Low Gain Feedback Properties, Design Methods and Applications Zongli Lin University of Virginia Shanghai Jiao Tong University The 32nd Chinese Control Conference July 28, 213
Outline A review of high gain feedback Introduction to low gain feedback Low gain feedback vs high gain feedback Low gain feedback design methods Applications: small control magnitude Applications: slow rate in control input Applications: small control energy Compositions of low gain and high gain feedback Low gain in characteristic model based adaptive control Concluding remarks and Q&A Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 2
A review of high gain feedback Consider { ẋ = Ax + Bu + Dw, x R n, u R, w R, z = Cx, z R, where x is the state, u is the control input, z is the controlled output, w is the disturbance and 1 d 1 1 d 2 A =......., B =., D =., 1 d n 1 a 1 a 2 a 3 a n 1 d n C = [ 1 ]. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 3
A review of high gain feedback Let F = [ ] f 1 f 2 f n be such that 1. λ..... 1 +. [ ] f1 f 2 f n = {λ 1, λ 2,, λ n }, Reλ i <. 1 A high gain feedback is given as: u = F H (ε)x := [ 1 1 1 1 ε f n 1 f ε n 1 2 f ε 2 n 1 ε f n under which the closed-loop system is given by ] x, ε (, 1], ẋ 1 = x 2 + d 1 w,. ẋ n 1 = x n + d n 1 w, ẋ n = ( a 1 + 1 ) ε f n 1 x1 + ( a 2 + 1 ) f ε n 1 2 x2 + + ( a n + 1 ε f n) xn + d n w, z = x 1. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 4
A review of high gain feedback Let x 1 = x 1, x 2 = εx 2,, x n = ε n 1 x n and τ = t ε. We have d dτ x 1 = x 2 + εd 1 w,. d dτ x n 1 = x n + ε n 1 d n 1 w, d dτ x n = (ε n a 1 + f 1 ) x 1 + ( ε n 1 a 2 + f 2 ) x2 + + (εa n + f n ) x n + ε n d n w, z = x 1. Observation The effect of the disturbance w can be reduced to zero as ε ; The closed-loop system is robust to plant parameter variations (feedback domination in recent nonlinear control literature); lim ε F H (ε) =. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 5
A review of high gain feedback Limitations of high gain feedback High feedback gain causes actuators to saturate. High gain causes peaking phenomenon [Sussmann & Kokotovic, IEEE TAC 91]. High gain feedback places restrictions on open loop systems (e.g., requirement for the minimum phase property, as demonstrated by the classical root-locus design method). Imaginary Axis 6 4 2 2 4 Root Locus 6 4 3 2 1 1 Real Axis Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 6
A review of high gain feedback Roles of low gain feedback To complement high gain feedback when high gain feedback cannot achieve certain control objectives, or cannot achieve certain control objectives by itself. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 7
Introduction to low gain feedback Consider ẋ = Ax + Bu, x R n, u R. Assumption The pair (A, B) is asymptotically null controllable with bounded controls (ANCBC), i.e., (A, B) is stabilizable; Reλ i (A), i. A low gain feedback can be constructed as u = F L (ε)x, ε (, 1], where F L (ε) is such that λ i (A + BF L (ε)) = ε + λ i (A), Reλ i (A) =. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 8
Introduction to low gain feedback Basic properties (a) F L (ε) αε; (b) e (A+BF L(ε))t β e εt ε r 1 2, t ; (c) F L (ε)e (A+BF L(ε))t γ i εe ε 2 t, F L (ε)(a + BF L (ε)) i e (A+BF L(ε))t γ i εe ε 2 t, t, i = 1, 2, where r is the largest algebraic multiplicity of the jω eigenvalues of A. Observation lim F L(ε) =. ε Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 9
