Lecture 1: Introduction to Dynamical Systems

Lecture 1: Introduction to Dynamical Systems

• General introduction

• Modeling by differential equations (inputs, states, outputs)

• Simulation of dynamical systems

• Equilibria and linearization

• System interconnections and block diagrams Related Reading

Dynamical Systems

2/45 In a dynamical system the effects of action do not occur immediately.

This phenomenon can be observed in all systems surrounding us:

• Mechatronic systems

Pushing gas pedal in car does only gradually increase the velocity

• Heating systems

Temperature does not rise immediately when a heater is switched on

• Biological systems

A headache disappears slowly when medicine is taken

• Business systems: An investment leads to future gains or losses Variables in a dynamical systems evolve in time. The time evolution depends on theexternal excitation, which itself changes over time.

Mathematical Models

One way to analyze the behavior of a dynamical system is by means of a mathematical model. Such models are often described by (ordinary or partial) differential equations.

Example: Mass-Spring-Damper System

Newton’s second law leads to

mq¨+c( ˙q) +kq= 0.

• q denotes the position of the mass (in a chosen coordinate system) and varies with time. q˙ and q¨are the velocity and the acceleration.

• kq is the spring restoring force (assumed to satisfy Hooke’s law) and

Inputs

4/45 The previous system is said to beautonomoussince it is not exposed to external influences. Non-autonomous systems do have external inputs. Example: Mass-Spring-Damper System

With an external force u(.) acting on the mass, we obtain

mq¨+c( ˙q) +kq=u(t).

The external force u(.) typically varies with time. Depending on the circumstances it can be interpreted as follows:

• If we are allowed to manipulateu(.)then it is called acontrol input.

• If u(.) is generated by nature and cannot be influenced/changed by us then it is called a disturbance input.

Outputs

Often not all variables that appear in a model are of interest. We choose outputs in order to describe those quantities that get focus.

Example: Mass-Spring-Damper System

If we are only interested in the po-sition of the mass, the output y is

mq¨+c( ˙q) +kq=u, y=q.

For some control input y(.) and along a system trajectory, the output

y(.) will be a function of time. Interpretations of outputs:

• y is a variable that can be measured (through sensors).

• y indicate a variable which we would like to monitor in order to investigate/analyze the properties of the system (in simulation).

Interpretation of the Model

6/45 Letu(t) be aninput signal defined for t_≥0and let an initial position

q0as well as an initial velocityv0 be given. Solve the differential equation mq¨(t) +c( ˙q(t)) +kq(t) =u(t) with q(0) =q0, q˙(0) = v0.

If u(.) is continuous and c(_·) is C1, we know that this initial value problem has a uniquesolutionq(t)andq˙(t)that is defined fort_∈[0, t1)

with some t1 >0.

Warning and Reminder: Solutions might not exist for all t_≥0. Since the values forq(0) andq˙(0)specify the solution of the differential equation (for a fixed input) uniquely, we call

x= x1 x2 = q ˙ q and x(t) = x1(t) x2(t) = q(t) ˙ q(t) thestate-vectorandstate-responseof the system. The system’s out-put (response) is then just given by y(t) = q(t).

Demo: Free and Driven Response

q(t) (full) and v(t) (dotted)

−5 0 5 −2 −1 0 1 2 Graph of function c(.) −5 0 5 −4 −2 0 2 4

Demo: Effect of Nonlinearity

q (blue) and v (green)

−5 0 5 −2 −1 0 1 2 Graph of function c(.) −20 −10 0 10 20 −20 −10 0 10 20

Demo: Finite Escape Time

q(t) (full) and v(t) (dotted)

−5 0 5 −25 −20 −15 −10 −5 0 Graph of function c(.) −5 0 5 10 15 −5 0 5 10

Demo: Stability and Instability

q(t) (full) and v(t) (dotted)

−5 0 5 −2 −1 0 1 2 Graph of function c(.) −10 −5 0 5 10 −10 −5 0 5

Demo: Stable Limit Cycle

q(t) (full) and v(t) (dotted)

−5 0 5 −10 −5 0 5 10 Graph of function c(.) −10 −5 0 5 10 −10 −5 0 5 10

Interpretation of the Model

12/45 If we write down a system like

mq¨+c( ˙q) +kq=u, y=q,

we are actually interested in its behavior, the set of all input-, state-and output-signalsu(.), (q(.),q˙(.)) and y(.) that satisfy these laws.

