform constant β > γ ,while uLICQ assumes that the smallest non-trivial singularvalue of the Jacobian is uniformly bounded below by β / .Thus, these are strengthened versions of SOSC and LICQ.We require uSOSC and uLICQ because, under SOSC andLICQ, the smallest eigenvalue of reduced Hessian or thesmallest non-trivial singular value of the Jacobian maybecome arbitrarily close to 0 as the horizon length N is extended (e.g., see Shin et al. (2021, Example 4.18)).Under uSOSC and uLICQ, on the other hand, the lowerbounds are independent of N . Assumption 5.
Given twice continuously differentiable func-tions { (cid:96) i ( · ) } Ni =0 , { f i ( · ) } N − i =0 and base data d (cid:63) − N , thereexists a primal-dual solution w (cid:63) − N of P N ( d (cid:63) − N ) atwhich L -uBLH, γ -uSOSC, and β -uLICQ are satisfied.The following lemma is a well-known characterization ofsolution mappings of parametric NLPs (Robinson, 1980;Dontchev and Rockafellar, 2009). Lemma 6.
Under Assumption 5, there exist neighbor-hoods D − N of d (cid:63) − N and W − N of w (cid:63) − N and continuous w †− N : D − N → W − N such that for any d − N ∈ D − N , w †− N ( d − N ) is a local solution of P N ( d − N ). Proof.
From Shin et al. (2021, Lemma 3.3). (cid:4)
We can thus see that there exists a well-defined solutionmapping w †− N ( · ) around the neighborhood of d (cid:63) − N . Wenow study stage-wise solution sensitivity by characterizingthe dependence of w † i ( · ) on the data d − N . Theorem 7.
Under Assumption 5, there exist uniform con-stants Υ > ρ ∈ (0 ,
1) (functions of
L, γ, β ) and neigh-borhoods D (cid:63) − N of d (cid:63) − N and W (cid:63) − N of w (cid:63) − N such that(2) holds for any d − N , d (cid:48)− N ∈ D (cid:63) − N and i ∈ I [ − ,N ] . Proof.
We observe that P N ( · ) is graph-structured (in-duced by G N = ( V N , E N ), where V N = {− , , · · · , N } and E N = {{− , } , { , } , · · · , { N − , N } ), and the maximumgraph degree D = 2. From uBLH, uLICQ, and uSOSC, onecan see that assumptions in Shin et al. (2021, Theorem.9) are satisfied. This implies that the singular valuesof ∇ w N w N L N ( w (cid:63) − N ; d (cid:63) − N ) are uniformly upper andlower bounded and those of ∇ w N d − N L N ( w (cid:63) − N ; d (cid:63) − N )are uniformly upper bounded (uniform constants givenby functions of L, β, γ ; see Shin et al. (2021, Equation(4.15))). We then apply Shin et al. (2021, Theorem 3.5)to obtain Υ > ρ ∈ (0 ,
1) as functions of the upperand lower bounds of the singular values (see Shin et al.(2021, Equation (3.17))). This allows expressing Υ , ρ asfunctions of
L, β, γ . (cid:4) Theorem 7 establishes EDS under the regularity conditionsof Assumption 5. It is important that Υ , ρ can be deter-mined solely in terms of