Low gain feedback design methods Remark Property (a) indicates the asymptotic nature of the low feedback, i.e., lim F L(ε) =. ε Property (b) reveals that the closed-loop system under low gain feedback will peak slowly to a magnitude of order O(1/ε r 1 ), with r being the largest algebraic multiplicity of the eigenvalues of A. Property (c) implies that, for any given bounded set of initial conditions, the control input and all its derivatives can be made arbitrarily small by decreasing the value of the low gain parameter ε. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 1
Low gain feedback vs high gain feedback Small vs large control forces ẋ = Ax + Bu. Under high gain feedback: F H (ε) as ε ; u(t) = F H (ε)e (A+BFH(ε))t x() as ε. Under low gain feedback: F L (ε) = O(ε) as ε ; u(t) = F L (ε)e (A+BF L(ε))t x() r x() εe ε 2 t as ε. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 11
Low gain feedback vs high gain feedback Slow vs fast control actions ẋ = Ax + Bu. Under high gain feedback: du(t) = FH (ε)(a + BF H (ε))e (A+BF H (ε))t x() as ε. dt Under low gain feedback: u (i) (t) = FL (ε)(a + BF L (ε)) i e (A+BF L (ε))t x() γi x() εe ε 2 t, as ε, i = 1, 2,. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 12
Low gain feedback vs high gain feedback Slow vs fast time scales ẋ = Ax + Bu, A = [ I ] [, B = I Under high gain feedback: u = diag { 1 1 ε,,, 1 n ε ε} Fx; n 1 ε x = d x = (A + BF ) x, x = diag { 1, ε,, ε n 1} x. d t ε Under low gain feedback: u = diag { ε n, ε n 1,, ε } Fx; 1 x ε = d x dεt = (A + BF ) x, x = diag { ε n 1, ε n 1,, ε, 1 } x. ]. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 13
Low gain feedback vs high gain feedback Small energy in control input vs small energy in controlled output ẋ = Ax + Bu, z = Cx. Low gain feedback: making u L2 u 2 L 2 = = u T (t)u(t)dt = small x T (t)f T L (ε)f L (ε)x(t)dt x T ()e (A+BF L (ε))t F T L (ε)f L (ε)e (A+BF L (ε))tt x()dt γ x() 2 2 ε 2 e ε 2 t dt = O(ε) as ε. High gain feedback: making z L2 z L2 = = small z T (t)z(t)dt x T (t)c T Cx(t)dt as ε. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 14
Low gain feedback vs high gain feedback Disturbance rejection in control input vs in controlled output { [ ] [ ] [ ẋ = Ax + Bu + Dω, I D1 A =, B =, D = z = Cx, I ]. Low gain feedback: making L 2 gain from ω to u small u L2 sup as ε. ω ω L2 High gain feedback: making L 2 gain from ω to z small z L2 sup as ε. ω ω L2 Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 15
Low gain feedback design methods Consider the linear system ẋ = Ax + Bu, x R n, u R m, where x is the state and u is the input. Assumption The pair (A, B) is asymptotically null controllable with bounded controls (ANCBC), i.e., (A, B) is stabilizable; Reλ i (A), i. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 16
Low gain feedback design methods Method 1: Eigenstructure assignment based design Design algorithm [Lin & Saberi, SCL 93; Lin, Springer 98]: Step 1. Find nonsingular transformation matrices T S and T I such that A 1 B 1 B 12 B 1l A 2 B 2 B 2l T 1 S AT S =......., T 1 S BT I =....... A l A B l B 1 B 2 B l where A contains all open left-half plane eigenvalues of A, and all eigenvalues of each A i are on the jω axis and hence (A i, B i ) is controllable as given by 1. A i =..... 1, B i =.. a i,ni a i,ni 1 a i,1 1, Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 17
