A Typical Feedback System. Disturbance. Reference. Input Controller Input. Plant. Sensor. Noise

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1 A Typical Feedback System Disturbance Reference Input Error Controller Input Plant Output Sensor Noise

2 Typical Control Objectives Uncontrolled system (plant) may not behave satisfactorily Design a control system that yields satisfactory behavior for the controlled system Typical properties desired of a controlled system Stability Input-output stability: Bounded inputs should give bounded outputs Internal stability: All internal variables remain bounded in the absence of inputs Tracking: (Output Input) 0 as t Regulation: Output 0 as t Disturbance/Noise Rejection: Satisfactory performance in the presence of plant disturbances and measurement noise Robustness: Satisfactory performance inspite of unmodelled dynamics and parameter uncertainty/change

3 Review of Continuous-Time Systems All signals are analog signals A linear, time invariant, single-input-single-output (SISO) system is typically described by a 0 y (n) + a 1 y (n 1) + + a (n) y = b 0 u (m) + b 1 u (m 1) + + b m u Solution = initial condition response + input response Input response = u impulse response (convolution) Transfer function = L(impulse response) L(y) = T.F. L(u) for zero initial conditions Transient response decided by poles and zero; poles decide stability Frequency response analysis: Harmonic Analysis, Bode, Nyquist

4 An Overview of Control Activities ANALYSIS: Relate system theoretic properties to system behaviour. Eg. Poles and stability Need analysis tools, eg. Routh-Hurwitz test CONTROLLER DESIGN: Translate specs to system properties and design a controller (control law) that assigns these properties to the controlled system Eg. Pole placement controller Need design tools, eg. pole placement technique IMPLEMENTATION: Sensors, actuators, filters, processors

5 A Digital Controller Digital Controller Input (From sensor) A/D Digital Processor D/A Output (To actuator) Discrete time signals (Sequences of numbers) Continuous time signals (Functions of time)

6 Discrete-Time Signals Sequence {u(k)} k=0 of real numbers A real-valued function k u(k) of integers Right-sided sequence (signal) u(0), u(1), u(2),... Two-sided sequence (signal)..., u( 2), u( 1), u(0), u(1), u(2),...

7 Operators on Discrete-Time Signals Identity operator 1 Shift or unit delay operator S (Su)(k) = u(k 1), k 1 = 0 k 1 Unit advance operator S 1 (S 1 u)(k) = u(k + 1), k 0 Difference operator u(k) = u(k) u(k 1) = (u Su)(k) = 1 S, S = 1

8 Some Basic Discrete-Time Signals Unit pulse/impulse signal δ(k) = 1, k = 0, = 0, k > 0, Unit step signal s(k) = 1, k 0 s = δ Harmonic signals Exponential signals u(k) = sin(kθ) u(k) = r k Harmonic signals with exponential amplitudes u(k) = r k sin(kθ) = Re (re jθ ) k

9 Linear Difference Equations y(k) + a 1 y(k 1) a n y(k n) = b 0 u(k) + b 1 u(k 1) b m u(k m) In terms of the shift operator y(k) + a 1 Sy(k) a n S n y(k) = b 0 u(k) + b 1 Su(k) b m S m u(k) D(S)y = N(S)u Auto-Regressive Moving Averages (ARMA) model Causal: Output independent of future input Strictly causal if output depends only the past inputs Shift invariant (time invariant) Shifted input Su produces shifted output Sy Linear (Superposition + Homogeneity) To solve need n initial conditions + input

10 An Example To numerically compute y(t) = t u(τ)dτ 0 ẏ(t) = u(t), y(0) = 0 At instants 0, T, 2T,..., kt,..., y(kt ) = y((k 1)T ) + kt u(τ)dτ (k 1)T Use forward rectangular rule to approximate the integral y(kt ) = y((k 1)T ) + T u((k 1)T ), y(0) = 0 e(t) y = T Su, y(0) = 0 e(t) u(t) t k-1 k t

11 Vector Spaces A vector space V is a set whose elements can be added in some manner whose elements can multiplied by scalars in some manner which contains a zero element For example: V = set of all functions of time V = set of all right-sided sequences

12 Linear Independence A linear combination is a finite sum of the form α 1 v α n v n Linear independence every linear combination involving atleast one nonzero scalar is nonzero {v 1, v 2,..., v n } V forms a basis for V if v s are linearly independent and every vector in V is a linear combination of v s If V has a basis of n elements for some n, then V is n-dimensional, else infinite-dimensional A linear operator is a linear function V V

13 Vector Space of Discrete Signals The set of all discrete signals is a vector space with (y 1 + y 2 )(k) = y 1 (k) + y 2 (k), (αy)(k) = αy(k), Zero element y 0 y 1,..., y n are linearly dependent iff α 1,..., α n such that α 1 y 1 (k) α n y n (k) = 0 k For λ 1 λ 2 nonzero real, {λ k 1}, {λ k 2} are linearly independent For λ complex, {Re λ k } and {Im λ k } are l. i. For λ 1 λ 2 complex, {Re λ k 1,2} and {Im λ k 1,2} are l. i. For λ 1 nonzero real and λ 2 complex, λ k 1, {Re λ k 2} and {Im λ k 2} are l. i. No finite basis possible Linear operators S, D(S),, D( )

14 Homogeneous Linear Difference Equations y(k) + a 1 y(k 1) a n y(k n) = 0 D(S)y = 0 y( 1), y( 2),..., y( n) Zero initial conditions imply solution is zero The set of all solutions is a vector space since y 0 is a solution D(S)(α 1 y 1 + α 2 y 2 ) = α 1 D(S)y 1 + α 2 D(S)y 2 Theorem: The vector space of solutions has dimension n (a n 0) To prove, need to find a basis consisting of n solutions

15 A Basis of Solutions Idea Solutions sets of initial conditions If w i form a basis for all initial conditions, the corresponding solutions form a basis for all solutions Choose n sets of initial conditions as follows n y w 1 y w y n w n Claim: y 1,..., y n form a basis for all solutions...

16 A Basis of Solutions (cont d) y 1,..., y n are linearly independent If α 1 y 1 (k) + + α n y n (k) = 0 k, then k = i α i = 0 Every solution is a linear combination of y 1,..., y n Let y be any solution and consider y(k) = y( 1)y 1 (k) + + y( i)y i (k) + + y( n)y n (k) y is a solution satisfying the same initial conditions as y Hence y = y is a linear combination of y 1,..., y n

17 Solution of Linear Difference Equations Choose a basis of initial conditions and use corresponding solutions as a basis of solutions Try a solution of the form y(k) = λ k (Sy)(k) = λ k 1 = λ 1 y(k) for k 1 (S 2 y)(k) = λ 2 y(k) for k 2 D(S)y(k) = D(λ 1 )y(k) for k n y(k) = λ k is a solution of D(S)y = 0 if λ satisfies D(λ 1 ) = 0, that is 1 + a 1 λ 1 + a 2 λ a n λ n = 0 = λ n + a 1 λ n 1 + a 2 λ n a n = 0 λ n D(λ 1 ) = characteristic polynomial