L, γ, β (and do not depend onthe horizon length N ). Practical DO problems typicallyhave additional equality/inequality constraints that arenot considered in (1). Thus, Theorem 7 may not be directlyapplicable to those problems. However, the results in Shinet al. (2021) are applicable to such problems as long asthe DO problem is a graph-structured NLP. Specifically,under uniform strong SOSC and uLICQ, we can establishEDS using Shin et al. (2021, Theorem 3.5, 4.9). Although uSOSC and uLICQ are standard notions of NLPsolution regularity, they are not intuitive notions froma system-theoretic perspective. However, we now showthat uSOSC and uLICQ can be obtained from uniformcontrollability and observability. We begin by defining: Q i := ∇ x i x i L i ( z (cid:63)i , λ (cid:63)i − i ; d (cid:63)i ) R i := ∇ u i u i L i ( z (cid:63)i , λ (cid:63)i − i ; d (cid:63)i ) S i := ∇ x i u i L i ( z (cid:63)i , λ (cid:63)i − i ; d (cid:63)i ) E i := ∇ x i d i L i ( z (cid:63)i , λ (cid:63)i − i ; d (cid:63)i ) F i := ∇ u i d i L i ( z (cid:63)i , λ (cid:63)i − i ; d (cid:63)i ) A i := ∇ x i f i ( z (cid:63)i ; d (cid:63)i ) B i := ∇ u i f i ( z (cid:63)i ; d (cid:63)i ) G i := ∇ d i f i ( z (cid:63)i ; d (cid:63)i ) . Definition 8. ( { A i } N − i =1 , { B i } N − i =0 ) is ( N c , β c )-uniformlycontrollable with N c ∈ I ≥ and β c > N )if, for any i, j ∈ I [0 ,N − with | i − j | ≥ N c , C i : j C (cid:62) i : j (cid:23) β c I holds, where C i : j := [ A i +1: j B i · · · A j B j − B j ] . Definition 9. ( { A i } N − i =0 , { Q i } Ni =0 ) is ( N o , γ o )-uniformly ob-servable with N o ∈ I ≥ and γ o > N ) iffor any i, j ∈ I [0 ,N − with | i − j | ≥ N o , O (cid:62) i : j O i : j (cid:23) γ o I holds, where O i : j := Q j A i : j − . . . Q i +1 A i Q i . Here A a : b := (cid:26) A b A b − · · · A a +1 A a , if a ≤ bA b A b +1 · · · A a − A a , otherwise.Note that uniform controllability and observability arestronger versions of their standard counterparts. One canestablish the following duality between uniform controlla-bility and observability. Proposition 10. ( { A i } Ni =1 , { B i } Ni =0 ) is ( N , α )-uniformlycontrollable if and only if ( { A (cid:62) i } i = N , { B (cid:62) i } i = N ) is ( N , α )-uniformly observable (here, note that the orders of se-quences { A (cid:62) i } i = N , { B (cid:62) i } i = N are inverted). Proof.
For i, j ∈ I [0 ,N − with | i − j | ≥ N : C i : j = [ A i +1: j B i · · · A j B j − B j ] = B (cid:62) i A (cid:62) j : i +1 ... B (cid:62) j − A (cid:62) j B (cid:62) j (cid:62) = O (cid:62) j : i . Duality follows from O (cid:62) j : i O j : i = C i : j C (cid:62) i : j . (cid:4) The following technical lemma is needed to show thatuniform controllability implies uLICQ.
Lemma 11.
Consider a block row/column operator U with L -uniformly bounded above block V of the form: U := IV I . . . I , I VI . . . I . We have that
U, U − are ( L +1)-uniformly bounded above. Proof.
Observe that: U − = I − V I . . . I , I − VI . . . I One can easily see that (cid:107) U (cid:107) , (cid:107) U − (cid:107) ≤ (cid:107) V (cid:107) ≤ L .Thus, U, U − are ( L + 1)-uniformly bounded above. (cid:4) Lemma 12. K -uniform upper boundedness of { A i } N − i =0 and { B i } N − i =0 , T T (cid:62) (cid:23) δI for uniformly lower bounded δ >
0, and ( N c , β c )-uniform controllability of ( { A i } N − i =1 , { B i } N − i =0 )implies (6), where β > K, δ, N c , β c . Proof.