Low gain feedback design methods Step 2. For each (A i, B i ), let F i (ε) R 1 n i be such that λ(a i + B i F i (ε)) = ε + λ(a i ), ε (, 1]. Step 3. Construct a family of low gain state feedback laws as where Remark u = F L (ε)x, F 1 (ε) F 2 (ε) F L (ε) = T I....... F l (ε) T 1 S. F L (ε) is a polynomial matrix in ε and lim ε F L (ε) =. This family of state feedback laws are referred to as low gain feedback, and ε the low gain parameter. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 18
Low gain feedback design methods Example. Let A = 1 1 1 1 2, B = 1. λ(a) = { j, j, j, j}; u = [ ε 4 +2ε 2 4ε 3 +4ε 6ε 2 4ε ] x, ε (, 1]. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 19
Low gain feedback design methods Theorem (Lin, Springer 98) Consider a single input pair (A, B) in the control canonical form 1... A =....... 1, B =.. a n a n 1 a 1 1 Let all eigenvalues be on the jω axis. Let F L (ε) be such that λ(a+bf L (ε))= ε+λ(a). Then, there exists an ε (, 1] such that, ε (, ε ], (a) (b) (c) F L (ε) αε, e (A+BF L (ε))t β ε r 1 e ε 2 t, t, F L (ε)(a+bf L (ε)) i e (A+BF L (ε))t γi εe ε 2 t, t, i =, 1, 2,, where r is the largest algebraic multiplicity of the eigenvalues of A, and α, β and γ i s are some positive scalars independent of ε. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 2
Low gain feedback design methods Method 2: ARE based designs Two different ARE based low gain design algorithms: The H 2 ARE based [Lin, Stroorvogel & Saberi, Automatica 96] The H ARE based [Teel, IEEE TAC 95] H 2 design algorithm: Step 1. Solve the following ARE for the unique positive definite solution P(ε) A T P + PA PBB T P + Q(ε) =, ε (, 1], where Q(ε) >, ε (, 1], and satisfies lim Q(ε) =. ε Step 2. Construct a family of low gain state feedback laws as u = F L (ε)x, where F L (ε) = B T P(ε). Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 21
Low gain feedback design methods Theorem (Lin, Springer 98) For each ε (, 1], there exists a unique matrix P(ε) > that solves the ARE. Moreover, such a P(ε) satisfies, 1 lim ε P(ε) = ; 2 There exists a constant α >, independent of ε, such that, for i =, 1, 2,, P 1 2 (ε)a i P 1 2 (ε) α i, ε (, 1]. Remark The advantage is directly resulting in a Lyapunov function V (x) = x T P(ε)x; The disadvantage is that no analytic solution of P(ε) can be obtained. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 22
Low gain feedback design methods Method 3: Lyapunov equation based design Design algorithm [Zhou, Duan & Lin, IEEE TAC 8]: Step 1. Solve the Lyapunov equation for the unique W (ε) >, W (A + ε ) T 2 I + (A + ε ) 2 I W = BB T, ε (, 1]. Step 2. Compute the matrix P(ε) as, P(ε) = W 1 (ε). ( A T P + PA PBB T P = εp ) Step 3. Construct a family of low gain state feedback laws as where F L (ε) = B T P(ε). u = F L (ε)x, Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 23
Low gain feedback design methods Theorem (Zhou, Duan & Lin, IEEE TAC 8) 1 The matrix A BB T P(ε) is asymptotically stable and satisfies A BB T P(ε) = P 1 (ε) ( A T εi ) P(ε). That is, the eigenvalues of the matrix A BR 1 B T P(ε) are those of A shifted to the their left by ε. 2 The closed-loop system converges to the origin no slower than e ε 2 t. 3 lim ε + P(ε) =. 4 If m = 1, then P(ε) is a polynomial matrix, and, if m > 1, then P(ε) is a rational matrix but is generally not a polynomial matrix. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 24