18 Another Basis of Solutions Characteristic polynomial/equation factor as C(λ) = (λ p 1 ) m 1 (λ p 2 ) m 2... (λ p l ) m l The following functions form a basis for the solutions of the LDE For p i real, {p k i }, {kpk i }, {k2 p k i },..., {kmi 1 p k i } For p i = re jθ and p i complex, {r k sin(kθ)}, {kr k sin(kθ)}, {k 2 r k sin(kθ)},..., {k mi 1 r k sin(kθ)} {r k cos(kθ)}, {kr k cos(kθ)}, {k 2 r k cos(kθ)},..., {k mi 1 r k cos(kθ)} Initial conditions determine the constants in the linear combination

19 Stability of Initial Condition Response Real characteristic root p {p k } decays iff p < 1 {k j p k } decays iff p < 1 bounded if p = 1 unbounded if p > 1 unbounded if p 1 Complex characteristic root p {Re p k }, {Im p k } decay iff p < 1 {Re k j p k }, {Im k j p k } decay iff p < 1 Theorem bounded if p = 1 unbounded if p > 1 unbounded if p 1 All solutions are bounded iff all characteristic roots lie in the closed unit disc {λ : λ 1} and all roots with unit magnitude are simple (unrepeated) All solutions decay iff all characteristic roots lie in the open unit disc {λ : λ < 1}

20 Convolution Convolution of two right sided sequences u and g is the sequence k (u g)(k) = u(l)g(k l) l=0 = u(0)g(k) + u(1)g(k 1) + + u(k 1)g(1) + u(k)g(0) u g = g u (α 1 u 1 + α 2 u 2 ) g = α 1 (u 1 g) + α 2 (u 2 g) For a fixed g, u g is a linear operator on u S(u g) = u Sg = g Su D(S)(u g) = u D(S)g = g D(S)u (u g) = u g = g u u δ = u (u δ)(k) = u(0)δ(k) u(k 1)δ(1) + u(k)δ(0) = u(k)

21 Pulse Response and Input Response Pulse response g = zero initial condition response to a unit pulse D(S)g = N(S)δ, 0 = g( 1) = g( 2) = Fact: The response y of a linear time invariant system to an arbitrary input u under zero initial conditions is given by y = u g Proof: To show D(S)y = N(S)u D(S)y = D(S)(u g) = u D(S)g = u N(S)δ = N(S)(u δ) = N(S)u Step response = s g

22 Bounded-Input-Bounded-Output (BIBO) Stability A system is BIBO stable if the output to every bounded input is bounded A sequence y is said to be bounded if there exists M such that y(k) < M, k For a bounded sequence y, define Fact: y = sup y(k) = least upper bound of {y(k)} k>0 A system is BIBO stable if and only if there exists N such that for every nonzero input u, the corresponding output y satisfies Theorem: y u < N A system is BIBO stable iff the input response g is absolutely summable, that is, g(k) < k=0

23 BIBO Stability and Pulse Response Suppose the pulse response is absolutely summable k y(k) = (u g)(k) u(l) g(k l) l=0 k u g(k l) u l=0 l=0 Suppose the pulse response is not absolutely summable u k (l) = sign g(k l) l k, = 0, l > k g(l) < u k = 1 y k y k (k) = (u k g)(k) = y k k g(k) l=0

24 Output Zeroing Inputs D(S)y = N(S)u y(k) + a 1 y(k 1) + + a n y(k n) = b i u(k i) + + b m u(k m) An output zeroing input produces no response under under zero initial conditions, that is, satisfies u g = 0 Must satisfy the difference equation N(S)u = 0, that is, b i u(k i) + b m u(k m) = 0 Set of all null inputs is a vector space A basis can be found from the characteristic zeros, solutions of b i λ m i + b i+1 λ m i b m = 0 Zero z i of multiplicity m i contributes {zi k }, {kzi k },, {k mi 1 zi k } Dimension of this vector space is m i

25 Impulse Response and Initial Condition Responses Let d be the impulse response of the system D(S)y = u, that is, D(S)d = δ d is an initial condition response of the system D(S)d = N(S)u since (D(S)d)(k) = 0, k > 0 d is a linear combination of the initial condition responses corresponding to the characteristic roots d = α 1 y 1 + α 2 y α n y n Impulse response g of D(S)y = N(S)u is g = N(S)d g = α 1 N(S)y 1 + α 2 N(S)y α n N(S)y n Characteristic root p i affects g iff it is not a characteristic zero g(k) = α 1 N(S)p k 1 + α 2 N(S)kp k 1 + α 3 k 2 p k 1 + α 4 N(S)p k 2 g decays iff roots p 1 are also a zeros of equal or greater multiplicity

26 Roots, Zeros and BIBO Stability p k, k=0 kp k < k=0 p < 1 p k, kp k decay Since g involves p k, kp k, g is absolutely summable iff g decays System is BIBO stable iff pulse response decays Theorem: System is BIBO stable iff every characteristic root with p 1 is also a characteristic zero of equal or greater multiplicity

27 Z Transform The Z transform of a sequence is a function of the complex variable z Given a sequence y, its right sided Z transform is Z(y) : Y (z) = y(0) + y(1) z = y(k)z k k=0 + y(2) z 2 Z(y) is the Laurent expansion of the complex function Y + y(3) z 3 + Z(y) agrees with Y only in the region of convergence of the Laurent series Recall that if x < 1, then 1 + x + x 2 + = 1 1 x If z 1 < 1, then 1 + z 1 + z 2 + z 3 + = Z transform is linear: Z(α 1 y 1 + α 2 y 2 ) = α 1 Z(y 1 ) + α 2 Z(y 2 ) 1 1 z = z 1 z 1

28 Z Transforms of Some Common Sequences Unit pulse δ Z(δ) = 1 Unit step s Exponential sequence {p k } Harmonic signal {sin(kθ)} S(z) = 1 + z 1 + z 2 + = 1 + pz 1 + p 2 z 2 + = 1 1 z = z 1 z 1, z > pz = z 1 z p, pz 1 < 1 z sin θ z 2 2z cos θ + 1, z > 1 Exponentially modulated harmonic signals {r k sin(kθ)} rz sin θ z 2 2rz cos θ + r2, z > r

29 Properties of the Z Transform Delay Z(Su) = u( 1) + z 1 U(z) Z(S 2 u) = Su( 1) + z 1 Z(Su) = u( 2) + z 1 u( 1) + z 2 U(z) Z(S n u) = u( n) + z 1 u( n + 1) + + z n 1 u( 1) + z n U(z) Z(D(S)u) = D(z 1 )U(z) Advance Z(S 1 u) = zu(z) zu(0), Z(S 2 u) = z 2 U(z) z 2 u(0) zu(1) Z(S n u) = z n U(z) z n u(0) z n 1 u(1) zu(n 1) Difference Convolution Z( u) = Z(u) Z(Su) = z 1 U(z) u( 1) z Z(u g) = G(z)U(z)