The Jacobian J N has the following form: J N = T − A − B I . . . − A N − − B N − I − A N − − B N − I By inspecting the block structure of J N and Shin et al.(2021, Lemma 4.15), one can see that it suffices to showthat the smallest non-trivial singular value of S − A i − B i I . . . − A j − − B j − I − A j − B j (7)is β / -uniformly bounded below for S = T or I andfor any i, j ∈ I [0 ,N − with N c ≤ | i − j | ≤ N c , where0 < β ≤ K, δ, N c , β c . This follows fromthe observation that one can always partition I [0 ,N − intoa family of blocks with size between N c and 2 N c . For now,we assume S = I . By applying a set of suitable block rowand column operations (in particular, first apply block rowoperations to eliminate A i , · · · , A j , and then apply blockcolumn operations to eliminate − B i , · · · , − A i : j − B j − )and permutations, one can obtain the following: (cid:20) I − A i +1: j B i · · · − A j B j − − B j (cid:21) . (8)he lower-right blocks constitute the controllability ma-trix C i : j ; from uniform controllability, the smallest non-trivial singular value of the matrix in (8) is uniformly lowerbounded by min(1 , β / c ). Here, we have applied block-rowand block-column operations as the ones that appear inLemma 11 (each multiplied block is uniformly boundedabove due to K -uniform boundedness of { A i } N − i =0 and { B i } N − i =0 ). Also, we have applied such operations onlyuniformly bounded many times (the number of operationsis independent of N since the number of blocks in thematrix in (7) is bounded by 4(2 N c + 1)( N c + 1), which isuniformly bounded above). We thus have that the smallestnon-trivial singular value of the matrix in (7) is uniformlylower bounded with uniform constant β / , and β > K, N c , β c . Now we consider the S = T case. One can observe that the smallest non-trivialsingular value of the matrix in (7) with S = T is lowerbounded by that with S = [ (cid:101) T ; T ] (here, (cid:101) T (cid:62) is a null spacematrix of T ); and again, it is lower bounded by δ / timesthat with S = I . We thus have that the smallest non-trivial singular value of the matrix in (7) with S = T isuniformly lower bounded by β / δ / . Therefore, we havethat the smallest non-trivial singular values of the matricesin (7) with S = I or T are β / -uniformly lower boundedfor any i, j ∈ I [0 ,N − with N c ≤ | i − j | ≤ N c , where β := min( β , δβ , (cid:4) If T ∈ R × n x , the assumption T T (cid:62) (cid:23) δI for uniformlylower bounded δ > δ > Notation in Section 1.We now show that uniform observability implies uSOSC.
Lemma 13. K -uniform boundedness of { A i } N − i =0 , { B i } N − i =0 ,and { Q i } Ni =0 , Q i (cid:23) S i = 0, R i (cid:23) rI ( r > N ), and ( N o , γ o )-uniform observability of( { A i } N − i =0 , { Q i } Ni =0 ) implies (5), where γ > K, N o , γ o , r . Proof.
The primal Hessian H N has the following form: H N Q R . . . Q N − R N − Q N . By inspecting the block structure of H N and J N andShin et al. (2021, Lemma 4.14), one can observe that itsuffices to show that: first, ReH (cid:18)(cid:20) Q i : j R i : j − (cid:21) , [ A i : j B i : j − ] (cid:19) (9)has γ -uniformly lower bounded smallest eigenvalue with γ > i, j ∈ I [0 ,N − with N o ≤ | i − j | ≤ N o ,where: A i : j := − A i I . . . . . . − A j − I , B i : j − := − B i . . . − B j − Q i : j := Q i Q i +1 . . . Q j , R i : j − := R i R i +1 . . . R j − , and second, R i (cid:23) γI for any i ∈ I [0 ,N − . This followsfrom the observation that one can always partition I [0 ,N − into a family of blocks with size between N c and 2 N c . Weconsider x i : j , u i : j − such that A i : j x i : j + B i : j − u i : j − = 0holds. By uniform positive definiteness of { R i } N − i =0 anduniform boundedness of { B i } N − i =0 , for κ := r/ K , we have12 u (cid:62) i : j − R i : j − u i : j − ≥ κ u (cid:62) i : j − B (cid:62) i : j − B i : j − u i : j − = κ x (cid:62) i : j A (cid:62) i : j A i : j x i : j , where the equality follows from A i : j x i : j + B i : j − u i : j − =0. Furthermore, from Q i : j (cid:23)
