Systems & Control Fundamentals

Systems & Control Fundamentals Gianmaria De Tommasi1 1 Associazione EURATOM/ENEA/CREATE DIS – Università degli Studi di Napoli Federico II detommas@unina.it November 20th 2009 GOTiT High Level Course Magnetic Control of Tokamak Plasmas 4th G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 1 / 62 Outline 1 Preliminaries Linear Systems Transfer function Block diagrams Frequency response Graphical representations of G(jω) 2 Feedback Control Systems The control problem Stability margins Nyquist Criterion Root locus G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 2 / 62 Preliminaries Linear Systems Dynamical Systems State-space model A finite dimensional continuous-time dynamical system can be described by the following differential equations:
ẋ(t) = f x(t) , u(t) , t , t0 , x(t0 ) = x0
y (t) = η x(t) , u(t) , t , t0 (1a) (1b) where: x(t) ∈ Rn is the system state x(t0 ) ∈ Rn is the initial condition u(t) ∈ Rm is the input vector y (t) ∈ Rp is the output vector G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 3 / 62 Preliminaries Linear Systems Linear time-invariant systems A linear time-invariant (LTI) continuous-time system is described by ẋ(t) = Ax(t) + Bu(t) , x(0) = x0 (2a) y (t) = Cx(t) + Du(t) (2b) where A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n and D ∈ Rp×m . A dynamical system with single-input (m = 1) and single-output (p = 1) is called SISO, otherwise it is called MIMO. Matlab commands sys = ss(A,B,C,D) creates a state-space model object. y = lsim(sys,u,t) simulates the the time response of the LTI system sys. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 4 / 62 Preliminaries Linear Systems Equilibria of nonlinear dynamical systems Consider a nonlinear and time-invariant system
ẋ(t) = f x(t) , u(t) , x(0) = x0
y (t) = η x(t) , u(t) (3a) (3b) If the input is constant, i.e. u(t) = ū, then the equilibria xe1 , xe2 , . . . , xeq of such a system can be computed as solutions of the homogeneous equation f xe , ū) = 0 , Given an equilibrium xei the correspondent output is given by
yei = η xei , ū . G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 5 / 62 Preliminaries Linear Systems Linearization around a given equilibrium If x0 = xe + δx0 and u(t) = ū + δu(t), with δx0 , δu(t) sufficiently
small, then the behaviour of (3) around a given equilibrium ū , xe is well described by the linear system δ ẋ(t) = δy (t) = ∂f ∂f δx(t) + δu(t) , x = x ∂x ∂u x = xe e u = ū u = ū δx(0) = δx0 ∂η ∂η δx(t) + δu(t) x = x ∂x ∂u x = xe e u = ū u = ū (4a) (4b) The total output can be computed as
y (t) = η xe , ū + δy (t) . G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 6 / 62 Preliminaries Linear Systems Example - Pendulum Mass m Length L Friction coefficient b G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 7 / 62 Preliminaries Linear Systems Example - Pendulum Nonlinear model Let x(t) =  θ(t) θ̇(t)  u(t) = F (t) y (t) = θ(t) then ẋ1 (t) = x2 (t) b 1 g cos x1 (t)u(t) x2 (t) + ẋ2 (t) = − sin x1 (t) − 2 L mL mL y (t) = x1 (t) G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 8 / 62 Preliminaries Linear Systems Example - Pendulum Equilibria
If ū = mg, solving f xe , ū = 0 we get  π  4 + kπ xek = 0 G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 9 / 62 Preliminaries Linear Systems Example - Pendulum
T Around the equilibria xe = π4 0 the behaviour of the pendulum is well described by the linear system δ ẋ1 (t) = δx2 (t) √ 2g b 1 δ ẋ2 (t) = − δx1 (t) − δx2 (t) + √ δu(t) 2 L mL 2mL δy (t) = δx1 (t) G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 10 / 62 Preliminaries Linear Systems The script pendulum.m - 1 % System parameters m = 10; b = 15; g = 9.81; L = 1; % Equilibria xbar = [pi/4 0]; ubar = m*g; % Output at the equilibria ybar = xbar(1); % Variation of the initial conditions dx0 = [0.3 3]; % Input variation: Am*sin(t) Am = 10; % Simulation time interval tfin = 15; G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 11 / 62 Preliminaries Linear Systems The script pendulum.m - 2 % Linearized model matrices A = [ 0 1 ; -sqrt(2)*g/L -b/(m*Lˆ2) ]; B = [ 0 ; 1/(sqrt(2)*m*L) ]; C = [ 1 0 ]; D = 0; % Linearized model sys_l = ss(A,B,C,D); % Time vector tlin = 0:.001:tfin; % Linear simulation du = Am*sin(tlin); ylin= lsim(sys_l,du,tlin,dx0); G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 12 / 62 Preliminaries Linear Systems The script pendulum.m - 3 % Nonlinear simulation [ t,x_nl ]=ode45(’p_nl’,[0 tfin],xbar+dx0,[],m,b,g,L,ubar,Am); y_nl = x_nl(:,1); % Plots figure(1) plot(t,y_nl*180/pi,’-’,tlin,(ylin+ybar)*180/pi,’-’) grid