30 Properties of the Z Transform (Contd.) Scaling in the complex plane Z({r k u(k)}) = U(z/r) Complex differentiation Initial value Final value theorem Z({ku(k)}) = z du dz (z) lim k provided the limit on the left exists u(0) = lim z U(z) u(k) = lim(z 1)U(z) z 1

31 Transfer Functions The transfer function G of the system D(S)y = N(S)u is the Z transform of its pulse response g G(z) = Z(g) If y is the input response (zero i.c.) to the input u, then y = (g u) Y (z) = G(z)U(z) G(z) = Y (z) U(z) transfer function = Z(output) Z(input) zero initial conditions To calculate the transfer function of D(S)y = N(S)u, take the Z transform on both sides Z(D(S)y) = Z(N(S)u) G(z) = Y (z) U(z) = N(z 1 ) D(z 1 )

32 Transfer Functions of Common Operators Unit Delay: y = Su Y (z) = Z(Su) = u( 1) + z 1 U(z), Y (z) U(z) = z 1 zero i.c. Unit advance: y = S 1 u Non causal. y(k) = u(k + 1) Difference operator: y(k) = u(k) u(k 1) Causal but not strictly causal Pulse Response = Z 1 (z 1 ) = {0, 1, 0, 0,...} Y (z) = zu(z) u(0), Y (z)/u(z) = z D(λ) = 1, N(λ) = 1 λ, T.f. = N(z 1 ) D(z 1 ) = 1 z 1 Impulse Response = {1, 1, 0, 0,...}

33 Inverse Z Transform Laurent expansion Perform long division for rational Y (z) Partial fraction expansion followed by look-up table Convolution property Y (z) = Y 1 (z)y 2 (z) = y = y 1 y 2 Solve numerically by forming a linear difference equation Y (z) = N(z 1 ) D(z 1 ) = y = pulse response of D(S)y = N(S)u

34 Partial Fractions Y (z) = N(z 1 ) D(z 1 ) = N(z 1 ) (1 p 1 z 1 )(1 p 2 z 1 ) 2 Unrepeated factor 1 pz 1 contributes to the expansion A 1 pz 1 Repeated factor (1 pz 1 ) m contributes A m (1 pz 1 ) + A m 1 m (1 pz 1 ) + + A 1 m 1 (1 pz 1 ) Unrepeated quadratic factor 1 2rz 1 cos θ + z 2 contributes Az 1 + B 1 2rz 1 cos θ + z 2 Repeated quadratic (1 2rz 1 cos θ + z 2 ) m factor contributes A m z 1 + B m (1 2rz 1 cos θ + z 2 ) m + + A 1 z 1 + B 1 1 2rz 1 cos θ + z 2 Expand in terms of z 1 (not z) in the usual fashion Inverse transform each term in the expansion using tables

35 The s z Correspondence y(t) = e σt : Y (s) has a pole at s = σ y(kt ) = e σkt = (e σt ) k = r k : Y (z) has a pole at z = r = e σt y(t) = e σt sin ωt: Y (s) has a pole at s = σ ± ıω y(kt ) = e σkt sin ωkt = (e σt ) k sin k(ωt ) = r k sin kθ: Y (z) has poles at z = re ±ıθ = e σt e ±ıωt = e (σ±ıω)t Suggests the correspondence z = e st for mapping poles of a s-domain signal to the poles of its sampled sequence in z-domain Where should z poles lie to get good transient behaviour (ζ, ω n )? Locate s poles using s domain experience for desired ζ, ω n Map s poles to z poles using z = e st

36 Jury s Test for Stability a 0 z n + a 1 z n a n 1 z + a n = 0, a 0 > 0 is said to be Hurwitz if all roots lie in the OLHP, Schur if all roots lie in the OUD z n z n 1 z n 2 z 2 z z 0 a 0 a 1 a 2 a n 2 a n 1 a n a n a n 1 a n 2 a 2 a 1 a 0 b 0 b 1 b 2 b n 2 b n 1 b n 1 b n 2 b n 3 b 1 b 0 c 0 c 1 c 2 c n 2 c n 2 c n 3 c n 4 c 0 b k = 1 a 0 a 0 a n a n k a k..., k = 0, 1,..., n 1, c k = 1 b 0 b 0 b n 1 b n 1 k b k, k = 0, 1,..., n 2 Stable if a 0 > 0, b 0 > 0, c 0 > 0,...

37 Stability through s z Transformation z = 1 + st/2 1 st/2, s = 2 T (z 1) (z + 1) OLHP open unit disc imaginary axis unit circle Given a polynomial p(z), p(z) = p ( ) 1 + st/2 1 st/2 = n(s) d(s) zeros of p(z) zeros of n(s) p is Schur iff n is Hurwitz Apply Routh-Hurwitz test to n(s)

38 BIBO Stability of Transfer Functions A system given by a transfer function G(z) is BIBO stable if and only if the impulse response g is absolutely summable if and only if the impulse response g decays if and only if all poles (after cancellation) of G(z) lie in the interior of the unit disc, the open unit disc (OUD) {z : z < 1} We call a transfer function stable if all its poles lie in the OUD

39 Step Response Y (z) = G(z)(1 z 1 ) 1 Bounded if (after pole-zero cancellation) all poles of G(z) lie in {z : z 1} no repeated poles on the unit circle no pole at z = 1 Approaches a limit if all poles of G(z) lie in {z : z < 1} Decays to zero if Steady state value = lim k y(k) = lim z 1 (z 1)Y (z) = G(1) all poles of G(z) lie in {z : z < 1} and z = 1 is a zero, that is, G(1) = 0 For asymptotic tracking of a step input, need stability + G(1) = 1 For asymptotic rejection of a step disturbance, need stability + G(1) = 0

40 Harmonic Response of Stable Transfer Functions u(k) = sin(kωt ) U(z) = z sin ωt z 2 2z cos ωt + 1 = z 1 sin ωt (1 e jωt z 1 )(1 e jωt z 1 ) Y (z) = G(z)U(z) = a 1 1 e jωt z 1 + a 2 1 e jωt z 1 + b 1 1 p 1 z 1 + a 1 = G(z)U(z)(1 e jωt z 1 ) z=e jωt = 1 2j G(ejωT ) = 1 2j rejφ a 2 = ā 1 = 1 2j re jφ a 1 Y ss (z) = 1 e jωt z + a e jωt z 1 y ss (k) = a 1 (e jωt ) k + a 2 (e jωt ) k = Im re jφ (e jωt ) k y ss (k) = r sin(kωt + φ), r = G(e jωt ), φ = G(e jωt ) Amplification at ω is G(e jωt ), phase difference is G(e jωt ) Frequency response is periodic in frequency

41 Digital Analog Conversion Same response at ω and ω + ω s A/D Computer D/A A/D r(t) T {r(kt)} Sampler D/A ZOH Zero Order Hold ZOH