0, we have that: x (cid:62) i : j Q i : j x i : j = ( Q / i : j x i : j ) (cid:62) Q i : j ( Q / i : j x i : j ) ≤ K x (cid:62) i : j Q i : j x i : j , where the inequality follows from that the largest eigen-value of Q i : j is bounded by K . Thus, x (cid:62) i : j Q i : j x i : j + u (cid:62) i : j − R i : j − u i : j − is not less than:min(1 /K, κ ) x (cid:62) i : j (cid:2) Q i : j A (cid:62) i : j (cid:3) (cid:20) Q i : j A i : j (cid:21) x i : j + r (cid:107) u i : j − (cid:107) . Observe that (cid:2) Q i : j A (cid:62) i : j (cid:3) can be permuted to: Q j I − A j − Q j − I . . . − A (cid:62) i +1 Q i +1 I − A (cid:62) i Q i . (10)We apply block row and column operations (as those ofLemma 6) uniformly bounded many times to obtain: (cid:20) I A (cid:62) j − i Q j · · · A (cid:62) i Q i +1 Q i (cid:21) . (11)From Proposition 10 and the ( N o , γ o )-uniform observabil-ity of ( { A i } N − i =0 , { Q i } Ni =0 ), we have that ( { A (cid:62) i } i = N − , { Q i } i = N )is ( N o , γ o )-uniformly controllable. We thus have that thematrix in (11) has min(1 , γ o )-uniformly lower boundedsmallest non-trivial singular value. This implies that thesmallest non-trivial singular value of the matrix in (10)is uniformly lower bounded by γ (cid:48) , where γ (cid:48) is given by afunction of K , N o , γ o . Therefore, we have that: x (cid:62) i : j Q i : j x i : j + u (cid:62) i : j − R i : j − u i : j − ≥ γ ( (cid:107) x i : j (cid:107) + (cid:107) u i : j − (cid:107) ) , where γ := min( γ (cid:48) /K, κγ (cid:48) , r/ R i (cid:23) γI for any i ∈ I [0 ,N − . Consequently, the smallesteigenvalues of the matrix in (9) for any i, j ∈ I [0 ,N − with N o ≤ | i − j | ≤ N o and R i are γ -uniformly lower bounded.Thus, by Shin et al. (2021, Lemma 4.14), (5) holds. Onecan confirm that γ is a function of K, N o , γ o , r . (cid:4) We now show that uniform boundedness of system matri-ces implies uBLH.
Lemma 14. If { Q i } Ni =0 , { R i } N − i =0 , { S i } N − i =0 , { A i } N − i =0 , { B i } N − i =0 , { E i } N − i =0 , { F i } N − i =0 , { G i } N − i =0 , and T are K -uniformly bounded above, (3) holds, where L < ∞ is afunction of K . Proof.
Uniform boundedness of the system matrices im-plies that for any i, j ∈ I [ − ,N ] , ∇ w i ξ i L N ( w (cid:63) − N ; d (cid:63) − N )re uniformly bounded above by 4 K (Shin et al. (2021,Lemma 4.6)). Furthermore, by inspecting the problemstructure, we can see that (cid:107)∇ w i ξ j L N ( w (cid:63) − N ; d (cid:63) − N ) (cid:107) =1 for i (cid:54) = j (there is only one identity block). Thus, (cid:107)∇ w i ξ j L N ( w (cid:63) − N ; d (cid:63) − N ) (cid:107) ≤ max(4 K, D = 2 and ap-plying Shin et al. (2021, Lemma 4.5), we have that ∇ w − N ξ − N L N ( w (cid:63) − N ; d (cid:63) − N ) is 4 max(4 K, L := 4 max(4 K, (cid:4) We now state EDS in terms of uniform controllability andobservability.
Assumption 15.
Given twice continuously differentiablefunctions { (cid:96) i ( · ) } Ni =0 , { f i ( · ) } N − i =0 , and data d (cid:63) − N , thereexists a primal-dual solution w (cid:63) − N of P N ( d (cid:63) − N ) atwhich the assumptions in Lemma 12, 13, 14 hold. Corollary 16.
Under Assumption 15, there exist uniformconstants Υ > ρ ∈ (0 ,
1) (functions of K , r , N c , β c , N o , γ o , δ ) and neighborhoods D (cid:63) − N of d (cid:63) − N and W (cid:63) − N of w (cid:63) − N such that (2) holds for any d − N , d (cid:48)− N ∈ D (cid:63) − N and i ∈ I [ − ,N ] . Proof.