on ylabel(’[deg]’) xlabel(’tempo [s]’) title(’theta’) legend(’NL’,’L’) G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 13 / 62 Preliminaries Linear Systems The function p_nl.m function xdot = p_nl(t,x,flag,m,b,g,L,ubar,Am) % % Returns the state derivative % u = ubar + Am*sin(t); xdot = [x(2); -g/L*sin(x(1)) - b/(m*Lˆ2)*x(2) +... u/(m*L)*cos(x(1))]; Download Matlab example G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 14 / 62 Preliminaries Linear Systems Asymptotic stability of LTI systems Asymptotic stability This property roughly asserts that every solution of ẋ(t) = Ax(t) tends to zero as t → ∞. Note that for LTI systems the stability property is related to the system and not to a specific equilibrium. Theorem - System (2) is asymptotically stable iff A is Hurwitz, that is if every eigenvalue λi of A has strictly negative real part
ℜ λi < 0 , ∀ λi . Theorem - System (2) is unstable if A has at least one eigenvalue λ̄ with strictly positive real part, that is
∃ λ̄ s.t. ℜ λ̄ > 0 .
Theorem - Suppose that A has all eigenvalues λi such that ℜ λi ≤ 0, then system (2) is
unstable if there is at least one eigenvalue λ̄ such that ℜ λ̄ = 0 which corresponds to a Jordan block with size > 1. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 15 / 62 Preliminaries Linear Systems Equilibrium stability for nonlinear systems For nonlinear system the stability property is related to the specific equilibrium. Theorem - The equilibrium state xe corresponding to the constant input ū a nonlinear system (3) is asymptotically stable if all the eigenvalues of the correspondent linearized system (4) have strictly negative real part. Theorem - The equilibrium state xe corresponding to the constant input ū a nonlinear system (3) is unstable if there exists at least one eigenvalue of the correspondent linearized system (4) which has strictly positive real part. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 16 / 62 Preliminaries Transfer function Transfer function of LTI systems Given a LTI system (2) the corresponding transfer matrix from u to y is defined as Y (s) = G(s)U(s) , with s ∈ C. U(s) and Y (s) are the Laplace transforms of u(t) and y (t) with zero initial condition (x(0) = 0), and
−1 B+D. G(s) = C sI − A (5) For SISO system (5) is called transfer function and it is equal to the Laplace transform of the impulsive response of system (2) with zero initial condition. Matlab commands sys = tf(num,den) creates a transfer function object. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 17 / 62 Preliminaries Transfer function Transfer function Given the transfer function G(s) and the Laplace transform of the input U(s) the time response of the system can be computed as the inverse transform of G(s)U(s), without solving differential equations. As an example, the step response of a system can be computed as:   1 −1 G(s) . y (t) = L s Matlab commands [y,t] = step(sys) computes the step response of the LTI system sys. [y,t] = impulse(sys) computes the impulse response of the LTI system sys. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 18 / 62 Preliminaries Transfer function Poles and zeros of SISO systems Given a SISO LTI system , its transfer function is a rational function of s G(s) = Πi (s − zi ) N(s) =ρ , D(s) Πj (s − pj ) where N(s)
and D(s) are
polynomial in s, with deg N(s) ≤ deg D(s) . We call pj poles of G(s) zi zeros of G(s) Matlab commands sys = zpk(z,p,k) creates a zeros-poles-gain object. p = eig(sys) or p = pole(sys) return the poles of the LTI system sys. z = zero(sys) returns the zeros of the LTI system sys. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 19 / 62 Preliminaries Transfer function Poles and eigenvalues of a LTI system Every pole of G(s) is an eigenvalue of the system matrix A. However, not every eigenvalue of A is a pole of G(s). If all the poles of G(s) have strictly negative real part – i.e. they are located in the left half of the s-plane (LHP) – the SISO system is said to be Bounded–Input Bounded–Output stable (BIBO). A system is BIBO stable if every bounded input to the system results in a bounded output over the time interval [0, ∞). G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 20 / 62 Preliminaries Transfer function Time constants, natural frequencies and damping factors A transfer function can be also specified in terms of time constants (τ ,T ) natural frequencies (ωn ,αn ) damping factors (ξ,ζ) gain (µ) system type (i.e. number of poles/zeros in 0, g)   2 ζ Πi (1 + Ti s)Πj 1 + 2 αnj s + αsn j j .  