Low gain feedback design methods Remark The resulting feedback gain is parameterized explicitly in terms of the low gain parameter ε; It directly results in a quadratic Lyapunov function V (x) = x T P(ε)x; The convergence rate of the resulting closed-loop system is explicitly specified. Example. Let A = λ(a) = { j, j, j, j}. 1 1 1 1 2, B = 1. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 25
Low gain feedback design methods Step 1. Solve the parametric Lyapunov equation to obtain ( ) 4 5ε 2 + 4 W (ε) = 1 ( ) 2 5ε 2 + 4 ( ) α (ε) 4 ε 4 ε 2 4 ( ) ε 4 1ε 2 24 ( ) 2 5ε 2 + 4 ( ) 2 3ε 4 + 6ε 2 + 8 ( ) 3ε 4 + 6ε 2 + 8 ε 6 2ε 4 12ε 2 16 ( ) 4 ε 4 ε 2 4 ( ) 3ε 4 + 6ε 2 + 8 ( ) 2 ε 6 + 4ε 4 + 1ε 2 + 8 ( ) ε 6 +4ε 4 +1ε 2 +8 ( ) ε 4 1ε 2 24 ε 6 2ε 4 12ε 2 16 ( ) ε 6 + 4ε 4 + 1ε 2 + 8 ε 8 +8ε 6 +3ε 4 +44ε 2 +16 where α (ε) = ε 3 ( ε 6 + 12ε 4 + 48ε 2 + 64 ). Step 2. Compute P(ε) = W 1 (ε) ( ) ( ) ( ) ( ) ε 6 + 4ε 4 + 6ε 2 + 4 ε 3ε 4 + 8ε 2 + 6 ε 2 3ε 4 + 6ε 2 + 4 ε ε 2 + 2 ε 2 ( ) ( ) ( ) ( ) 3ε 4 + 8 ε 2 + 6 ε 2 2 5ε 4 + 8ε 2 + 2 ε 11ε 2 + 1 ε 2 4 ε 2 + 1 ε = ( ) ( ) ( ) 3ε 4 + 6 ε 2 + 4 ε 11ε 2 + 1 ε 2 2 7ε 2 + 2 ε 6ε 2 ( ) ( ) ε 2 + 2 ε 2 4 ε 2 + 1 ε 6ε 2 4ε. Step 3. The low gain feedback laws are given by u = B T P(ε)x = [ ε 4 + 2ε 2 4ε 3 + 4ε 6ε 2 4ε ] x, ε (, 1]. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 26
Applications: small control magnitude Application 1: Semi-global stabilization in the presence of actuator saturation x 2 u x 2 x x 1 x 1 Theorem (Lin & Saberi, SCL 93) Consider a linear system subject to actuator saturation ẋ = Ax + Bsat(u), where (A, B) is ANCBC. Then, the low gain feedback law u = F L (ε)x semi-globally asymptotically stabilizes the system at the origin. Remark Global stabilization can be achieved only if (A, B) is ANCBC [Sussmann, Sontag & Yang, IEEE TAC 94]. In general, non-linear feedback is needed to achieve global stabilization [Fuller, IJC 69]. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 27
Applications: small control magnitude Example. Consider ẋ = Ax + Bsat(u), where 1 A = 1 1, B =, λ(a) = {j, j, j, j}, r = 2. 1 2 1 F L (ε) = [ ε 4 2ε 2 4ε 3 4ε 6ε 2 4ε ] λ(a + BF L (ε)) = ε + λ(a). 5 ε =.1 ε =.1 6 4 2 States x(t) States x(t) 2 4 5 1 2 3 4 5 6 7 8 Time 6 2 4 6 8 1 12 14 16 Time 1.5.1 Control u(t).5 1 Control u(t).1 1.5 1 2 3 4 5 6 7 8 Time 2 4 6 8 1 12 14 16 Time Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 28
Applications: small control magnitude Application 2: Stabilization in the presence of time delays in the input Consider a linear system subject to time delay in the input ẋ = Ax + Bu(t τ), where (A, B) is ANCBC. Let F L (ε) be designed by eigenstructure assignment. Theorem (Lin and Fang, IEEE TAC 7) For any given, arbitrarily large τ >, there exists an ε > such that the feedback law u(t) = F L (ε)e Aτ x(t), ε (, ε ], asymptotically stabilizes the system. If there is no non-zero imaginary eigenvalues, then the stabilizing feedback law can be simplified to u(t) = F L (ε) e\\\\ Aτ x(t). The above feedback laws also achieve semi-global stabilization if the system is also subject to input saturation. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 29