• Signals are vector-valued functions of time, also called trajectories. Tacitly assumed to be piece-wise continuous.

• The laws to-be-satisfied are often described by differential equations. Impose hypothesis that guarantee existence/uniqueness of i.v.p.

• Different system descriptions (laws) can lead to the same behavior. We then call the system (descriptions) equivalent.

In the so-called behavioral approach to dynamical systems all this is developed in a mathematically precise fashion.

Sample Questions in Control

mq¨+c( ˙q) +kq =u, y =q

we are e.g. interested in the following issues:

• If there is no external excitation how does the state/output behave?

• Can we steer the system from one position to another?

• Is it possible to determine the input from knowledge of the output?

• Can we modify the system in order to create a desired behavior? Some of these (yet rough) questions are related toanalyzingthe system for its properties, while others involve synthesizing control inputs or modify the system laws in order to change the system’s behavior.

From Second-Order to First-Order Models

14/45 The system description

mq¨+c( ˙q) +kq=u

involves the first and second derivative of q. If m ₆= 0 it is called a second order differential equation.

If we introduce the state-variables x1 =q and x2 = ˙q we conclude

˙ x1 = ˙q=x2, ˙ x2 = ¨q=− k mq− c( ˙q) m + 1 mu=− k mx1− 1 mc(x2) + 1 mu.

This can be written more compactly as ˙ x1 ˙ x2 = x2 −k mx1− 1 mc(x2) + 1 mu =f(x1, x2, u).

From High-Order to First-Order Models

Supposeris a general real-valued nonlinear function ofn+1arguments. Lemma 1 The system

q(n)+r(q(n−1), q(n−2), . . . ,q, q, u˙ ) = 0 is equivalent to ˙ x=       ˙ x1 ˙ x2 .. . ˙ xn−1 ˙ xn      =       x2 x3 .. . xn −r(xn, xn−1, . . . , x2, x1, u)      =:f(x, u). Proof. Introduce x1 = q, x2 = ˙q, . . . , xn−1 = q(n−2), xn = q(n−1), substitute the variables and write down the resulting vector equation.

Non-Uniqueness

16/45 It is important to note that there is no unique way to do this. Under-standing this issue will be an essential topic later.

Lemma 2 The system

q(n)+r(q(n−1), q(n−2), . . . ,q, q, u˙ ) = 0 is equivalent to ˙ z =       ˙ z1 ˙ z2 .. . ˙ zn−1 ˙ zn      =       −r(z1, z2, . . . , zn−1, zn, u) z1 z2 .. . zn−1      = ˆf(z, u).

Proof. Introduce z1 = q(n−1), z2 =q(n−2), . . . , zn−1 = ˙q, zn =q and proceed as before.

Generic System Description in Control

Physical modeling often leads to a system of higher-order differential equations. As seen above, these can typically (though not always!) be equivalently written as a first-order vector differential equation

˙

x=f(x, u) and y =h(x, u).

with maps f :X_×U _→Rn _{andh:X×U →}

Rk(X ⊂Rn,U ⊂Rm).

This compact system description forms the starting point of control. More explicitly these equations read as

˙ x1 = f1(x1, . . . , xn, u1, . . . , um) .. . ˙ xn = fn(x1, . . . , xn, u1, . . . , um) y1 = h1(x1, . . . , xn, u1, . . . , um) .. . yk = hk(x1, . . . , xn, u1, . . . , um) Note that these functions could as well depend explicitly on time!

System Response and Differential Equations

18/45 For f :X_×U _→Rn, h:X _×U _→Rk with open X _{⊂ R}n, U _{⊂ R}m

consider

˙

x=f(x, u) and y =h(x, u).

Ifu(.) :I _→Uis an input function on an open intervalI _⊂Rcontaining 0 and if ξ _{∈ R}n _{is an initial condition for the state, then the} state-response x(.) is obtained by solving the initial value problem

˙

x=f(x, u(t)) with initial condition x(0) =ξ.

The output response is just defined by y(t) :=h(x(t), u(t)) fort _∈I.

• Existence and uniqueness of the state-response on some open sub-interval ofI is guaranteed iff is continuous and Lipschitz continuous in x and ifu(.) is continuous.