G(s) = µ ξl s2 g s Πk (1 + τk s)Πl 1 + 2 ωn s + ωn l G. De Tommasi (CREATE) Systems & Control Fundamentals l GOTiT - Nov. 2009 21 / 62 Preliminaries Block diagrams Block diagrams When dealing with transfer functions, it is usual to resort to Block diagrams which permit to graphically represent the interconnections between system in a convenient way. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 22 / 62 Preliminaries Block diagrams Series connection Matlab commands sys = series(sys1,sys2) makes the series interconnection between sys1 and sys2. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 23 / 62 Preliminaries Block diagrams Parallel connection Matlab commands sys = parallel(sys1,sys2) makes the parallel interconnection between sys1 and sys2. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 24 / 62 Preliminaries Block diagrams Feedback connection Matlab commands sys = feedback(sys1,sys2,[+1]) makes the feedback interconnection between sys1 and sys2. Negative feedback is the default. If the third parameter is equal to +1 positive feedback is applied. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 25 / 62 Preliminaries Block diagrams Stability of interconnected systems Given two asymptotically stable LTI systems G1 (s) and G2 (s) the series connection G2 (s)G1 (s) is asymptotically stable the parallel connection G1 (s) + G2 (s) is asymptotically stable the feedback connection G. De Tommasi (CREATE) G1 (s) 1±G1 (s)G2 (s) is not necessarely stable Systems & Control Fundamentals GOTiT - Nov. 2009 26 / 62 Preliminaries Frequency response Frequency response Given a LTI system the complex function
−1 G(jω) = C jωI − A B+D, with ω ∈ R is called frequency response of the system. G(jω) permits to evaluate the system steady-state response to a sinusoidal input. In particular if u(t) = A sin(ω̄t + ϕ) , then the steady-state response of a LTI system is given by   y (t) = G(j ω̄) A sin ω̄t + ϕ + ∠G(j ω̄) . G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 27 / 62 Preliminaries Graphical representations of G(jω) Bode plot Given a LTI system G(s) the Bode diagrams plot the magnitude of G(jω) (in dB, G(jω) dB = 20 log10 G(jω) ) and the phase of G(jω) (in degree) as a function of ω (in rad/s) in a semi-log scale (base 10). Bode plots are used for both analysis and synthesis of control systems. Matlab commands bode(sys) plots the the Bode diagrams of the LTI system sys. bodemag(sys) plots the Bode magnitude diagram of the LTI system sys. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 28 / 62 Preliminaries Graphical representations of G(jω) Bode plot - Example Let consider 1+s 1+s  = 10 G(s) = 10  2 s s(0.0025s2 + 0.03s + 1) s 400 + 2 0.3 20 s + 1 Matlab commands s = tf(’s’); sys = 10*(1+s)/(s*(sˆ2/400+0.6*s/20+1)); bode(sys); grid G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 29 / 62 Preliminaries Graphical representations of G(jω) Bode plot - Example G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 30 / 62 Preliminaries Graphical representations of G(jω) Minimum phase systems A stable system is said to be a minimum phase system if it has not time delays or right-half plane (RHP) zeros. For minimum phase systems there is a unique relationship between the gain and phase of the frequency response G(jω). This may be quantified by the Bode’s gain-phase relationship Z 1 +∞ d ln |G(jω)| ω + ω̄ dω ln . ∠G(j ω̄) = π −∞ d ln ω ω − ω̄ ω The name minimum phase refers to the fact that such a system has the minimum possible phase lag for the given magnitude response |G(jω)|. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 31 / 62 Preliminaries Graphical representations of G(jω) Nyquist plot The Nyquist is a polar plot of the frequency response G(jω) on the complex plane. This plot combines the two Bode plots - magnitude and phase - on a single graph, with frequency ω, which ranges in (−∞ , +∞), as a parameter along the curve. Nyquist plots are useful to check stability of closed-loop systems (see Nyquist stability criterion ). Matlab commands nyquist(sys) plots the Nyquist plot of the LTI system sys. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 32 / 62 Preliminaries Graphical representations of G(jω) Nyquist plot - Example G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 33 / 62 Preliminaries Graphical representations of G(jω) Nichols plot It is similar to the Nyquist plot, since it plots both the magnitude and the phase of G(jω) on a single chart, with frequency ω as a parameter along the curve. As for the Bode plot the magnitude |G(jω)| is expressed in dB and the phase ∠G(jω) in degree. Nichols charts are useful for the design of control systems, in particular for the design of lead, lag, lead-lag compensators. Matlab commands nichols(sys) plots the Nichols chart of the LTI system sys. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 34 / 62 Preliminaries Graphical representations of G(jω) Nichols plot - Example G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 35 / 62 Feedback Control Systems The control problem The control problem The objective of a control system is to make the output of a plant y (t) behave in a desired way by manipulating the plant input u(t). A good controller should manipulate u(t) so as to counteract the effect of a disturbance d(t) (regulator problem) keep the output close to a given reference input r (t) (servo problem) In both cases we want the control error e(t) = y (t) − r (t) to be small. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 36 / 62 Feedback Control Systems The control problem Why feedback ? A major sources of difficulty are that 1 the plant model G(s) and the disturbance model Gd (s) may be affected by uncertainty and/or may change with time 2 the disturbance is not always measurable 3 the plant can be unstable It turns out that e feed-forward approach is not robust enough and/or is not always a viable solution. A feedback approach can guarantee the desired degree of robustness. However design a feedback control system is not straightforward: instability is around the corner! G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 37 / 62 Feedback Control Systems The control problem Performance and stability A good controller must guarantee: Nominal stability - The system is stable with no model uncertainty Nominal Performance - The system satisfies the performance specifications with no model uncertainty Robust stability The system is stable for all perturbed plants about the nominal model up to the worst case model uncertainty Robust performance The system satisfies the performance specifications for all perturbed plants about the nominal model up to the worst case model uncertainty G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 38 / 62 Feedback Control Systems The control problem One degree-of-freedom controller The input to the plant is given by
U(s) = K (s) R(s) − Y (s) − N(s) . The objective of control is to manipulate design a controller K (s) such that the control error e(t) = r (t) − y (t) remains small in spite of disturbances d(t). G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 39 / 62 Feedback Control Systems The control problem Terminology and Notation L(s) = G(s)K (s) is called loop transfer function
−1 S(s) = I + L(s) is called sensitivity function
−1 T (s) = I + L(s) L(s) is called complementary sensitivity function It is straightforward to note that T (s) + S(s) = I . G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 40 / 62 Feedback Control Systems The control problem One DOF controller Exploiting the composition rules for block diagrams, it turns out that Y (s) = T · R(s) + SGd · D(s) − T · N(s) (6a) U(s) = KS · R(s) − K (s)S(s)Gd · D(s) − KS · N(s) (6c) E(s) = −S · R(s) + SGd · D(s) − T · N(s) G. De Tommasi (CREATE) Systems & Control Fundamentals (6b) GOTiT - Nov. 2009 41 / 62 Feedback Control Systems The control problem One DOF controller Remark - S(s) is called sensitivity because it gives the relative sensitivity of the closed-loop transfer function T (s) to the relative plant model error. In particular, at a given frequency ω we have for a SISO plant that dT T dG G = S. Remark - Equations (6) are written in matrix form because they apply to MIMO systems. For SISO systems we may write G. De Tommasi (CREATE) S(s) = 1 , 1 + L(s) T (s) = L(s) . 1 + L(s) Systems & Control Fundamentals GOTiT - Nov. 2009 42 / 62 Feedback Control Systems The control problem The control dilemma Let consider Y (s) = T · R(s) + SGd · D(s) − T · N(s) . In order to reduce the effect of the disturbance d(t) on the output y (t), the sensitivity function S(s) should be made small (particularly in the low frequency range) In order to reduce the effect of the measurement noise n(t) on the output y (t), the complementary sensitivity function T (s) should be made small (particularly in the high frequency range) However, for all frequencies it is T +S =I. Thus a trade-off solution must be achieved. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 43 / 62 Feedback Control Systems The control problem Feedback may cause instability One of the main issues in designing feedback controllers is stability. If the feedback gain is too large then the controller may overreact and the closed-loop system becomes unstable. Download Simulink example G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 44 / 62 Feedback Control Systems The control problem Bandwidth and crossover frequency Usually the frequency response of loop transfer function |L(jω)| has a low-pass behaviour. The crossover frequency ωc is the frequency such that |L(jωc )| = 1. In most of the cases the crossover frequency is a good estimation of the closed-loop bandwidth G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 45 / 62 Feedback Control Systems Stability margins Gain and phase margin The frequency response of the loop transfer function L(jω) can be used to estimate the stability margins. Gain margin (GM) 1/|L(jω180 | , where ω180 is the phase corssover frequency. Phase margin (PM) ∠L(jωc ) + 180◦ , where ωc is the crossover frequency. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 46 / 62 Feedback Control Systems Stability margins Gain Margin The GM is the factor by which the loop gain |L(jω)| may be increased before the closed-loop system becomes unstable. The GM is thus a direct safeguard against steady-state gain uncertainty. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 47 / 62 Feedback Control Systems Stability margins Phase Margin The phase margin tells how much phase lag can added to L(s) at frequency ωc before the phase at this frequency becomes 180◦ which corresponds to closed-loop instability (see Nyquist stability criterion ). The PM is a direct safeguard against time delay uncertainty. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 48 / 62 Feedback Control Systems Nyquist Criterion Nyquist Criterion - Preliminaries The Nyquist Criterion permits to check the stability of a closed loop system by using the Nyquist plot of the loop frequency response L(jω). The criterion is based on the fact the the close-loop poles are equal to the zeros of the transfer function D(s) = 1 + L(s) . Hence, if D(s) has at least one zero z̄ such that ℜ(z̄) > 0 the closed-loop system is unstable. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 49 / 62 Feedback Control Systems Nyquist Criterion Nyquist Criterion Consider a loop frequency response L(jω) and let P be the number of poles of L(s) with strictly positive real part Z be the number of zeros of L(s) with strictly positive real part The Nyquist plot of L(jω) makes a number of encirclements N (clockwise) about the point (−1 , j0) equal to N =Z −P. It turns out that the closed-loop system is asymptotically if and only if the Nyquist plot of L(jω) encircle (counter clockwise) the point (−1 , j0) a number of times equal to P. The criterion is valid if the Nyquist plot of L(jω) do not intersect the point (−1 , j0). G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 50 / 62 Feedback Control Systems Nyquist Criterion Nyquist plot - Example G(s) = 1 , s3 + 3s2 + 3s + 1 K (s) = Kp 1+s . s Download Simulink example G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 51 / 62 Feedback Control Systems Nyquist Criterion Nyquist Criterion - Remarks 1 If the loop transfer function L(s) has a zero pole of multiplicity l, then the Nyquist plot has a discontinuity at ω = 0. Further analysis indicates that the zero poles should be neglected, hence if there are no other unstable poles, then the loop transfer function L(s) should be considered stable, i.e. P = 0. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 52 / 62 Feedback Control Systems Nyquist Criterion Nyquist Criterion - Remarks 1 2 If the loop transfer function L(s) has a zero pole of multiplicity l, then the Nyquist plot has a discontinuity at ω = 0. Further analysis indicates that the zero poles should be neglected, hence if there are no other unstable poles, then the loop transfer function L(s) should be considered stable, i.e. P = 0. If the loop transfer function L(s) is stable, then the closed-loop system is unstable for any encirclement (clockwise) of the point -1. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 52 / 62 Feedback Control Systems Nyquist Criterion Nyquist Criterion - Remarks 1 2 3 If the loop transfer function L(s) has a zero pole of multiplicity l, then the Nyquist plot has a discontinuity at ω = 0. Further analysis indicates that the zero poles should be neglected, hence if there are no other unstable poles, then the loop transfer function L(s) should be considered stable, i.e. P = 0. If the loop transfer function L(s) is stable, then the closed-loop system is unstable for any encirclement (clockwise) of the point -1. If the loop transfer function L(s) is unstable, then there must be one counter clockwise encirclement of -1 for each pole of L(s) in the right-half of the complex plane. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 52 / 62 Feedback Control Systems Nyquist Criterion Nyquist Criterion - Remarks 1 2 3 4 If the loop transfer function L(s) has a zero pole of multiplicity l, then the Nyquist plot has a discontinuity at ω = 0. Further analysis indicates that the zero poles should be neglected, hence if there are no other unstable poles, then the loop transfer function L(s) should be considered stable, i.e. P = 0. If the loop transfer function L(s) is stable, then the closed-loop system is unstable for any encirclement (clockwise) of the point -1. If the loop transfer function L(s) is unstable, then there must be one counter clockwise encirclement of -1 for each pole of L(s) in the right-half of the complex plane. If the Nyquist plot of L(jω) intersect the point (−1 , j0), then deciding upon even the marginal stability of the system becomes difficult and the only conclusion that can be drawn from the graph is that there exist zeros on the imaginary axis. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 52 / 62 Feedback Control Systems Root locus Location of the poles of a closed-loop system The time behaviour of a closed-loop system is strictly related to the position of its poles on the complex plane. For example, for a second order closed-loop system it is possible to relate the features of the step response such as rise time overshoot settling time to the location of its poles. The Root Locus design method permits to evaluate how changes in the loop transfer function L(s) affect the position of the closed-loop poles. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 53 / 62 Feedback Control Systems Root locus The Root Locus The closed-loop poles are given by the roots of 1 + L(s) . (7) Assuming that L(s) = ρL′ (s) the Root Locus plot the locus of all possible roots of (7) as ρ varies in the range [0 , ∞). The Root Locus can be used to study the effect of additional poles and zeros in L′ (s), i.e. in the controller K (s). The Root Locus can be effectively used to design SISO controllers. Matlab commands rlocus(sys) - plots the root locus for the loop transfer function specified by sys. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 54 / 62 Feedback Control Systems Root locus Root Locus - Example L′ (s) = G. De Tommasi (CREATE) 1+s 1 1 · 3 = . s s + 3s2 + 3s + 1 s(s + 1)2 Systems & Control Fundamentals GOTiT - Nov. 2009 55 / 62 Feedback Control Systems Root locus Root Locus - Example Consider the unstable loop transfer function L′ (s) = 1 (s − 2)2 It is not possible to stabilize the system with a simple proportional controller. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 56 / 62 Feedback Control Systems Root locus Root Locus - Example It is not possible to stabilize the system with a simple proportional controller. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 57 / 62 Feedback Control Systems Root locus Root Locus - Example Add a pole in 0 to have zero steady-state error L′ (s) = 1 s(s − 2)2 It is still not possible to stabilize the system with a simple proportional controller. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 58 / 62 Feedback Control Systems Root locus Root Locus - Example Add two zeros to draw the poles in the LHP L′ (s) = The controller K (s) = ρ (s+10)2 s G. De Tommasi (CREATE) (s + 10)2 s(s − 2)2 can stabilize the plant but is not causal. Systems & Control Fundamentals GOTiT - Nov. 2009 59 / 62 Feedback Control Systems Root locus Root Locus - Example Add an high frequency pole to have a proper controller L′ (s) = (s + 10)2 s(s + 100)(s − 2)2 (s+10)2 The controller K (s) = ρ s(s+100) can stabilize. G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 60 / 62 Feedback Control Systems Controller design Hands on system Let’s try to design our own controller (with Matlab)! G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 61 / 62 Appendix Suggested textbooks F. M. Callier and C. A. Desoer Linear System Theory Springer-Verlag, 1991 G. F. Franklin, J. D. Powell and A. Emami-Naeini Feedback Control of Dynamic Systems Pearson Prentice Hall, 2008 S. Skogestad and I. Postlethwaite Multivariable Feedback Control - Analysis and Design John Wiley and Sons, 2006 K. Zhou and J. C. Doyle Essentials of Robust Control Prentice Hall, 1998 G. De Tommasi (CREATE) Systems & Control Fundamentals GOTiT - Nov. 2009 62 / 62