# Control theory Summary

## Full text

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### B. Hut 2nd November 2010

Note: This course uses the book ‘Modern Control Systems’ (eleventh edition) of Richard C. Dorf and Robert H. Bischop. Pagenumbers or equationnumbers do refer to that book. The course also uses slides of lectures of prof.

dr. ir. J.M.A. Scherpen. Lecture- and slidenumbers refer to her presentations, where the first presentation is on 8th of September 2010. If you have any comments on these notes, please contact me: boudewijn@astro.rug.nl

• The Lagrangian L(q, ˙q) is the total kinetic energy T (q, ˙q) minus the total potential energy

V (q). 3rd lecture; 23th slide

L(q, ˙q) = T (q, ˙q) − V (q)

Where q is the gegeneralized position. To calculate the energies of elements of a system, see p. 44, table 2.2.

• The number of energy storage elements equals the number of (minimal) states.3rd lecture; 36th slide

• The Euler-Lagrange equations are given by 3rd lecture; 23th slide

d dt

 ∂L

∂ ˙qi



− ∂L

∂qi

= Fid+ Fie i = 1, 2, ..., n

Where Fid, Fieare resp. the dissipative forces and external forces.

• The Rayleigh dissipation function D( ˙q) is given by 3rd lecture; 34th slide

Fid = −∂D( ˙q)

∂ ˙qi , i = 1, 2, ..., n

Where D( ˙q) is the Rayleigh dissipation function or dissipated energy.

• The state space model, based on the Euler-Lagrange equations if of the form x

 x = Ax + Bu˙ y = Cx + Du

Where u is the systems ‘entrance’ and y the systems ‘output’. And in this course, A ∈ Rm×n, B ∈ Rm×1, C ∈ R1×n, D ∈ R.

• Using this linear form, the transfer function H(s) is given by 5th lecture; 25th slide

H(s) = Y (s)

U (s) = C(sIn− A)−1B + D Where Y (s), U (s) are the Laplace transform of resp. y(t), u(t).

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• To calculate the frequency respons, one can use s = jw x

Where j ≡√

−1.

• The A matrix can be used to determine if a system is stable. x

The system is stable, if the real part of the eigenvalues λi of matrix A are smaller or equal to zero:

Re(λi) ≤ 0 ∀ i = 1, 2, ...

Note: the eigenvalues of A = the poles of the system.

• To get natural fequency ωn and damping ζ out of the transfer function, use x

H(s) = A

s2+ 2ζωns + ω2n

• The kinds of damping are x

ζ > 1 Overdamped system ζ = 1 Critically damped system 0 ≥ ζ ≤ 1 Underdamped system

ζ < 0 Unstable system

• Poles and roots of transfer function. x

A root can cancel a pole. Placement of poles determine stability (characteristics) of system:

Re(s) < 0 stable -, Re(s) = 0 asymptotically stable -, Re(s) > 0 unstable system.

Recall: Poles of system = eigenvalues of A matrix.

Poles and roots can determine order of system (= number of states to describe transfer function)

• In an equilibrium point, the derivative of x is zero: x

˙ x = 0

• Around an equilibrium point, the system van be linearized: 5th lecture; 9th - 12th slide

A =

" ∂f

1

∂x1

∂f1

∂x2

∂f2

∂x1

∂f2

∂x2

# x,¯u)

, B =

 ∂f1

∂f∂u2

∂u

 x,¯u)

, C =  ∂h

∂x1

∂h

∂x2



x,¯u), D = ∂h

∂u x,¯u)

With 

˙

x = f (x, u)

y = h(x, u) and

 x = A˜˙˜ x + B ˜u

˜

y = C ˜x + D ˜u Where ˜x, ˜u, ˜y are of the new linearized system.

Also possible: Taylor series expansion around ¯x, ¯u, one gets

˙

x = x + ∆ ˙˙¯ x = f (x, u) = f (¯x, ¯u) + ∂f∂x

x,¯u)∆x + ∂f∂u

x,¯u)∆u +O(x, u)

˙

y = y + ∆ ˙˙¯ y = h(x, u) = h(¯x, ¯u) + ∂h∂x

x,¯u)∆x + ∂h∂u

x,¯u)∆u +O(x, u) Because the operating point(¯x, ¯u) is chosen to be an equilibrium, the higher order terms O are neglitible around that point. Also, f (¯x, ¯u) = 0. Therefore,

 ∆ ˙x = A∆x + B∆u, ∆x(t0) = ∆x0

∆y = C∆x + D∆u

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• Some transfer functions. x

Open loop transfer function L(s) = D(s)G(s)