42 Analysis of Sample-and-Hold Operation r(t) T {r(kt)} ZOH r(t) Let s(t) = unit step function, s(t kt ) = unit step function delayed by kt r(t) = r(0)[s(t) s(t T )] + r(t )[s(t T ) s(t 2T )] + = r(kt )[s(t kt ) s(t kt T )] k=0 L(s(t)) = s 1, L(s(t kt )) = s 1 e skt [ ] [ ] 1 e R(s) = r(kt )(e st ) k st s k=0 }{{}}{{} R G (s) ZOH (s) Z(R(s)) def = Z(sampled sequence of r(t)) R (s) = Z(R(s)) z=e st

43 Sampler as an Impulse Modulator Let δ(t) = unit impulse in continuous time L(δ(t)) = 1, L(δ(t kt )) = e skt ( ) R (s) = L r(kt )δ(t kt ) = L(r (t)) Define δ T (t) = k=0 δ(t kt ) k=0 δ T is an impulse train r (t) = r(t)δ T (t) r is a modulated impulse train δ T (t) r * (t) Ideal sampler = impulse modulator

44 An Ideal ZOH G ZOH (s) = 1 s e st s = L[s(t) s(t T )] = L[ impulse response of ZOH] ZOH NOTE: No transfer function possible for a ZOH

45 Frequency Domain Analysis of an Impulse Train δ T (t) is a periodic function = expand in a Fourier series δ T (t) = c n e j2πnt/t n= c n = 1 T T/2 δ T (t)e j2πnt/t dt T/2 Fourier transform of δ T (t) = 1 T δ T = n= 1 T ejnω st 3ω s 2ω s ω s 0 ω s 2 ω s 3 ω s

46 Frequency Domain Analysis of a Modulated Impulse Train R (s) = L(r (t)) = L(r(t)δ T (t)) = 1 r(t)δ T (t)e st dt T = 1 T? = 1 T = 1 T 0 0 r(t) n= n= e jnωst e st dt n= 0 r(t)e (s jω sn)t dt R(s jω s n) Fourier transform R (jω) is periodic in ω with period ω s R (jω) obtained by superimposing scaled copies of R(jω) shifted by multiples of ω s

47 Aliasing R(jω) R *(jω) ω 2ω s ω s ω s 2ω s ω Contributions at ω due to R(jω), R(jω ± njω s ) Frequencies ω ± nω s aliases of ω, show up at ω after sampling

48 An Example of Aliasing y 1 (t) = sin t, y 2 (t) = sin 7t, ω s = 6, T = π/3 y 1 (kt ) = y 2 (kt ) = sin kt

49 Anti-Aliasing R(jω) R(jω) ω m R *(jω) ω ω m R *(jω) ω 2ω s ω s ω s 2ω s ω 2ω s ω s ω s 2ω s ω 2ω m < ω s 2ω m > ω s Nyquist s/shannon s Sampling Theorem: A signal can be recovered from its samples if the sampling frequency is more than twice the highest frequency in the signal To minimise the effect of aliasing, sampling is preceded by a low-pass antialias filter Eliminates frequencies above the Nyquist frequency

50 Signal Reconstruction from Samples Possible if signal is band limited and ω m < ω s /2 To recover R(jω) from R (jω), need a filter Lsuch that R(jω) = L(jω)R (jω) R (jω) = 1 T R(jω) + 1 R(jω njω s ) T n=,n 0 }{{} frequencies>ω s /2 L(jω) = T, ω [ ω s /2, ω s /2], L(jω) = 0 = 0, elsewhere everywhere L(j ω) T ω s ω s ω

51 Impulse Response of an Ideal Low-Pass Filter Inverse Fourier transform of L(jω) l(t) = 1 2π = 1 2π ωs /2 ω s /2 = sin(ω st/2) ω s t/2 L(jω)e jωt dω T e jωt dω = sinc(ω s t/2) sinc(π t/t) t/t Note: L is a noncausal filter

52 Reconstruction Using a Low-Pass Filter r(t) = (l r )(t) = = r(τ)δ T (τ)l(t τ)dτ r(kt )sinc(ω s (t kt )/2) k= RHS is the unique band limited signal that has ω m < ω s /2 Same samples as r Reconstruction is noncausal present value depends on future samples Cannot be implemented online, can be used for offline reconstruction

53 Antialias Filtering R(jω) Antialias ω Sample R *(jω) 2ω s ω s ω s 2ω s ω

54 Frequency Domain Analysis of Zero-Order Hold G ZOH (jω) = 1 e jωt jω = T e jωt/2sin(ωt/2) (ωt/2) = T e jωt/2 sinc(ωt/2) sincx = sin x x, x 0, = 1, x = 0 Magnitude : G ZOH (jω) = T sinc(ωt/2) Phase : G ZOH (jω) = ωt 2 + π at every sign change of sinc

55 Frequency Response of G ZOH 1 sinc(ω T/2) sinc(ω T/2) Phase/π ω T/2π

56 Harmonic Response of Sample and Hold R(jω) ω ω R *(jω) 2ω s ω s ω s 2ω s ω Input sinusoid of frequency ω < ω s /2 R(jω) First harmonic of output has magnitude : sinc(ωt/2) phase : ωt/2 First harmonic Imposter First harmonic is sinc(ωt/2) sin ω(t T/2)

57 Harmonic Response of Sample and Hold: An Example r(t)

58 Harmonic Response of Sample and Hold: An Example r(t) r(t)

59 Harmonic Response of Sample and Hold: An Example r(t) First harmonic r(t)

60 Harmonic Response of Sample and Hold: An Example r(t) r(t) First harmonic

61 Higher-Order Hold Functions Interpolation using Taylor series r(t) = r(nt ) + ṙ(nt )(t nt ) r(nt )(t nt )2 +, nt t < (n + 1)T Zero-order hold: Truncate at first term r(t) = r(nt ), nt t (n + 1)T First-order hold: Truncate at first-order term r(t) = r(nt ) + ṙ(nt )(t nt ), nt t (n + 1)T To find ṙ(nt ), extrapolate to (n 1)T t (n + 1)T, put t = (n 1)T ṙ(nt ) = r(nt ) r((n 1)T ) T

62 First-Order Hold Pulse Response (n 1)T nt Pulse Response = s(t) + t T s(t) 2 s(t T ) 2 (t T ) s(t T ) T + s(t 2T ) + 1 (t 2T ) s(t 2T ). T G FOH (s) = 1 s 1 s 2e st + 1 s e 2sT + 1 T s 2(1 2e st + e 2sT ) ( ) (1 + T s) 1 e st 2 = T s

63 Analysis of a Sampler and First-Order Hold r(t) T {r(kt)} FOH r(t) R(s) = R (s)g FOH (s) 1.5 G FOH (jω) Magnitude G ZOH (jω) ω T/2π Phase G FOH (jω) G ZOH (jω) ω T/2π

64 Analysis of a Sample, Process and Hold e(t) e(kt) H(z) u(kt) ZOH u(t) U(s) = U (s)g ZOH (s) = U(z) z=e st G ZOH (s) = [H(z)E(z)] z=e st G ZOH (s) = H (s)e (s) }{{} G ZOH (s) convolved impulse trains