Recap: Global Existence and Uniqueness Theorem

˙

x=g(t, x), x(τ) = ξ

forg :D_→Rn with D_⊂R×Rn.

Assumptions. Suppose D _{⊂ R×R}n _{is open and g : D →}

Rn,

(t, x)_→g(t, x) is continuous and Lipschitz-continuous in x:

For each (τ, ξ) _∈ D there exists a neighborhood U _⊂ D of (τ, ξ) and some L >0 with

kg(t, x)₋g(t, y)_{k ≤}L_kx₋y_k for all (t, x),(t, y)_∈U.

• Is satisfied if bothg and ∂xg are continuous onD.

• It suffices that g _∈C1_(D, Rn).

Recap: Global Existence and Uniqueness Theorem

20/45 Theorem 3 Under the above assumptions and for every(τ, ξ)_∈D

there exists a maximal interval of existence I(τ,ξ) = (t−, t+)3τ and

a unique C1_{-solution x(., τ, ξ) :I}

(τ,ξ) →Rn of the ivp

˙

x=g(t, x), x(τ) = ξ. (1)

Maximality ofI(τ,ξ)means: If the initial value problem has aC1-solution xJ : J → Rn on the interval J 3 τ, then J ⊂ I(τ,ξ) and xJ is the restriction of x ontoJ.

Moreover, for t+ <∞ exactly one of the following properties holds:

• The solution “escapes”: limt→t+, t<t+kx(t)k=∞.

• There solution “approaches the boundary ofD”: For all sequences

tν ∈ (t−, t+) with tν → t+ and limν→∞x(tν) = x+, the point

Generic Description of Linear Systems

f(x, u) = Ax+Bu and h(x, u) =Cx+Du

with matrices A_∈Rn×n, B _∈Rn×m,C _∈Rk×n, D_∈Rk×m. Hence a generic linear (time-invariant) system is described as

˙

x=Ax+Bu and y =Cx+Du.

This is the system description mostly considered in this course.

The behavior of many dynamical systems in engineering can often be (approximately) modeled by linear systems. On the other hand, physical modeling often results in descriptions with nonlinear dynamics.

Simulation

22/45 A system description in terms of the differential and output equation

˙

x(t) =f(x(t), u(t)), x(0) =ξ, y(t) = h(x(t), u(t))

allows the numerical computation of the response (to an input and an initial condition) by ode-solvers (such as ode45 or ode15s in Matlab). By integrating over time, the description is equivalent to

x(t) = ξ+ Z t

0

f(x(τ), u(τ))dτ, y(t) =h(x(t), u(t)).

This leads to the general diagram for a Simulink simulation:

Signal Generator Scope f(x,u) Right-hand side of differential equation h(x,u) Output function 1 s Integrator with initial condition x0 u x

Example: Segway (Cart-Pendulum)

Modeling with the Lagrange technique ([F] p.30-31) leads to (M+m)¨p₋mlcos(θ)¨θ+cp˙+mlsin(θ) ˙θ2 = F

−mlcos(θ)¨p+ml2θ¨+γθ˙₋mglsin(θ) = 0

where, next to the parameters defined in the figure, c and γ are the coefficients describing the viscous friction of the cart and the pendulum.

Example: First-Order System Description

24/45 More insightful and compact description:

(M+m) ₋mlcos(θ) −mlcos(θ) ml2 ¨ p ¨ θ + cp˙+mlsin(θ) ˙θ2 γθ˙₋mglsin(θ) = F 0 .

Introduce the abbreviations

U(θ) = (M +m) ₋mlcos(θ) −mlcos(θ) ml2 , v(θ,p,˙ θ˙) = cp˙+mlsin(θ) ˙θ2 γθ˙₋mglsin(θ) .

Then the system can be described as

U(θ) ¨ p ¨ θ +v(θ,p,˙ θ˙) = F 0 .

Since the matrix U(θ) is always invertible (why?) this is the same as ¨ p ¨ θ =₋U(θ)−1v(θ,p,˙ θ˙) +U(θ)−1 F 0 = w1(p, θ,p,˙ θ, F˙ ) w2(p, θ,p,˙ θ, F˙ ) with yet another abbreviation of the right-hand side.