Sensitivity function S(s) = 1+D(s)G(s)1 = 1+L(s)1 Complementary sensitivity function T (s) = 1+D(s)G(s)D(s)G(s) = 1+L(s)L(s)

Error function E(s) = S(s)R(s)

Closed loop transfer function 1 + L(s)

*This error function E(s) is without disturbance W (s) and noise V (s). Otherwise, see 6/49

• System type 0. x

s→∞lim L(s) = Kp6= 0 and Kp< ∞

For step input: ess= lim

s→∞sE(s) = lim

s→∞

1 1 + L(s)

1 s = lim

s→∞S(s) = 1 1 + Kp

For faster references: ess= lim

s→∞sE(s) → ∞

• System type 1. x

s→∞lim sL(s) = Kv6= 0 and Kv< ∞ For step input: ess= lim

s→∞sE(s) = 0 For ramp input: ess = lim

s→∞sE(s) = lim

s→∞

1 1 + L(s)

1 s2 = lim

s→∞S(s)1 s = 1

Kv

For faster references: ess= lim

s→∞sE(s) → ∞

• System type 2. x

s→∞lim s2L(s) = Ka6= 0 and Ka< ∞ For step input: ess= lim

s→∞sE(s) = 0 For ramp input: ess= lim

s→∞sE(s) = 0 For parabola input: ess= lim

s→∞sE(s) = lim

s→∞

1 1 + L(s)

1 s3 = lim

s→∞S(s)1 s2 = 1

Ka For faster references: ess= lim

s→∞sE(s) → ∞

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• . x

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• Summary of the course. 14th lecture; 45th-56th slide

1. Modeling based on energy storage

First step in engineering practice: understanding the system.

– Mathematical models are a tool, and the Eurler Lagrange framework offers a mod- eling framework for dynamical systems based on energy and power storage.

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– In general: if physics are known, modeling based on energy-/mass-/force-,.. bal- ances (physical laws) is useful.

– For model based control, model is requisted.

– Time domain model basis for control.

– If model is nonlinear, and interest is in small neighborhood of equilibrium point:

linearization.

2. Linearization, LTI system dscriptions, Transfer function

3. Time domain, Laplace domain, Frequency domain

4. Specifications for 2nd order systems (designs), both in time and Laplace/frequency domain

5. Stability

Equilibrium points can be instable, marginally stable or asymptotically stable.

Characteristic equation yields λi = si, i = 1, 2, ...n:

– Poles of transfer function – Eigenvalues of A matrix – Unstable: Reλi > 0

– Marginally stable: Reλi = 0 – Asymptotically stable: Reλi < 0 6. Open loop and closed loop control

7. Pole placement, PID control, Lead control, Lag control 8. Control design

Goal: yss= r, or ess= c (c ∈ R).

Hence, asymptotically stability of error signal is required.

Y (s) = T (s)R(s) + G(s)S(s)W (s) − T (s)V (s) U (s) = D(s)S(s)R(s) + S(s)W (s) − D(s)S(s)V (s) E(s) = S(s)R(s) − G(s)S(s)W (s) + T (s)V (s)

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Sensitivity function S(s) = 1

1 + D(s)G(s) = 1 1 + L(s) Complementary Sens. Func. T (s) = D(s)G(s)

1 + D(s)G(s)= L(s) 1 + L(s) Loop transfer function L(s) = D(s)G(s)

Characteristic equation always defined by 1 + D(s)G(s) = 1 + L(s) = 0. Thus, 1 + L(s) = 0 ⇒ L(s) = −1

Thus, the zeros of 1 + L(s) are poles of closed loop system. For ess= 0 (yss= r), type of system important, i.e., response closed loop system on certain reference signals.

Furthermore, we need poles with real part < 0 (= Left Half Plane (LHP)). Possible method to check for location poles depending on controller gain K: Root locus.

So far, focus on pole placement in Laplace domain. How about specs?

Many systems accurately described by 2nd order systems: specifications in terms of damping ζ, natural frequency ωn, settling time ts, peak time tp, rise time tr, overshoot Mp. There are some relations between different specifications.

Important for design of open loop system, but also for designing controller such that closed loop system fulfills specifications.

In frequency domain (s = α + jω): Loop shaping.

Again, stability of 1 + L(s) in Niquist plot, encirclements of (-1, 0) by L(jω).

For 2nd order systems via Bode plot. There is a direct relation with the Nyquist plot. The bode plot gives help in specification via Gain Margin GM and Phase Margin P M (robustness), cross-over frequency ωc, bandwidth frequency ωBW, resocance peak, resonance overshoot and relations to specifications in time domain.

Robustness is important for control design: Adding P M and/or GM and/or changing ωc: Lead or Lag control design.