Applications: small control magnitude Proposition (Lin and Fang, IEEE TAC 7) The system ẋ = αx + u(t τ), u = kx, α >, is asymptotically stable if and only if k > α and τ < arccos(α/k) < 1. k2 α 2 Remark The feedback law u(t) = F L (ε)e Aτ x(t) is obtained from the classical predictor feedback law u(t) = F L (ε)e Aτ x(t) + F L (ε) t t τ ea(t τ s) Bu(s)ds. The idea of dropping off the distributed term of the predictor feedback was earlier used in [Mazenc, Mondie & Niculescu, SCL, 4] and [Fang & Lin, IEEE TAC 6] for the second order oscillator system. Further extension by using Lyapunov equation based low gain feedback and by allowing time varying delay was made [Zhou, Lin & Duan, Automatica 12], where the design is termed truncated predictor feedback. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 3
Applications: slow rate in control input Consider a linear system subject to simultaneous magnitude and rate saturation in the input { ẋ = Ax + Bsat(v), v = sat( v + u), where x is the plant state and v is the actuator state. Theorem (Lin, SCL, 97) The family of linear feedback laws u = 1 ε 2 F L(ε)x achieves semi-global asymptotic stabilization. ( ) 1 ε 2 1 v, ε (, 1], Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 31
Applications: small control energy Application 1: Semi-global stabilization under control energy constraint Consider a linear system ẋ = Ax + Bu, x R n, u R m, where (A, B) is ANCBC. Let u = F L (ε)x, ε (, 1]. Theorem (Zhou, Lin & Duan, IEEE TAC, 11) For any given, arbitrarily large, bounded set Ω R n, and any given, arbitrarily small, scalar α >, there is an ε > such that, for all ε (, ε ], the closed-loop system is asymptotically stable and u(t) 2 L 2 = F L (ε)e (A+BF L (ε))t x() 2 dt < α 2, x() Ω. Other applications: L 2 almost disturbance decoupling problem for linear systems with zeros on the imaginary axis [Chen, Lin & Hang, IJC, 98; Lin & Chen, Automatica, ; Lin, Springer 98]. L 2 almost disturbance decoupling problem for weakly non-minimum phase nonlinear systems [Lin, SCL, 98]. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 32
Compositions of low gain and high gain feedback Observation Combinations of low gain and high gain feedback solve various control problems: Additive low-and-high gain feedback Composite nonlinear feedback (CNF) design Embedded low-and-high gain design Combination 1: Additive low-and-high gain feedback [Lin & Saberi, IJRNC 95, 97; Saberi, Lin & Teel, IEEE TAC 96] ẋ = Ax + Bsat(u), x R n, u R m. Assumption The pair (A, B) is asymptotically null controllable with bounded controls (ANCBC), i.e., (A, B) is stabilizable; Reλ i (A), i.. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 33
Compositions of low gain and high gain feedback Step 1: Low gain design. u L = F L (ε)x, ε (, 1]. Let P(ε) > be such that (A + BF L (ε)) T P(ε) + P(ε) (A + BF L (ε)) Q(ε), for some Q(ε) >. Remark If an ARE or Lyapunov function based low gain design is adopted, then such a P(ε) is the solution of the ARE. If the eigenstructure assignment based low gain feedback design is adopted, a procedure for constructing such a P(ε) can be found in [Lin, Springer 98]. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 34
Compositions of low gain and high gain feedback Step 2: High gain design. u H = ρb T P(ε)x := F H (ε, ρ)x, ε (, 1], ρ, where ρ is referred to as the high gain parameter. Step 3: Low-and-high gain design. where F LH (ε, ρ) = F L (ε) + F H (ε, ρ) u = u L + u H = F LH (ε, ρ)x, ( = (1 + ρ)b T P(ε), for ARE or Lyapunov function based design). Let us define an ellipsoid Ω(P(ε)) = { x R n : x T P(ε)x 1 }, and the region where the control input does not saturate L(F L (ε)) = {x R n : F L (ε)x 1}. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 35