From Theorem 7 and Lemma 12, 13, 14. (cid:4)
Assume now that the system is time-invariant and focuson a region around a steady-state. A corollary of Theorem7 for such a setting is derived. We present this result sincethis setting is of particular interest in the MPC literature.Consider a time-invariant system with a stage-cost func-tion (cid:96) ( · ), initial regularization function (cid:96) b ( · ), terminal costfunction (cid:96) f ( · ), and dynamic mapping f ( · ). The DO prob-lem is given by (1) with f i ( · ) = f ( · ) for i ∈ I [0 ,N − , (cid:96) i ( · ) = (cid:96) ( · ) for i ∈ I [1 ,N − , (cid:96) ( x, u ) = (cid:96) ( x, u ) + (cid:96) b ( x ),and (cid:96) N ( x ) = (cid:96) f ( x ).The steady-state optimization problem is:min x,u (cid:96) ( x, u ; d ) s.t. x = f ( x, u ; d ) | λ. (12)For given d s and an associated primal-dual solution w s :=[ x s ; u s ; λ s ] of (12), we define: Q := ∇ xx L ( w s ; d s ) , S := ∇ xu L ( w s ; d s ) ,R := ∇ uu L ( w s ; d s ) , A := ∇ x f ( z s ; d s ) , B := ∇ u f ( z s ; d s ) , where L ( w ; d ) := f ( z ; d ) − λ (cid:62) x + λ (cid:62) f ( z ; d ); for the initialand terminal cost functions (cid:96) b ( · ) and (cid:96) f ( · ), we define: λ b := ∇ x (cid:96) b ( x s ; d s ) , Q b := ∇ xx (cid:96) b ( x s ; d s ) λ f := ∇ x (cid:96) f ( x s ; d s ) , Q f := ∇ xx (cid:96) f ( x s ; d s ) . The quantities defiend above ( Q , R , etc.) are independentof N since w s can be determined independently of N . Assumption 17.
Given twice continuously differentiable (cid:96) ( · ), (cid:96) b ( · ), (cid:96) f ( · ), f ( · ), and data d s , there exists a steady-state solution w s , at which Q f (cid:23) Q (cid:23) Q b (cid:23) S = 0, R (cid:31)
0, (
A, B ) controllable, (
A, Q ) observable,
T T (cid:62) (cid:31) λ b + λ s ∈ Range( T (cid:62) ) and λ f = λ s hold. Corollary 18.
Under the time invariance setting and As-sumption 17, there exist uniform constants Υ > ρ ∈ (0 ,
1) such that the following holds: for any N ∈ I ≥ , thereexist neighborhoods D s − N of d s − N := [ T x s ; d s ; · · · ; d s ] and W s − N of w s − N := [ λ s − ; w s ; · · · ; w s ; x s ] such that(2) holds for any d − N , d (cid:48)− N ∈ D s − N , where λ s − is thesolution of T (cid:62) λ s − = λ b + λ s . Proof.
From the existence (follows from (cid:96) b + (cid:96) s ∈ Range( T (cid:62) )) and uniqueness (follows from T T (cid:62) (cid:31)
0) ofthe solution of T (cid:62) λ s − = λ b + λ s , we have well-defined λ s − .From T (cid:62) λ s − = λ b + λ s and the optimality of w s for (12), w s − N satisfies the first-order optimality conditions for P N ( d s − N ). Furthermore, all the assumptions in Lemma14 are satisfied with some uniform constant K because (cid:96) ( · ), (cid:96) b ( · ), (cid:96) f ( · ), f ( · ), T , w s , and d s are independent of N ; thus,by Lemma 14, we have (3) for a uniform constant L < ∞ .Moreover, T T (cid:62) (cid:23) δI holds for some uniform constant δ >
0, and R i (cid:23) rI for i ∈ I [0 ,N − with some uniformconstant r >
0, since (cid:96) ( · ), w s , d s , T are independent of N .Similarly, ( A, B ) controllability implies ( N c , β c )-uniformcontrollability of ( { A i } N − i =1 , { B i } N − i =0 ) with some uniformconstant N c , β c , and ( A, Q ) observability implies ( N o , γ o )-uniform observability of ( { A i } N − i =0 , { Q i } Ni =0 ) for some uni-form constants N o , γ o (for now, we assume that Q b = 0and Q f = Q ). From Lemma 12, 13, we have (5) and (6)for uniform β, γ >