Lead- and lad designs are variations on PD control to prevent amplification at higher frequencies.

If only low frequencies are important, then PID is ok:

With PD: Pole placement arbitrary,

With I: Robustness, no steady state error ess for higher type inputs, at cost of more oscillations.

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• . x

• Design / Engineering design. p. 16

Design is the process of conceiving or inventing the forms, parts, and details of a system to achieve a specified purpose.

• Trade-off. p. 16 or 2nd lecture; 11th slide

There is always a trade of between simplicity of a model and the approximation of reality by the model.

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• White box. 2nd lecture; 13th slide

Model is based on underlying physics, laws of nature and known parameters.

• Black box. 2nd lecture; 13th slide

Model is based on data, no information on internal structure and relations identification.

• Grey box. 2nd lecture; 13th slide

Model is partly white, partly black. So model is based on partly known, partly based on data.

• Control system design process. p. 17-18

The design of control system is a specific example of engineering design. The goal of control engineering design is to obtain the configuration, specifications, and identification of the key parameters of a proposed system to meet an actual need.

This is an illustration of the control system design process:

1. Establishment of goals, variables to be controlled, and specifications:

(a) Establish the control goals.

(b) Identify the variables to be controlled.

(c) Write the specifications.

2. System definition and modeling:

(a) Establish the system configurations.

(b) Obtain a model of the process, the actuator, and the sensor.

3. Control system design, simulation, and analysis:

(a) Describe a controller and select key parameters to be adjusted.

(b) Optimize the parameters and analyze the performance.

• System. 2nd lecture; 21th slide

A system is an object (or set of objects) from which we want to study the properties.

A system relates signals that exist in a part of the environment that we want to study.

A model of a system is a tool we use to anwer questions about the system without having to do an experiment.

There are different types of mathematical models.

⇒ Deterministic models. 2nd lecture; 25th slide

Exact realationship between measurable and derived variables; no uncertainty.

⇒ Stochastic models. 2nd lecture; 25th slide

Contains quantities described using stochastig(/uncertain) variables or processes.

⇒ Dynamic models. 2nd lecture; 26th slide

Variables changing with time, differential or deference equations.

⇒ Static models. 2nd lecture; 26th slide

⇒ Continuous time models. 2nd lecture; 26th slide

t ∈ R; relations with help of differential equations.

⇒ Descrete time models. 2nd lecture; 26th slide

k ∈ Z; realations with help of difference equations.

⇒ Lumped (binned) models. 2nd lecture; 27th slide

Oridnary differential equations, finite number of variables, descrete set of variables in space.

⇒ Distributed (non-binned) models. 2nd lecture; 27th slide

Parial differential equations, infinite number of variables, continuous set of variables in space.

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⇒ Change oriented models. 2nd lecture; 27th slide

Difference or differential equations.

⇒ Discrete event models. 2nd lecture; 27th slide

Event driven equations (bij productieprocessen).

⇒ Hybrid models. 2nd lecture; 27th slide

A mix of Change oriented models and discrete event models.

• Domains of models. 2nd lecture; 47th slide

Domain Effort Flow Generalized position Generalized momentum

e f q p

Electric voltage current charge flux

u [V ] i [A] q [C] ϕ [V s]

Translation force velocity displ. mom.

F [N ] v [ms−1] x [m] p [N s]

Rotation torque ang. vel. ang. displ. rot. mom.

Hydraulic pressure vol. flow volume mom. flowtube

p [N m−2] Q [m3s−1] V [m3] Γ [N sm−2] Thermodynamic temp. entropy flow entropy

T [K] fT [W K−1] S [J K−1] -

• Modeling independent of engineering domain. 3rd lecture; 10th slide

That is, because in the practical sense, there is always a mix of different domains.

The main property in common is energy storage.

• State space systems. 3rd lecture; 16th slide

The input-state-output system, nonlinear o. d. e. form:

˙

x(t) = f (x(t), u(t), t) y(t) = h (x(t), u(t), t)

Where x(t) is the position of the system and u(t) is the ‘entrance’ of the system at time t.

y(t) is the output of the system, it depends also on x(t), u(t) and t.

If there is no time-dependence, the equations become: x(t) = f (x(t), u(t)) and y(t) =˙ h (x(t), u(t))

• Hamilton’s principle / Principle of least action. 3rd lecture; 20th slide

Hamilton’s principle is the principle of least action. The principle results in equations of motion via the calculus of variation.