Compositions of low gain and high gain feedback Remark If ε can be chosen such that Ω(P(ε)) L(F L (ε)), then the closed-loop system remains linear within Ω(P(ε)), and Ω(P(ε)) is a contractively invariant set of the closed-loop system under input saturation. Thus, the closed-loop system is asymptotically stable at x = with Ω(P(ε)) enclosed in the domain of attraction. For all the low gain feedback designs, such an Ω(P(ε)) can be made large enough to enclose any given bounded set as a subset by choosing the value of ε sufficiently small. With the addition of the high gain component, the ellipsoid Ω(P(ε)) remains contractively invariant. Thus, once the low gain parameter ε is fixed, the high gain parameter ρ can be tuned high enough to obtain closed-loop performances beyond the large stability region. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 36
Compositions of low gain and high gain feedback Example [Lin, SCL 97]. Consider a linear system in the presence of actuator satuation/deadzone nonlinearities and input additive uncertainties and disturbances, ẋ = 1 1 x + 1 σ(u + g(x) + d(t)). σ(v) v Actuator nonlinearity σ Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 37
Compositions of low gain and high gain feedback The low-and-high gain feedback law: u = (ε 3 +.5ε 2 ρ)x 1 (3ε 2 +.8125ερ)x 2 (3ε+.4375ρ)x 3. 4 3 States 2 1 1 1 2 3 4 5 6 7 8 1 Actuator Output.5.5 1 1 2 3 4 5 6 7 8 9 1 g(x) = x 1 x 2 2 + x 3, d(t) = 2 sin t, ε =.1, ρ = 1 Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 38
Compositions of low gain and high gain feedback Combination 2: Composite nonlinear feedback (CNF) design [Lin, Pachter & Banda, IJC 98] { ẋ = Ax + Bsat(u), x() = x R 2, u R, y = Cx, y R. Assumption The pair (A, B) is controllable and hence, without loss of generality, can be assumed to be in the following form [ ] [ ] 1 A =, B =. a 1 a 2 1 The output matrix is given by where c 1. C = [ c 1 c 2 ], Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 39
Compositions of low gain and high gain feedback Step 1: Linear feedback design. u Lin = Fx + Gr, F = [f 1 f 2 ] is such that A + BF is Hurwitz; The closed-loop system C(sI A BF ) 1 B has a small damping ratio; G = (a 1 + f 1 )/c 1. Lemma (Lin, SCL 97) Let P > be such that (A + BF ) T P + P(A + BF ) = Q, Q >. Let c > be such that x {x : x T Px c} = Fx 1, (, 1). Then, the linear feedback law u Lin will cause the output y to asymptotically track a step command input of amplitude r, as long as x() and r satisfy ( x 1 () r ), x 2 () {x : x T Px c}, and a 1 c 1 r c 1. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 4
Compositions of low gain and high gain feedback Step 2: Nonlinear feedback design. u NLin = ρ(x, r)b T P ( [ r/c1 x Step 3: Composition of linear and nonlinear feedback. ]), ρ(x, r) >. u CNF = u Lin + u NLin = (F ρ(x, r)b T P)x + a 1 f 1 + ρ(x, r)p 12 c 1 r. Theorem (Lin, SCL 97) For any nonnegative function ρ(x, r), locally Lipschitz in x, the nonlinear feedback law u CNF will cause the system output to asymptotically track the step command input of amplitude r from an initial state x(), provided that x() and r satisfy ( x 1 () r ), x 2 () {x : x T Px c}, and a 1 c 1 r c 1. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 41