Example: First-Order System Description

x1 x2 = p θ , x3 x4 = ˙ p ˙ θ and u=F.

The system is then described as     ˙ x1 ˙ x2 ˙ x3 ˙ x4    =     x3 x4 w1(x1, x2, x3, x4, u) w2(x1, x2, x3, x4, u)    =     f1(x1, x2, x3, x4, u) f2(x1, x2, x3, x4, u) f3(x1, x2, x3, x4, u) f4(x1, x2, x3, x4, u)    .

The given representation of f(x, u) can be coded exactly as described. This is much less error-prone than explicit computations as e.g. given in [AM] p.37, in particular for more complicated situations.

However, let us stress again we just obtained another representation of the very same system as defined by the equations on slide 23.

Equilibria

26/45 Definition 4 All pairs of vectors (xe, ue)∈X×U which satisfy

0 =f(xe, ue)

are called equilibria of the system x˙ =f(x, u).

If the system is driven with the constant control input u(t) = ue and if the state is initialized asxe(0) =xe, then the state-response is given by

x(t) = xe and thus x˙e(t) = 0 for all t≥t0.

If starting at an equilibrium the state-trajectory stays there. Equilibria are in general not unique. Not even for linear systems. Often (but by no means always!) our interest is in trajectories that “do not deviate too much” from equilibria.

Example: Segway

27/45 The equilibria of the system on slide 23 are determined by

0 =F and mglsin(θ) = 0.

Solutions are θe = 0 (upright position) and θe = π (downright position), while pe is arbitrary.

Four simulations, including perturbation of initial condition (dash-dotted):

0 1 2 3 4 5 −1 0 1 p 0 1 2 3 4 5 0 5 θ

A controller should keep the Segway in the upright position which while the car follows a given position trajectory. Hence the system trajectory

Linearization

28/45 Iff andhareC1, their first-order Taylor expansions around(xe, ue)are

f(x, u) _≈ f(xe, ue) +∂xf(xe, ue) | {z } A (x₋xe) +∂uf(xe, ue) | {z } B (u₋ue) h(x, u) _≈ h(xe, ue) | {z } ye +∂xh(xe, ue) | {z } C (x₋xe) +∂uh(xe, ue) | {z } D (u₋ue). The approximation is good if x_≈xe and u≈ue.

Definition 5 Suppose that f(xe, ue) = 0. If f and h are C1, the linearization of x˙ =f(x, u), y=h(x, u) at (xe, ue)is

˙

x∆=Ax∆+Bu∆, y∆=Cx∆+Du∆.

Letu(t)_≈ue and x(0) ≈xe lead to the nonlinear response y(t)≈ye. If one drives the linearization withu∆(t) :=u(t)−ue and for the initial value x∆(0) :=x(0)−xe, we hopethaty∆(t) +ye approximatesy(t).

Example: Segway

Nonlinear (blue) versus linearized (green) simulation at(pe, θe) = (0, π):

0 2 4 6 8 10 0 0.5 1 F 0 2 4 6 8 10 0 0.5 1 p 0 2 4 6 8 10 3 3.2 3.4 θ Good match.

Example: Segway

30/45 Nonlinear (blue) versus linearized (green) simulation at(pe, θe) = (0,0):

0 0.5 1 1.5 0 0.5 1 F 0 0.5 1 1.5 0 2 4 p 0 0.5 1 1.5 0 10 20 θ

System Interconnections and Modularity

One of the most important concepts in modeling dynamical systems is modularity. Complex models are built by interconnecting simple system components in a hierarchical manner.

Dynamical systems are interconnectedby equating signals.

• Allows to use software libraries with standard system components and interfaces between different physical domains.

• Many concrete modeling packages are available (Matlab, Modelica).

• Even for complex models, the hierarchical (nested) structure renders bookkeeping manageable.

• Individual components can be adapted while preserving the overall interconnection structure.

Example: Mass-Spring-Damper System

32/45 The simple mass-spring-damper systemmq¨+c( ˙q)+kq =udoes actually result from Newton’s laws as follows:

F = ˙p with p=mv.

The total force acting on the mass (assumed to be constant) is

F =F1+F2+F3

with the externally exerted force

F1 =u,

the friction force (assumed to depend on velocity)

F2 =c(v),

and the spring restoring force (assumed to depend linearly on elongation)

Example: Pendulum on Cart

Suppose the cart force is generated by an electric DC motor attached to a one wheel of radius r.

