Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL Dr Glenn Vinnicombe HANDOUT 3. Stability and pole locations.

Transcription

1 Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL Dr Glenn Vinnicombe HANDOUT 3 Stability and pole locations asymptotically stable marginally stable unstable Imag(s) repeated poles + Asymptotically stable Left half plane Unstable Marginally stable Right half plane Unstable Real(s) Imaginary axis

2 55 Summary Stability, or the lack of it, is the most fundamental of system properties. When designing a feedback system the most basic of requirements is that the feedback system be stable. There are different ways of defining stability. In this handout we shall: Define the following notions: Asymptotic stability Marginal stability Instability Relate stability of a system to the poles of its transfer function In addition, we shall: Relate the transient response of a system to the poles of its transfer function Contents 3 Stability and pole locations Asymptotic Stability Bounded input bounded output stability Asymptotic Stability and Pole locations Marginal Stability Instability Stability Theorem Polesandthetransientresponse Comparison with second order systems KeyPoints... 69

3 3.. Asymptotic Stability Asymptotic Stability Definition: An LTI system is asymptotically stable if its impulse response g(t) satisfies the condition g(t) dt < Examples:. LCR circuit: g(t) = e t sin(3t + 2) e t g(t) g(t) dt e t dt = < 2. Delay line with lossy reflections: g(t) = δ(t kt ) 2k k= t g(t) dt = k= t δ(t kt ) dt 2k = k= 2 k = 2 <

4 3.. Asymptotic Stability Bounded input bounded output stability Why is asymptotic stability defined as above? If g(t) dt = A< (Asymptotically stable) and u(t) U for all t (Bounded Input) then t y(t) = g(τ)u(t τ)dτ t U t g(τ) u(t τ) dτ g(τ) dτ UA( Bounded Output) Asymptotically stable systems always give bounded outputs in response to bounded inputs Conversely, if there exists a T such that g(t) dt were not bounded then for any A g(t) dt > A. In, this case let if t>t u(t) =, if t T and g(t t) >., if t T and g(t t) So u(t) is bounded (U = ) but y(t) = u(τ)g(t τ)dτ = g(t τ) dτ = g(t) dt > A. Non asymptotically stable systems can give unbounded outputs in response to bounded inputs

5 3.. Asymptotic Stability Asymptotic Stability and Pole locations Consider an LTI system with a rational transfer functiong(s). That is, it can be written as the ratio of two polynomials G(s) = n(s) d(s) Also assume that G(s) is proper, thatis deg[n(s)] deg[d(s)]. e.g. G(s) = s (a differentiator) is not proper (This condition will always be satisfied for physically realizable systems. Moreover, any system whose transfer function violates this condition is not asymptotically stable.) Also assume (temporarily) that the poles of G(s) are distinct i.e. that d(s) has no repeated roots then G(s) = n(s) (s p )(s p 2 ) (s p n ) = λ + λ (s p ) + λ 2 (s p 2 ) + + λ n (s p n ) by partial fraction expansion, and so g(t) = λ δ(t) + λ e p t + λ 2 e p 2 t + +λ n e p nt. If we now write p k = σ k + jω k,foreachk =...n,then and so e p k t = e (σ k +jω k )t = e σ k t e jω k t = e σ k t e jω k t = e } {{ } σ k t g(t) λ δ(t) + λ e σ t + λ 2 e σ 2 t + + λ n e σ nt.

6 3.. Asymptotic Stability 59 Now, [ ] eσt dt = σ eσt, if σ< = σ, if σ Hence, if every pole has σ k <, then g(t) dt λ λ + + λ λ n and consequently the system is asymptotically stable. σ σ 2 REPEATED POLES: If G(s) has repeated poles, i.e. G(s) = (s p) l, σ n < where l denotes the multiplicity of the pole at s = p, then the partial fraction expansion of G(s) will be of the form G(s) = + λ (s p) + λ 2 (s p) λ l (s p) l +. Hence, the impulse response g(t) will be of the form g(t) = +λ e pt + λ 2 te pt + + λ l (l )! tl e pt + However, if p = σ + jω and σ< (ie Real(p) < ), then tk e pt dt = tk e σt dt < for any k. Hence the conclusion remains valid.

7 3.. Asymptotic Stability 6 So, An LTI system with transfer function G(s) is asymptotically stable if ALL the poles of G(s) have a negative real part (i.e. lie in the left half of the complex plane) Conversely, if a system is asymptotically stable then, for Real(s), G(s) = e st g(t)dt e st g(t) dt g(t) dt ( since e st for Re(s) ) = A< and so G(s) cannot have any poles on the imaginary axis or in the right half of the complex plane. In conclusion: An LTI system with transfer function G(s) is asymptotically stable Imag(s) ALL poles of G(s) lie in the LHP Left Half Plane LHP Right Half Plane RHP Real(s) Real(s) < Real(s) > Imaginary Axis Real(s) =

8 3.2. Marginal Stability Marginal Stability Definition: An LTI system is marginally stable if it is not asymptotically stable, but there nevertheless exist numbers A, B< such that g(t) dt < A + BT for all T Examples:. Integrator: g(t) = H(t) = T G(s) = /s jω-axis pole at s = 2. Undamped spring-mass system: g(t) = cos(3t) g(t) dt = T g(t) dt dt = T G(s) = s/(s 2 + 9) jω-axis poles at s =±3j 3. Delay line with lossless reflections: g(t) = δ(t k), k= g(t) dt T + 4. Something which cannot arise as the impulse response of any ODE: (g(t), but system is not asymptotically stable) g(t) = t + g(t) dt = log(t + ) T