Compositions of low gain and high gain feedback Theorem (Lin, SCL 97) Let ρ be a constant parameter. As ρ tends to +, one of the two closed-loop poles approaches ; the other approaches a finite location in the open left half plane; and hence, the damping ratio of the closed-loop system tends to +. Choice of ρ(x, r) The function ρ(x, r) can chosen to be a non-increasing function of the error signal y r such that, lim ρ(x, r) = and lim y r + ρ(x, r) = κ 1. y r Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 42
Compositions of low gain and high gain feedback Example. Consider an F-16 aircraft derivative, at an altitude of 2, feet and a Mach number of.9 [Blakelock, Wiley & Sons 9]: [ ] [ ] 1 A =, B =,.373 2.184 1 C = [ 32.8781 24.3282 ], where the output is the pitch rate and the control input is the elevator deflection scaled such that the maximum actuator capacity is unity. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 43
Compositions of low gain and high gain feedback Design 1. Choose F = [ 2.866 ], G =.74, resulting in a closed-loop system damping ratio of.7. Letting Q = I, we yield a CNF law, u CNF = (2.866 +.255ρ(x, r)) x 1.323ρ(x, r)x 2 (.74 +.625ρ(x, r)) r, with ρ(x, r) = 5e y r. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 44
Compositions of low gain and high gain feedback 2.5 2 Pitch Rate y 1.5 1.5 5 1 15 Time Design 1: The dotted line is the command input, the solid line represents the response due to the CNF law, and the dashed line the nominal linear feedback (ρ(x, r) = ). Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 45
Compositions of low gain and high gain feedback Design 2. Choose F = [ 7.8259 = ], G =.2267. This linear feedback law results in a damping ratio of.4. Letting Q = I, we obtain a CNF law, u CNF = (7.8259 +.671ρ(x, r)) x 1.2597x 2 (.2267 +.24ρ(x, r)) r, with ρ(x, r) = 3e 1 y r. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 46
Compositions of low gain and high gain feedback 3 2.5 2 Pitch Rate y 1.5 1.5 5 1 15 Time Design 2: The dotted line is the command input, the solid line represents the response due to the CNF law, and the dashed line the nominal linear feedback (ρ(x, r) = ). Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 47
Compositions of low gain and high gain feedback Generalizations Extensions have been made by several authors (Chen, Lan, Lee, Peng, Postlethwaite, Turner, Venkataramanan, Walker) on Higher order systems Singular systems Output feedback Discrete-time systems Applications in computer hard-disk drives Applications in flight control Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 48
Compositions of low gain and high gain feedback Combination 3: Embedded low-and-high gain design [Lin & Saberi, IJRNC 94] Consider a nonlinear system in a special normal form η = f (η, ξ j ), j {1, 2,, r + 1}, ξ 1 = ξ 2,. ξ j 1 = ξ j, ξ j = ξ j+1,. ξ r = u, where ξ r+1 = u. Assumption The zero dynamic η = f (η, ) is globally asymptotically stable at η =. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 49
