EG4321/EG7040. Nonlinear Control. Dr. Matt Turner

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1 EG4321/EG7040 Nonlinear Control Dr. Matt Turner

2 EG4321/EG7040 [An introduction to] Nonlinear Control Dr. Matt Turner

3 EG4321/EG7040 [An introduction to] Nonlinear [System Analysis] and Control Dr. Matt Turner

4 Syllabus Recap 1. Background: linear state-space systems; nonlinear concepts 2. Linearisation: methods and limitations 3. Lyapunov stability 4. Passivity 5. Absolute Stability 6. Nonlinear Design methods

5 Passivity - context I Course has focused on internal stability of nonlinear systems Input assumed zero No attention paid to output Focus on state boundedness/convergence 1. Lyapunov s 2nd method 2. Phase-portraits 3. Linearisation x 0 G u 0 y =?? ẋ = f(x) Some idea of input-output stability useful...? Input non-zero and general x 0 Output well-defined Given a sensible input, is the output also sensible? u 0 G ẋ = f(x,u) y = h(x,u)

6 Passivity - context II Also interested in interconnected systems u 1 + e 1 G 1 y 1 e y2 G 2 u 2 G 1 { ẋ1 = f 1 (x 1,e 1 ) y 1 = h 1 (x 1,e 1 ) G 2 { ẋ2 = f 2 (x 2,e 2 ) y 2 = h 1 (x 2,e 2 ) What properties should G 1 and G 2 have in order for: Interconnection to be stable/asymptotically stable when u 1 0 and u 2 0? Outputs y 1 and y 2 to be well-behaved if inputs u 1 and u 2 are well-behaved

7 Genesis of passivity concepts I The concept of passivity in control has its roots in circuit theory. Roughly speaking an electrical network is said to be passive if Energy supplied to network Energy Stored in network (over given period) (over given period) Ensures no accumulation of energy Instantaneous power supplied to a network from a voltage source is Power = v(t)i(t) Thus energy supplied over a period from 0 to time t is thus Energy = t 0 v(τ)i(τ)dτ Energy storage in magnetic fields (inductors) and electric fields (capacitors): 1 2 Li L(t) Cv c(t) 2

8 Genesis of passivity concepts II The network is passive if energy supplied equals or exceeds energy stored t 0 v(τ)i(τ)dτ E(t) E(0) Equivalently, differentiating v(t)i(t) Ė(t) Notes Key component of passivity definition is the energy function, E For electrical networks, E is a quadratic function E(t) = n C i=1 1 2 C iv 2 n L i + j=1 1 2 L ji 2 j Uncanny resemblance to Lyapunov function...

9 Example - electrical network + u(t) R L i(t) C Input Voltage: u(t) Output Current: y(t) = i(t) States: Current i(t) Capacitor voltage v c (t) Kirchoff s Voltage Law: u = v R +v c +v L = ir +v c +L di dt Re-arranging: di dt = R L i 1 L v c + 1 L u Also, from capacitor equation dv c dt = 1 C i

10 Example - electrical network In state-space form we thus have the (linear!) equations: dv c dt = 1 C i } = 1 L v c R L i + 1 L u state equation di dt y = i output equation Therefore with the energy storage function E = 1 2 Li Cv2 c we have Ė = Li di dt +Cv dv c c dt = Li ( 1L v c RL i + 1L ) u +Cv c ( 1 C i) = Ri 2 v c i +ui +v c i = Ri 2 +ui < ui System is passive

11 Passivity - Generalisation Consider the system: G = { ẋ = f(x,u) y = h(x,u) f(.) : R n R m R n h(.) : R n R m R m where f(.,.) and h(.,.) are well-behaved functions and such that f(0,0) = 0 and h(0,0) = 0. Ifthere existconstantsǫ,δ,ρ 0, apositivesemi-definitestoragefunction V(x) and a positive semi-definite function ψ(x) such that V x f(x,u)+ǫu u +δy y +ρψ(x) u y x,u R n R m the system G is said to be passive. ψ(x) = 0 x = 0 Sub-definitions Input strict passivity: ǫ > 0 Output strict passivity: δ > 0 State strict passivity: ρ > 0 Lossless-ness ǫ = δ = 0

12 Passivity and Lyapunov Close-relationship between passivity and Lyapunov s 2nd method. 1. Assume that system G is passive 2. Assume that V(x) is positive definite and radially unbounded. Then Origin of G is globally stable when u 0 V x f(x,0) δy y = δh(x,0) h(x,0) If G is output strictly passive (δ > 0) and such that y(t) = 0 x(t) = 0 then origin of G is globally asymptotically stable when u 0 If G is state strictly passive (ρ > 0) then then origin of G is globally asymptotically stable when u 0 This condition is known as zero-state-observability

13 Passivity and linear systems Ultimately, passivity used to prove stability of interconnections Often, a linear system is one element of that interconnection. Recall representations of linear systems: State-space (time-domain) ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t)+Du(t) Transfer function y(s) = [C(sI A)B +D] u(s) }{{} G(s) Questions: Is there an easy way to verify passivity of a linear system? Can we verify passivity of a linear system via its transfer function?

14 Positive realness An interesting transfer function concept is positive realness Roughly speaking... G(s) is positive real if The poles of G(s) have zero or negative real part G(jω)+G (jω) 0 ω R G(s) is strictly positive real if The poles of G(s) have strictly negative real part G(jω)+G (jω) > 0 ω R Close relations between passivity and positive realness G(s) positive real G passive G(s) strictly positive real G strictly passive

15 Positive realness - SISO systems For single-input-single-output (SISO) systems, positive realness can be checked graphically. In this case G(jω)+G (jω) = R[G(jω)]+I[G(jω)]+R[G(jω)] I[G(jω)] = 2R[G(jω)] Implications: Positive realness: R[G(jω)] 0 Strict positive realness: R[G(jω)] > 0 No constraint on I[G(jω)]. Passivity assessment via Nyquist Diagram! Imaginary Axis G1(s) not s.p.r. Nyquist Diagram G2(s) s.p.r Real Axis

16 Passivity - summary Lyapunov s 2nd method provides useful information about a system s internal stability Passivity says something about a system s input-output stability. Passivity tools can be used to prove stability of interconnected systems (to be studied soon) Passivity of linear systems can be assessed by 1. Looking at their transfer functions 2. Looking at the Nyquist Diagram (SISO case)