65 ZOH Equivalent u(kt) ZOH u(t) G(s) y(t) y(kt) Transfer Function possible Let u(kt ) = δ(kt ), unit pulse sequence ū(t) = s(t) s(t τ) y(t) = w(t) w(t τ), w = unit step response of.g(s) y(kt ) = w(kt ) w((k 1)T ) Y (z) = (1 z 1 )W (z) W (z) = Z[L 1 (s 1 G(s))] = Z(s 1 G(s)). Transfer Function = (1 z 1 )Z(s 1 G(s)) }{{} ZOH equivalent Y (s) = Y (z) z=e st = (1 e st )Z(s 1 G(s)) z=e st U (s) Y (s) = G(s)Ū(s) = G(s)G ZOH(s)U (s)

66 A Glossary of Notation Given U(s) G ZOH (s) = 1 e st s Z(U(s)) = Z transform of sampled u(t) U (s) = Z(U(s)) z=e st Given U(z) U (s) = U(z) z=e st Given G(s) G h0 (z) def = ZOH equivalent of G(s) = (1 z 1 )Z(s 1 G(s))

67 An Example e(t) u(kt) u(t) y(t) y(kt) H(z) ZOH G(s) No transfer function possible between y(t) and e(t) Can find Y (s) Transfer function possible between y(kt ) and e(kt )

68 An Example e(t) u(kt) u(t) y(t) y(kt) H(z) ZOH G(s) No transfer function possible between y(t) and e(t) Can find Y (s) Transfer function possible between y(kt ) and e(kt ) Y (s) = G(s)U(s) = G(s)G ZOH (s)u (s) = G(s)G ZOH (s)h (s)e (s)

69 An Example e(t) u(kt) u(t) y(t) y(kt) H(z) ZOH G(s) No transfer function possible between y(t) and e(t) Can find Y (s) Transfer function possible between y(kt ) and e(kt ) Y (s) = G(s)U(s) = G(s)G ZOH (s)u (s) = G(s)G ZOH (s)h (s)e (s) Y (z) E(z) = H(z)G h 0 (z) = H(z)(1 z 1 )Z(s 1 G(s))

70 Block Diagram Manipulation For Sampled Data System: An Example r(t) e(t) e(kt) C(z) u(kt) ZOH u(t) G(s) y(t) r(t) r(kt) u(kt) u(t) C(z) ZOH G(s) y(t) y(kt) u(kt) ZOH u(t) G(s) y(t) r(t) r(kt) C(z) G h0(z) y(kt) Y (z) E(z) = C(z)G h 0 (z) 1 + C(z)G h0 (z) Y (s) = G(s)G ZOH (s)u (s) = G(s)G ZOH (s)c (s)(r (s) Y (s))

71 Another Example r(t) e1(t) e2(kt) G1(s) H(z) e2(t) u(kt) u(t) y(t) ZOH G(s) r(t) y(t) r1(t) G1(s) H(z) ZOH G(s) G1(s) r1(kt) y1(kt) Y 1 (z) R 1 (z) = H(z)(GG 1) h0 (z) 1 + H(z)(GG 1 ) h0 (z), Y 1 (s) = ( Y1 (z) R 1 (z)) Y (s) = G(s)G ZOH (s)h (s)[r1(s) Y1 (s)] Discrete-Time Equivalents of Continuous-Time Controllers Design controller in continuous time R 1(s) = (GR 1 ) (s) G (s)r (s) Numerically implement a discrete-time equivalent z=e st R 1(s) Example H(s) = 1 s + a = ẏ + ay = u

72 = y(t) = t = y(kt ) = y(kt T ) + 0 [u(τ) ay(τ)]dτ kt (k 1)T [u(τ) ay(τ)]dτ Each numerical approximation for the integral gives a discrete-time equivalent

73 Backward Rectangular Rule kt (k 1)T u(τ)dτ u(kt )T y(kt ) = y(kt T ) + kt (k 1)T [u(τ) ay(τ)]dτ y(kt ) = y(kt T ) + T u(kt ) at y(kt ) Y (z) U(z) = T 1 z 1 + at = 1 ( ) 1 z 1 + a H B (z) = H(s) s=(1 z 1 )/T s 1 z 1, z 1 T 1 T s T

74 Stability Regions Under Backward Rule z 1 2 = T s 2 1 T s Re s < 0 = z 1 2 < 1 2 Im Image of OLHP 0 1 Re

75 Forward Rectangular Rule kt (k 1)T u(τ)dτ u(kt T )T y(kt ) = y(kt T ) + kt (k 1)T [u(τ) ay(τ)]dτ y(kt ) = y(kt T ) + T u(kt T ) at y(kt T ) Y (z) U(z) = T z 1 + at = 1 ( z 1 ) + a H F (z) = H(s) s=(z 1)/T s z 1 T, z 1 + T s T

76 Stability Regions Under Forward Rule z = 1 + T s Re s < 0 = Re z < 1 Re Im 0 1 Image of OLHP

77 Trapezoidal Rule kt (k 1)T u(τ)dτ T [u(kt T ) + u(kt )] 2 y(kt ) = y(kt T ) + kt (k 1)T [u(τ) ay(τ)]dτ y(kt ) = y(kt T ) + T 2 [u(kt T ) + u(kt )] at 2 [y(kt T ) + y(kt )] Y (z) U(z) = 1 ( ) 2 1 z 1 T + a 1+z 1 H T (z) = H(s) s= 2 T 1 z 1 1+z 1 : Tustin s Rule s 2 T 1 z T s/2, z 1 + z 1 1 T s/2

78 Stability Regions Under Tustin s Rule z = 1 + T s/2 1 T s/2 Re s < 0 = z < 1 ORHP Re Im 0 1 OLHP

79 Discrete-Time Equivalent by Impulse Invariance Find Ĥ(z) such that pulse response of Ĥ(z) is the sampled sequence of the impulse response of H(s) H(s) ^ H(z) SAME Ĥ(z) = Z(H(s))

80 Discrete-Time Equivalence by Step Invariance Find H(s) Ĥ(z) such that step response of Ĥ(z) is the sampled sequence of the step response of H(s) H(z) ^ ( ) H(s) Ĥ(z) = (1 z 1 )Z = H h0 (z) s SAME

81 Equivalence at a Frequency When will the steady state response of Ĥ(z) to {cos kωt } equal the sampled sequence of the steady state response of H(s) to cos ωt? H(s) H(z) ^ SAME If and only if H(jω) = Ĥ(ejωT )

82 Tustin s Rule and Equivalence at a Frequency H(s) = a s + a, H T(z) = 2 T a z 1 z+1 + a H(ja) = j, H T(e jat 1 ) = 1 + j 2 at tan at 2 The discrete equivalent does not match the original at the corner frequency Tustin s rule causes frequency distortion Distortion is reduced if at/2 << 1

83 Tustin s Rule with Pre-warping Pre-warp the continuous system such that on applying Tustin s rule, matching is obtained at the selected frequency Substitute Recover Tustin s rule if b = 2/T s = b z 1 z + 1 Same as applying Tustin s rule to the pre-warped transfer function H pre warped (s) = H(bT s/2) Choose b to get matching at the desired frequency