9 3.3. Instability Instability Definition: A system is unstable if it is neither asymptotically stable nor marginally stable. Examples:. Inverted pendulum: g(t) = e 4t + e 4t G(s) = s 4 2. Two integrators in series: + pole at s = 4 g(t) = t G(s) = double pole at s = s2 3. Oscillation of badly designed control system: g(t) = e.t sin(.3t) G(s) =.3 ( (s.) ) poles at s =. ±.3j Warning: Different people use different definitions of stability. In particular, systems which we have defined to be marginally stable would be regarded as stable by some, and unstable by others. For this reason we avoid using the term stable without qualification.

10 3.4. Stability Theorem Stability Theorem Let G(s) = λ (s p) l represent one term in the partial fraction expansion of a more complex transfer function, where p = σ + jω denotes a pole of that transfer function. Furthermore, let tl g(t) =L G(s) = λ (l )! ept. We already know that the system G(s) is asymptotically stable if and only if σ<. We shall now look at the difference between the cases σ = and σ > by evaluating g(t) dt for various values of σ and l. First,notethat, σ = and l = σ = and l> t l g(t) = λ (l )! eσt g(t) = λ (a constant) & g(t) dt = g(t) dt = λ T marginally stable t l g(t) = λ (l )! σ> g(t) dt = λ T l l! unstable g(t) dt = CeσT T l + unstable

11 3.4. Stability Theorem 64 It can be shown that: If any term in the partial fraction expansion is unstable, then the whole system is unstable. If any term in the partial fraction expansion is marginally stable, then the whole system is, at best, marginally stable. Hence, (for systems with proper rational transfer functions) wehavethe Stability Theorem:. A system is asymptotically stable if all its poles have negative real parts. 2. A system is unstable if any pole has a positive real part, or if there are any repeated poles on the imaginary axis. 3. A system is marginally stable if it has distinct poles on the imaginary axis, and all the remaining poles have negative real parts. asymptotically stable unstable unstable repeated jω-axis poles marginally stable

12 3.4. Stability Theorem 65 pole at s = 2 pole at s =.5 + 2j pole at s =.5 2j 5 4 zero at s = j G(s) 3 2 zero at s =.5 zero at s =.5.87j Real(s) Imag(s) 2 2 Imag(s) Real(s) Poles/zeros for G(s) = (s +.5)(s2 s + ) (s + 2)(s 2 +.s + 4) Note: this is an asymptotically stable system.

13 3.5. Poles and the transient response Poles and the transient response Impulse response: (assuming distinct poles) g(t) = λ δ(t) + λ e p t + λ 2 e p 2 t + +λ n e p nt Consider a term e pt. p real: gives a real exponential, with time constant /p. p< p> p complex: then p will also be a pole (complex poles always appear in conjugate pairs since they are roots of a real polynomial) furthermore, their coefficients in the partial fraction expansion will also be conjugate (because g(t) = g(t) ) The contributions from these two poles combine to give either a damped or a growing sinusoid: λ { { }} { L Ae jφ s p + λ { }} { Ae jφ } s p time constant = /σ 2π/ω = Ae jφ e pt + Ae jφ e p t = Ae σt { e j(ωt+φ) + e j(ωt+φ)} = 2Ae σt cos(ωt + φ) frequency = ω σ< σ> (c.f. Ce ω nct cos(ω n c 2 t + φ) Mechanics data book)

14 3.5. Poles and the transient response Comparison with second order systems Consider the 2nd order ODE ÿ ω 2 n + 2c ω n ẏ + y = x (c: damping factor (< ), ω n : undamped natural frequency) taking Laplace transforms (with zero initial conditions) s2 ω 2 + 2c s + ȳ(s) = x(s) n ω n giving the transfer function, from x(s) to ȳ(s), ȳ(s) = s 2 ω 2 n + 2c ωn s + x(s). The poles of this transfer function are the solutions to s 2 ω 2 n + 2c ω n s + = or s 2 + 2ω n cs + ω 2 n = that is s = 2ω nc ± 4ω 2 nc 2 4ω 2 n 2 = ω n c ± jω n c 2 If we let p denote either of these poles, then Real{p} = ω n c and p 2 = ω 2 n. Hence: c = Real{p} p and ω n = p.

15 3.5. Poles and the transient response 68 which results in the following important figures: Imag(s) p= σ + jω ω ω n c 2 ω n sin c σ ω n c cos c Im(s) Real(s) 2 c = / 2 c = c = Re(s) 2 2 ω n =.5 ω n =. ω n =.5 ω n = 2. 2 Contours of constant c and ω n.

16 3.6. Key Points Key Points The total impulse response of an LTI system is the sum of the terms due to each real pole, or pair of complex poles. The system s response to any input will also include these features. The following figure shows a selection of pole locations, with their corresponding contribution to the total response. This is an important figure. Note: The real part of the pole, σ, determines both stability and the time constant, /σ. The imaginary part, ω, determines the frequency of oscillation, ω (rad/sec). Imag(s) Real(s) Left half plane Right half plane Imaginary axis Pole locations and corresponding transient responses