• Newton’s law in general version. 3rd lecture; 21th slide

F = dp

dt = dmv

dt = ma + ˙mv

• Euler-Lagrange equations. 3rd lecture; 23th slide

d dt

 ∂L

∂ ˙qi



−∂L

∂qi = Fi= resulting forces, i = 1, 2, ..., n Where L is the Langrangian:

L(q, ˙q) = T (q, ˙q) − V (q) = Total kinetic energy − Total potential energy

By the way, the total kinetic energy T corresponds with the energy storage in inductive elements, the total potential energy V with the energy storage in capacitive elements.

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And the resulting forces can be written as: Fi= Fid+ Fie, where Fidare the dissipative forces and Fie are the external forces. see: ‘Dissipation and Euler-Lagrange eqs’ and ‘Rayleigh diss. func.’. The advantage of the EL-equations is that you only nead the kinetic- and potential energy to create a model.

• Euler-Lagrange eqs to the possible state space model for linear systems. 3rd l; 26th s

There is an expression for T ( ˙q) and V (q):

T ( ˙q) =1

2q˙TM ˙q, V (q) = 1 2qTQq

Then using L(q, ˙q) = T (q, ˙q) − V (q), and the Euler-Lagrange equations, we get:

M ¨q + Qq = Bu = external forces Where

M =

m1 0 0

0 m2 0

0 0 . ..

 and B =

 Im

0



= Fext

Then, choose

 x1= q x2= ˙q →

 x˙1= x2

˙

x2= M1(Bu − Qx1) So state space model:

˙

x = Ax + Bu, where u = Fext, A =

 0 1

Q

M 0



and B =

 Im

0



! There may be a sign-error.

• Non-linear systems. 3rd lecture; 28th-33th slide

For non-linear systems, the Euler-Lagrange equation becomes M (q)¨q + C(q, ˙q) + k(q) = Bu

C(q, ˙q) are the coriolis and centrifugal forces. For linear systems, C(q, ˙q) = 0

• Dissipation and Euler-Lagrange equations. 3rd lecture; 34th slide

d dt

 ∂L

∂ ˙qi



−∂L

∂qi

= Fie+ Fid= external forces + dissipative forces, i = 1, 2, ..., n

• Rayleigh dissipation function. 3rd lecture; 34th slide

Fid = −∂D

∂ ˙qi( ˙q), i = 1, 2, ..., n

Where D( ˙q) is the Rayleigh dissipation function. (or ‘dissipated energy ’)

• Number of states. 3rd lecture; 36th slide

The number of (minimal) states equals the number of storage elements.

Lagrangian and Hamiltonian frameworks so far require even number of states, that is, be- cause we choose always (in this course) an even number of variables(x1, x2, ..., xn).

• For an example to calculate Euler Lagrange equations for variables of the electric- and of the mechanical domain, see 4th lecture; 14th slide.

• Phonomena that only occur in the presence of nonlinearities. 4th lecture; 22th slide

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– Finite escape time (example: an explosion)

– Multiple isolated equilibria, stable or unstable, different characteristics (example: a pendulum)

– Limit cycles (i.e. an isolated periodic solution)

• Types of nonlinearities. 4th lecture; 28th slide

– Backlash

– Smooth nonlinearities in dynamics For examples, see 4th lecture; 29th-30th slide

• Linearization / Simplification. 5th lecture; 9th - 12th slide

Linearization consists three steps:

1. Determine operationg point(s). Often the points of equilibrium.

2. Approximate functions around the operating point(s).

3. Convert the system to local (linear) coordinates.

Determine an operating point ¯x, ¯u for the nonlinear system

 x = f (x, u)˙

y = h(x, u) and consider a small perturbation

 x = ¯x + ∆x u = ¯u + ∆u With Taylor series expansion around ¯x, ¯u, one gets

˙

x = x + ∆ ˙˙¯ x = f (x, u) = f (¯x, ¯u) + ∂f∂x

x,¯u)∆x + ∂f∂u

x,¯u)∆u +O(x, u)

˙

y = y + ∆ ˙˙¯ y = h(x, u) = h(¯x, ¯u) + ∂h∂x

x,¯u)∆x + ∂h∂u

x,¯u)∆u +O(x, u) Because the operating point is chosen to be an equilibrium, the higher order terms O are neglitible around that point. Also, f (¯x, ¯u) = 0. Therefore,

 ∆ ˙x = A∆x + B∆u, ∆x(t0) = ∆x0

∆y = C∆x + D∆u where

A = ∂f

∂x x,¯u)

B = ∂f

∂u x,¯u)

C = ∂h

∂x x,¯u)

D = ∂f

∂u x,¯u) In this course: A ∈ Rm×n, B ∈ Rm×1, C ∈ R1×n, D ∈ R.

• LTI system block diagram. 5th lecture; 14th slide

A LTI system block diagram can be used to visualize the linear time invariant state space system:

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