84 Pole-Zero Mapping Equivalent Map all poles of H(s) according to z = e st Map all finite zeros of H(s) by z = e st Map zeros at to zeros at 1 1 s + a 1 1 e at z 1 (s + a) 1 e at z 1 1 s 1 + z 1 To get a strictly causal system, map one s 1 factor to z 1 Choose gain factor to get matching at a specified frequency Usually ω = 0, that is, matching at DC H(jω) = H zp (e jωt )

85 Root Locus r(t) K* H(z) ZOH G(s) y(t) K* H(z) Gh0(s) y(kt) Root locus = locus of roots of 1 + KH(z)G h0 (z) = 0 as K varies fro 0 to Plotted in the same way as for continuous-time systems

86 Mapping Theorem Based on Mapping Theorem z traces a simple closed curve C clockwise in the complex plane The no. clockwise of encirclements of the origin by F (z) equals no. of zeros of F enclosed by C no of poles of F enclosed by C Application to closed-loop stability analysis Choose F (z) = 1 + G(z)H(z) = closed-loop characteristic polynomial Choose C to enclose all possible unstable poles

87 Nyquist Contour Choose C to enclose the exterior of the open unit disc Im ω=ω s /2 OUD Re All encirclements are contributed by portion along the unit circle

88 Nyquist Criterion + G(z) H(z) Nyquist Criterion: Loop transfer function L(z) = G(z)H(z) Z = N + P P = no. of unstable open-loop poles (unstable poles of L(z)) Z = no. of closed-loop unstable poles (unstable roots of 1 + L(z) = 0) N = no. of clockwise encirclements of 1 by L(e jωt ), ω [0, ω s ]

89 Gain and Phase Margins 1 1/GM PM

90 Frequency Response Analysis with W-Transform Frequency response in terms of Z-transform is Periodic in ω Difficult to draw by hand (s-domain rules do not apply) Use W-transform to map OUD into OLHP using w = 2 T (z 1) (z + 1), z = 1 + wt/2 1 wt/2 Ĝ(w) = G(z) z= 1+wT/2 1 wt/2 Bode plots of Ĝ(w) can be drawn using s-domain rules Nyquist criterion can be applied to Ĝ(w) as in s-domain Controller Ĥ designed for Ĝ can be transformed back and applied to G Ĝ and G yield the same gain and phase margins

91 Closed-Loop Asymptotic Tracking of Reference Inputs r(k) e(k) y(k) + H(z) G(z) Asymptotic Tracking: Want E(z) R(z) = G(z)H(z) lim e(k) = 0 k lim k e(k) exists if and only if all poles of E(z) lie in the OUD except possibly for one pole at z = 1 lim k e(k), if it exists, equals lim z 1 (z 1)E(z)

92 Tracking of Step Inputs r(k) = 1, R(z) = E(z) = z z 1 z 1 (z 1) [1 + G(z)H(z)] For lim k e(k) to exist, all closed-loop poles must lie in the OUD lim e(k) = lim k z 1 z 1 + G(z)H(z) = lim z 1 G(z)H(z) For lim k e(k) = 0, the (open) loop transfer function must have a pole at z = 1 No. of poles of G(z)H(z) at z = 1 is the type of the open-loop system Define position error constant For perfect tracking, need K p = K p = lim z 1 G(z)H(z) For perfectly tracking step inputs, need closed-loop stability + type 1 open-loop system

93 Tracking of Ramp Inputs E(z) = r(k) = kt, R(z) = T z (z 1) 2 T z (z 1) 2 1 [1 + G(z)H(z)] = T z (z 1)[z 1 + (z 1)G(z)H(z)] For lim k e(k) to exist, all closed-loop poles must lie in the OUD, and open-loop system must be of type 1 lim e(k) = lim k z 1 T z (z 1)[1 + G(z)H(z)] = T lim z 1 (z 1)G(z)H(z) For lim k e(k) = 0, the (open) loop transfer function must have at least two poles at z = 1 Define velocity error constant For perfect tracking, need K v = K v = lim z 1 (z 1)G(z)H(z)/T For perfectly tracking ramp inputs, need closed-loop stability + type 2 open-loop system

94 Tracking of Sinusoidal Inputs r(k) = A sin(kωt ), R(z) has poles at e ±jωt For e(k) to converge to a steady state behavior, closed-loop must be (BIBO) stable Steady-state error amplitude = G(e jωt )H(e jωt ) For lim k e(k) = 0, G(z)H(z) must have at least one pole at z = e jωt For perfectly tracking {A sin kωt }, need closed-loop stability + open-loop poles at z = e ±jωt

95 Internal Model Principle Requirements for closed-loop tracking Input to be tracked Requirements for tracking Input Input poles Open-loop poles Closed-loop poles Step z = 1 z = 1 OUD Ramp z = 1, 1 z = 1, 1 OUD Sinusoidal z = e ±jωt z = e ±jωt OUD Internal Model Principle: A closed-loop system will track an input perfectly asymptotically if and only if The closed-loop system is stable and The open-loop system contains a model of the input

96 Continuous-Time LTI State-Space Systems ẋ = Ax + Bu, y = Cx + Du, state dynamics measurement equation x = vector of state/internal variables y = vector of output measurements u = vector of inputs Linear: A, B, C, D independent of x, y Time Invariant: A, B, C, D independent of time To find output, need Initial state vector and input

97 Matrix Exponential Given a square matrix A, define e At def = I + At + 1 2! A2 t ! A3 t 3 + Well defined (series converges sufficiently nicely) Satisfies d e A0 dt eat e A(t+τ) = I = Ae At = e At A = e At e Aτ (e At ) 1 = e At Note: (e At ) ij e A ijt

98 Examples ÿ + ω 2 ny = 1 m u d ẏ = 0 1 ẏ + 1 k u dt y ωn 2 0 y 0 }{{}}{{} A B e At = cos ω 1 nt ω n sin ω n t ω n sin ω n t cos ω n t ÿ = u A = 0 1, e At = t 0 1

99 Solution of the State Equation State response Output response Impulse response Transfer matrix y(t) = x(t) = t e At x(0) + e }{{} A(t τ) Bu(τ)dτ 0 Natural Response }{{} Forced Response e At = state transition matrix t Ce At x(0) + Ce }{{} A(t τ) Bu(τ)dτ + Du(t) 0 Natural Response }{{} g(t) = Ce At B + Dδ(t) C(sI A) 1 B + D Forced Response

100 ZOH Equivalent of a State Space System u(kt) ZOH u(t) State Space System y(t) y(kt) x(k + 1) = e AT x(k) + kt [ T = e AT x(k) + u(t) = u(k), t [kt, kt + T ) kt +T 0 e A(kT +T τ) Bu(k)dτ ] e A(T σ) Bdσ u(k), σ = τ kt Discrete-time state space model of the ZOH equivalent x(k + 1) = Φx(k) + Γu(k) y(k) = Hx(k) + Ju(k) Φ = e AT, Γ = [ T 0 ] e A(T σ) Bdσ, H = C, J = D