Compositions of low gain and high gain feedback Remark The system is known to be not semi-globally stabilizable by linear high gain feedback of the states ξ i s if j > 1 [Byrnes & Isidori, IEEE TAC 91; Sussmann & Kokotovic, IEEE TAC 91]; Its semi-global asymptotic stabilizbility was first estabilized in [Teel, SCL 92] by using a feedback law of nested saturation type. Step 1: Low gain design. Design the low gain feedback laws that stabilize the states ξ 1, ξ 2,, ξ j 1, u L = ε j 1 c j 1 ξ 1 ε j 2 c j 2 ξ 2 εc 1 ξ j 1, where c 1, c 2,, c j 1 are such that the polynomial s j 1 + c 1 s j 2 + + c j 2 s + c j 1 is Hurwitz. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 5
Compositions of low gain and high gain feedback Under these low gain feedback laws, the state equations for ξ i s can be rewritten as follows ξ 1 = ξ 2, ξ 2 = ξ 3,. ξ j 1 = ε j 1 c j 1 ξ 1 ε j 2 c j 2 ξ 2 εc 1 ξ j 1 + ξ j, ξ j = ξ j+1, ξ j+1 = ξ j+2,. ξ r 1 = ξ r, ξ r = ε j 1 c j 1 ξ r j+2 + ε j 2 c j 2 ξ r j+3 + + εc 1 ξ r + u, where j 1 ξ k = ξ k + ε j l c j l ξ k j+l, k = j, j + 1,, r. l=1 Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 51
Compositions of low gain and high gain feedback Step 2: Low-and-high gain design. Design a family low-and-high gain feedback laws, also parameterized in ε, to stabilize the dynamic of the states ξ k, k = j, j + 1,, r, as follows, u LH = ε j 1 c j 1 ξ r j+2 ε j 2 c j 2 ξ r j+3 εc 1 ξ r d r j+1 ε r j+1 ξ j d r j ε r j ξ j+1 d 1 ε ξ r, where d 1, d 2,, d r j+1 are such that s r j+1 + d 1 s r j + + d r j s + d r j+1 is Hurwitz. Theorem This family of low-and-high gain feedback laws semi-globally asymptotically stabilizes the nonlinear system. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 52
Compositions of low gain and high gain feedback Examples. Example 8.2 of [Byrnes & Isidori, IEEE TAC 91]: η = (1 ηξ 2)η, ξ 1 = ξ 2, ξ 2 = u. Example 1.1 of [Sussmann & Kokotovic, IEEE TAC 91]: η =.5(1 + ξ 2)η 3, ξ 1 = ξ 2, ξ 2 = u. Semi-globally stabilizing feedback law: u = ξ 1 ε2 +1 ε ξ2. Example 4.1 of [Teel, SCL 92]: η = η + η 2 u, ξ 1 = ξ 2, ξ 2 = u. Semi-globally stabilizing feedback law: u = 2ε 2 ξ 1 3εξ 2. Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 53
Low gain in characteristic model based adaptive control Characteristic model based adaptive control [Wu, Hu, Xie, Meng, et al.] A characteristic model represents a higher order plant with a second order time varying difference equation, based on which adaptive control is designed. Example. A characteristic model based stabilizing adaptive control law: u(k) = u 1 (k) + u 2 (k) + u 3 (k) + u 4 (k), Maintaining/tracking control law u 1(k) = ˆf 1 (k)y(k) ˆf 2 (k)y(k 1) ĝ 1 (k)u 1 (k 1) ĝ (k)+λ 1 ; Golden section adaptive control law u 2(k) =.382ˆf 1 (k)y(k).618ˆf 2 (k)y(k 1)+ĝ 1 (k)u 2 (k 1) ĝ (k)+λ 1 ; Differential control law u 3(k) = d 1 y(k)+y(k 1) T ; Integral control law u 4(k) = d 2T k i=1 y(i). Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 54
Low gain in characteristic model based adaptive control 4 3 x1 x2 2 1.5 x1 x2 2 1 States x(t) 1 1 States x(t).5.5 2 1 3 1.5 4 2 4 6 8 1 12 14 16 Time(s) 2 5 1 15 2 25 3 Time(s) Control u(t) 15 1 5 5 Control u(t) 2 1.5 1.5.5 1 1 1.5 15 2 4 6 8 1 12 14 16 Time(s) 2 5 1 15 2 25 3 Time(s) G(s) = 1 s 2, λ 1 =.8, d 1 =.514, d 2 =.2 G(s) = 1 s 2, λ 1 =.8, d 1 =.514, d 2 =.2 Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 55
Concluding remarks and Q&A Design is about trade off. Both low gain and high gain feedback designs guide the trade off. Keep the humble low gain option open. There are many opportunities for research on low gain feedback. Thank you and questions? Low Gain Feedback The 32nd Chinese Control Conference July 28, 213 56