0. Now, observe that (5) for Q b = 0and Q f = Q implies (5) for any Q b (cid:23) Q f (cid:23) Q ;thus, we have (5) with uniform γ > Q b , Q f .Since the first and second order conditions of optimalityand constraint qualifications are satisfied, w s − N is a strictminimizer for P N ( d − N ). Since we have (3), (5), and (6)with uniform L, γ, β , we have uBLH, uLICQ, and uSOSCat ( w s − N , d s − N ). By applying Theorem 7, we can obtain(2). Lastly, since the parameters K, r, N c , β c , N o , γ o areindependent of N , so do Υ and ρ .Initial and terminal cost functions that satisfy Assumption17 can be constructed as: (cid:96) b ( x ) := − (( I − T + T ) λ s ) (cid:62) x(cid:96) f ( x ) := ( x − x s ) (cid:62) Q ( x − x s ) + ( λ s ) (cid:62) x, where ( · ) + is the pseudoinverse of the argument. One canobserve that (cid:96) b ( · ) can be set to constantly zero if T = I .3. NUMERICAL RESULTSWe illustrate the results of Theorem 7 and of Corollaries16, 18. In this study, we solve the problem with base data d (cid:63) − N to obtain the base solution w (cid:63) − N . We then solvea set of problems with perturbed data; in each of theseproblems, a random perturbation ∆ d j is introduced at aselected time stage j , while the rest of the data do not haveperturbation (i.e., ∆ d i = 0 for i (cid:54) = j ). The obtained solu-tions w †− N ( d (cid:63) − N + ∆ d − N ) for the perturbed problemsare visualized along with the base solution w (cid:63) − N . Thescripts can be found here https://github.com/zavalab/JuliaBox/tree/master/SensitivityNMPC .We consider a quadrotor motion planning problem (Hehnand D’Andrea, 2011) with the time-invariant setting; thecost functions are given by: (cid:96) ( z ; d ) :=( x − d ) (cid:62) Q ( x − d ) + u (cid:62) Ru(cid:96) f ( x ; d ) :=( x − d ) (cid:62) Q f ( x − d ) , (cid:96) b ( x ) = 0 , where Q := diag(1 , , , q, q, q, , , R, Q f := I ; weset T = I ; and the dynamic mapping is obtained bydiscretization of:
20 40 60 80 100 − . − . − . . . . . i x i [ ] − . − . . . . i x i [ ] − . − . . . . i u i [ ] − . . . . i u i [ ] − . − . . . . i λ i [ ] − . − . . . . . i λ i [ ] Fig. 1. Base and perturbed solutions. Left: Case 1 ( q = 1 , b = 1). Right: Case 2 ( q = 0 , b = 0). d Xdt = a (cos γ sin β cos α + sin γ sin α ) (13a) d Ydt = a (cos γ sin β sin α − sin γ cos α ) (13b) d Zdt = a cos γ cos β − g (13c) dγdt = ( bω X cos γ + ω Y sin γ ) / cos β (13d) dβdt = − bω X sin γ + ω Y cos γ (13e) dαdt = bω X cos γ tan β + ω Y sin γ tan β + ω Z , (13f)where the state and control variables are defined as: x :=( X, ˙ X, Y, ˙ Y , Z, ˙ Z, γ, β, α ) and u := ( a, ω X , ω Y , ω Z ). Weuse q and b as parameters that influence controllabilityand observability. In particular, the system becomes lessobservable if q becomes small and the system loses con-trollability as b becomes small (the effect of manipulationon ω X becomes weak). We have empirically tested thesensitivity behavior for q = b = 1 (Case 1) and q = b = 0(Case 2). One can see that some of the assumptions (e.g., S i = 0 in Corollary 16) may be violated, but one canalso see that, qualitatively, the system is more observableand controllable in Case 1 than in Case 2. The results arepresented in Figure 3. The base trajectories are shown asdashed lines, the perturbed trajectories are shown as solidgray lines, and the perturbed stages are highlighted usingvertical lines. We can see that, for Case 1 ( q = 1 , b = 1),the differences between the base and perturbed solutionsbecome small as moving away from the perturbation point(EDS holds). On the other hand, for Case 2 ( q = 0 , b = 0)one cannot observe EDS; this confirms that observabilityand controllability induce EDS.4. CONCLUSIONSWe have shown that exponential decay of sensitivity holdsfor DO problems under: uBLH, uSOSC, and uLICQ. Wehave also shown that these regularity conditions can beobtained from uniform controllability and observability. REFERENCESDontchev, A.L. and Rockafellar, R.T. (2009). Implicitfunctions and solution mappings , volume 543. Springer.Gr¨une, L., Schaller, M., and Schiela, A. (2019). Sensitivityanalysis of optimal control for a class of parabolic pdesmotivated by model predictive control.
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