101 State Space Realizations G(z) = b 0 + b 1 z 1 + b n z n 1 + a 1 z a n z n Let Ŷ (z) = U(z)[1 + a 1z a n z n ] 1, so that Y (z) = [b 0 + b 1 z 1 + b n z n ]Ŷ (z) ŷ(k) + a 1 ŷ(k 1) + + a n ŷ(k n) = u(k) Choose x(k) = ŷ(k n) ŷ(k n + 1). ŷ(k 1), x(k+1) = ŷ(k n + 1) ŷ(k n + 2). ŷ(k) = x 2 (k) x 3 (k). a n x 1 (k) a 1 x n (k) + u(k) y(k) = b 0 ŷ(k) + b 1 ŷ(k 1) + + b n ŷ(k n) = b 0 ŷ(k) + b 1 x n (k) + + b n x 1 (k) = (b n b 0 a n )x 1 (k) + (b n 1 b 0 a n 1 )x 2 (k) + + (b 1 b 0 a 1 )x n (k) + b 0 u(k)

102 State Space Realizations (cont d) First companion form x(k + 1) = y(k) = x(k) +. u(k) } a n a n 1 a n 2 {{ a 1 } 1 }{{} [ A ] B (b n b 0 a n ) (b 1 b 0 a 1 ) x(k) + [b 0 ] u(k) }{{}}{{} D C

103 Transfer Matrix from State Space Model State dynamics equations in Laplace domain Poles are eigenvalues of A zx(z) zx(0) = AX(z) + BU(z) X(z) = z(zi A) 1 x(0) }{{} Natural response Y (z) = CX(z) + DU(z) + (zi A) 1 BU(z) }{{} Forced response = Cz(zI A) 1 x(0) + [C(zI A) 1 B + D] U(s) }{{} Transfer matrix Transfer matrix = C(zI A) 1 B + D

104 Transformations of State Space Models State transformation ˆx = Sx Input-output relation is unchanged x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k) ˆx(k + 1) = SAS 1 }{{} Â y(k) = CS 1 }{{} Ĉ ˆx(k) + }{{} SB u(k) B ˆx(k) + }{{} D u(k) D C(zI A) 1 B + D = Ĉ(zI Â) 1 B + D Two state-space models are equivalent if they yield the same transfer matrix Every input-output system has several equivalent state space representations/realizations

105 State Evolution in Discrete-Time System x(k + 1) = Ax(k) + Bu(k) x(1) = Ax(0) + Bu(0) x(2) = A 2 x(0) + ABu(0) + Bu(1) x(3) = A 3 x(0) + A 2 Bu(0) + ABu(1) + Bu(2). x(k) = y(k) = A k x(0) + }{{} natural response CA k x(0) + }{{} natural response k 1 l=0 A k l 1 Bu(l) } {{ } forced response k 1 l=0 CA k l 1 Bu(l) + Du(k) } {{ } forced response

106 Impulse Response Response to impulse input u(k) = u 0 δ(k) y(k) = CA k 1 Bu 0 + Du 0 δ(k) Impulse response sequence = {Du 0, CBu 0, CABu 0,...} Impulse response matrix General response H(k) = D, k = 0 = CA k 1 B, k > 0 y(k) = CA k x(0) + (H u)(k) Compare with forced response in z-domain Y (z) = [C(zI A) 1 B + D]U(z) = Z(H) = C(zI A) 1 B + D = D + z 1 CB + z 2 CAB +

107 Reachable Sets Which states can be reached from a given initial condition by using all possible inputs? Reachable set from x 0 at step k R(k, x 0 ) = {states reachable from x 0 in k steps} { } k 1 = A k x 0 + A k l 1 Bu(l) : u(0), u(1),..., u(k 1) arbitrary l=0 System is controllable if every state can be reached from every other state in a finite (but possibly large) number of steps System is controllable iff R(k, x 0 ) = R n for sufficiently large k

108 Facts on Reachable Sets Fact 1 : R(k, x 0 ) = R(k, 0) + A k x 0 If a R(n, 0), then Fact 2 : R(n, 0) = Range C, C = [B AB A 2 B A n 1 B] a = Bu(n 1) + ABu(n 2) + + A n 1 Bu(0) = C If a Range C, then u(n 1). u(0) a = Cb = Bb 1 + ABb A n 1 Bb n R(n, 0) Range C

109 Facts on Reachable Sets (cont d) Fact 3 : R(k, 0) = R(n, 0), k n Clearly R(n, 0) R(k, 0) For k > n, an element of R(k, 0) is of the form Bu(k 1) + + A n 1 Bu(k n) + A n Bu(k n + 1) + + A k 1 Bu(0) By Cayley-Hamilton theorem, powers of A higher than n 1 can be written as combinations of powers of A upto n 1 Elements of R(k, 0) are contained in R(n, 0) System is controllable iff rank C = n

110 Unobservable Sets Can we guess the initial state by observing only the output? y 1 (k) = CA k x 1 + (H u)(k) y 2 (k) = CA k x 2 + (H u)(k) Can distinguish x 1 from x 2 iff CA k x 1 CA k x 2 for some k Unobservable set from x 0 at step k U(k, x 0 ) = {states that yield the same output as x 0 upto step k 1} = {x : CA i x = CA i x 0, i = 0, 1,, k 1} System is observable if every state can be distinguished from every other state in a finite (but possibly large) number of steps System is observable iff U(k, x 0 ) = {x 0 } for sufficiently large k

111 Facts on Unobservable Sets Fact 1 : U(k, x 0 ) = U(k, 0) + x 0 Fact 2 : U(k, 0) = U(n, 0), k n Fact 3 : U(n, 0) = kernel O, O = C CA. CA n 1 System is observable iff rank O = n

112 Hautus Test for Controllability Eigenvalue λ C of A is controllable if rank [λi A B] = n Fact: System is controllable iff every eigenvalue of A is controllable If λ is not controllable, then there exists x C n such that x A = λx, x B = 0 = x A i B = 0 = x C = 0 = rank C < n Controllability remains invariant under state transformation Uncontrollable eigenvalues are unaffected by control If λ is an uncontrollable eigenvalue, and the feedback u = Kx is used, then λ also appears as a closed-loop eigenvalue

113 Hautus Test for Observability Eigenvalue λ C of A is unobservable if rank λi A C = n Fact: System is observable iff every eigenvalue of A is observable If λ is not observable, then there exists x C n such that Ax = λx, Cx = 0 = CA i x = 0 = Ox = 0 = rank O < n Observability remains unchanged under state transformations Unobservable eigenvalues cannot be detected through the output

114 A Two-Dimensional Example A = C = λ 1 0, B = 0 λ 2 [ ] c 1 c 2 b 1 b 2 u b b 1 1 (z λ c 1 ) (z ) λ 2 1 c 2 + y By the Hautus test, need b 1 0 b 2 for controllability and c 1 0 c 2 for observability

115 Kalman Decomposition Theorem Fact: Every state space system can be transformed into A co A 12 A 13 x co (k) x(k + 1) = 0 A co A 23 x co (k) A c x c (k) [ ] y(k) = 0 C co C c x(k) + Du(k) B co B co 0 u(k) G(s) = C(zI A) 1 B + D = C co (zi A co ) 1 B co + D The controllable and observable part yields a smaller realization Eigenvalues that are either unobservable or uncontrollable are not poles

116 Minimal Realizations A minimal realization is one having the least no. of states Desirable for implementation A minimal realization has as many states as the number of poles A realization is minimal iff it is controllable and observable All minimal realizations are equivalent

117 Jordan Form Every matrix can be reduced to its Jordan form through a similarity transformation λ λ λ λ λ λ λ λ λ 4 characteristic polynomial = (z λ 1 ) 3 (z λ 2 ) 3 (z λ 3 ) 2 (z λ 4 ) λ 1 and λ 4 have 1 eigenvector each, λ 2 and λ 3 have 2 eigenvectors each λ 3 and λ 4 are semisimple, λ 4 is simple

118 Internal Stability Internal stability refers to the natural response of state (internal) variables A state space system is (internally) Lyapunov stable if every initial condition response is bounded Asymptotically stable if every initial condition response decays to zero Unstable if it is not Lyapunov stable Stability depends on the elements of J k x(k) = A k x(0) = T J k T 1 x(0), J = Jordan form

119 Powers of Jordan Blocks System is Lyapunov stable iff J = λ 0 = J k = λk 0 0 λ 0 λ k J = λ 1 = J k = λk kλ k 1 0 λ 0 λ k λ 1 0 λ k kλ k 1 k(k 1) 2 λ k 2 J = 0 λ 1 = J k = 0 λ k kλ k λ 0 0 λ k All eigenvalues CUD and All eigenvalues of unit magnitude are semisimple System is asymptotically stable iff all eigenvalues OUD

120 BIBO Stability and Internal Stability System is BIBO stable iff every input vector with bounded components gives an output vector with bounded components System is BIBO stable if and only if every pole OUD (internal) asymptotic stability = BIBO stability Converse does not hold in general Fact: A controllable, observable, BIBO stable system is asymptotically stable Fact: A system is BIBO stable iff every minimal realization is asymptotically stable

121 Positive-Definite Matrices P R n n, symmetric, is positive-definite (P > 0) if x T P x > 0 for every x C n, x 0 A symmetric positive-definite matrix has real eigenvalues that are positive Every symmetric positive-definite matrix gives rise to the quadratic function V P (x) = x T P x If P > 0, then the level sets of V P are hyper-ellipsoids, eg. P = x x 1 Level curves of V P

122 Lyapunov Function How does a given quadratic function change along the natural state response? x(k + 1) = Ax(k) V P (x(k)) = x T (k)p x(k) V P (x(k + 1)) = x T (k + 1)P x(k + 1) = x T (k)a T P Ax(k) V P (x(k + 1)) V P (x(k)) = x T (k)[a T P A P ]x(k) Idea: If P is positive definite and V P (x(k)) decreases with k, then x(k) 0 Such a V P is called a Lyapunov function Want P > 0 and A T P A P = Q, where Q > 0

123 Lyapunov Equation Fact: If there exist P > 0 and Q > 0 satisfying the Lyapunov equation below, then system is asymptotically stable A T P A P = Q Fact: System is asymptotically stable iff for every Q > 0, there exists a positive-definite solution P to the Lyapunov equation For an asymptotically stable system, the solution P is unique To prove stability or instability, pick Q > 0 (eg. definiteness of P Q = I), solve for P and check sign OR check the feasibility of the linear matrix inequalities (LMIs) A T P A + P > 0 P > 0 Can be done using efficient numerical algorithms

124 Full-State Feedback x(k + 1) = Ax(k) + Bu(k) Open loop system u(k) = Kx(k) + r(k) Full state feedback x(k + 1) = (A BK)x(k) + Br(k) Closed loop system Pole-placement problem Can we design a gain matrix K such that A BK has desired eigenvalues? Fact: Every uncontrollable open-loop eigenvalue is a closed-loop eigenvalue Assume Complete controllability Single input

125 Pole Placement using Companion Form Idea: Use state transformation ˆx = Sx such that  = SAS 1 = a n a n 1 a n 2 a 1, B = SB = }{{} controller canonical form Use feedback u = K ˆx = [ˆk n ˆkn 1 ˆk 1 ]ˆx  B K = a n ˆk n a n 1 ˆk n 1 a n 2 ˆk n 2 a 1 ˆk 1

126 Pole Placement using Companion Form (cont d) Characteristic polynomial of A and  z n + a 1 z n a n Characteristic polynomial of  B K z n + (a 1 + ˆk 1 )z n (a n + ˆk n ) Desired characteristic polynomial z n + α 1 z n α n Choose K = [α n a n α n 1 a n 1 α 1 a 1 ] Feedback in terms of original states Ackermann s Formula: u = K ˆx = }{{} KS x K K = e T nc 1 [A n + α 1 A n α n I] e T n = [0 0 1]

127 Proof of Ackermann s Formula Define s 1 = e T nc 1, and consider S = s 1 s 1 A. s 1 A n 1 Claim: SA = ÂS Claim: S is invertible Claim: SB = B S yields the transformation to the controller canonical form K = KS yields Ackermann s formula Multi-input case: redundant degrees of freedom in choosing K Can be used to assign eigenstructure

128 Estimation/Observation Online reconstruction of state vector from the output x(k + 1) = Ax(k) + Bu(k) } y(k) = Cx(k) {{ } Observable single output system output injection {}}{ L(y(k) ŷ(k)), x(k + 1) = A x(k) + Bu(k) + ŷ(k) = C ˆx(k) }{{} Luenberger observer u A, B x C A, B x^ C y y^ + O b s e r v L e r Error dynamics e(k + 1) = (A LC)e(k), e = x x

129 Observer Design by Pole Placement Can we choose observer gain matrix L such that A LC is Schur? Yes, if the system is observable (A, C) is observable iff (A T, C T ) is controllable There exists a gain matrix K such that A T C T K has desired eigenvalues K = e T n[c T C T A T C T A (n 1)T ] 1 [A nt + α 1 A (n 1)T + + α n I] Letting L = K T, A LC has desired eigenvalues L = [A n + α 1 A n α n I]O 1 e n

130 Dynamic Output-Feedback Compensation Idea: In a full-state feedback controller, use state estimate generated by an observer in place of the actual state r + u x A, B C y K + x^ A, B C y^ Dynamic output feedback controller L

131 Seperation Principle x(k + 1) = Ax(k) + Bu(k) x(k + 1) = A x(k) + Bu(k) + L(y(k) C x(k)) y(k) = Cx(k) u(k) = K ˆx(k) + r(k) x(k + 1) = A BK x(k) + B r(k) x(k + 1) LC A LC BK x(k) B }{{} Closed loop system Choose state vector as [x T e T ] T, e = x x x(k + 1) = A BK BK x(k) + B e(k + 1) 0 A LC e(k) 0 r(k) Seperation Principle: Output-feedback controller can be obtained by combining an independently designed Regulator that uses full state feedback with An observer