If the voltageV is applied to the motor, the winding currentI generates a torque; with the load torque T the motor dynamics is described by

Jφ¨+bφ˙ =kmI−T, LI˙+RI =V −keφ˙ with positive constants J, b, km, L, R, ke.

The load torque and the motor shaft-angle are assumed to be related toF and p by

T =F r and p=rφ. _{0 p}

φ r

Example: Pendulum on Cart

34/45 We obtain the following description of the interconnection:

LI˙+RI +ke r p˙ = V (M +m+ J r2)¨p−mlcos(θ)¨θ+ (c+ b r2) ˙p+mlsin(θ) ˙θ 2 − km r I = 0 −mlcos(θ)¨p+ml2θ¨+γθ˙₋mglsin(θ) = 0.

Note that the coupling is often called two-sided, since the dynamics of the interconnected systems influence each other. If ke = 0 then the coupling is one-sided.

It is not straightforward to predict the impact of the motor dynamics onto the overall dynamic response of the model.

Let us hence only analyze, by simulation, the influence of “large” and “small” values of L.

Example: Pendulum on Cart

0 1 2 3 4 5 0 0.5 1 F 0 1 2 3 4 5 0 1 2 p 0 1 2 3 4 5 3 3.2 3.4 θ

Example: Pendulum on Cart

36/45 Larger L slows the motor dynamics down:

0 1 2 3 4 5 0 0.5 1 F 0 1 2 3 4 5 0 1 2 p 0 1 2 3 4 5 3 3.2 3.4 θ

Series Interconnection

Definition 6 The series interconnection of the systems ˙

x1 =f1(x1, u1), y1 =h1(x1, u1), x˙2 =f2(x2, u2), y2 =h2(x2, u2)

is obtained if the output of the first system enters the second system as an input: u2 =y1. (The vector dimensions must be equal.)

The series interconnection is then described by ˙ x1 ˙ x2 = f1(x1, u1) f2(x2, h1(x1, u1)) , y2 =h2(x2, h1(x1, u1)). Linear systems: ˙ x1 = A1x1 +B1u1, y1 = C1x1+D1u1 ˙ x2 = A2x2+B2u2, y2 = C2x2+D2u2 η= x1 x2 ˙ η = A1 0 B2C1 A2 η+ B1 B2D1 u1, y2 = D2C1 C2 η+D2D1u1

Parallel Interconnection

38/45 Definition 7 The parallel interconnectionof the systems

˙

x1 =f1(x1, u1), y1 =h1(x1, u1), x˙2 =f2(x2, u2), y2 =h2(x2, u2)

is obtained by having them enter a common input and summing the outputs: u1 =u2 =u and y=y1+y2. (Dimensions must fit.)

The parallel interconnectionis then described by ˙ x1 ˙ x2 = f1(x1, u) f2(x2, u) , y =h1(x1, u) +h2(x2, u). Linear systems: ˙ x1 = A1x1 +B1u1, y1 = C1x1+D1u1 ˙ x2 = A2x2+B2u2, y2 = C2x2+D2u2 η= x1 x2 ˙ η= A1 0 0 A2 η+ B1 B2 u, y= C1 C2 η+ (D1+D2)u

Matlab’s Control System Toolbox

Matlab’s Control System Toolbox has so-called ss objects which are used to work with linear state-space systems as follows:

• System definition:sys1=ss(A1,B1,C1,D1), sys2=ss(A2,B2,C2,D2)

• Series interconnection:ser=sys1*sys2

• Parallel interconnection:par=sys1+sys2

• Simulation: y=lsim(sys,u,T,x0)

• Extraction of defining matrices: [A,B,C,D]=ssdata(sys)

Note that the (overloaded) operations for the series and parallel inter-connections remind us about the way how matrices act as mappings. Internally these operations are based on the above derived formulas. You should be actively exploring the possibilities of the toolbox.

Block-Diagrams

40/45 Like in Simulink,block-diagramsas generally used for the visualization of systems interconnections in control.

Generically an individual block is indicated as

Operation to be performed on input

Input Output

Static blocks are maps from _R into _R or from _Rp _into

Rq. Dynamic

Block-Diagrams

Dynamic blocks described by differential equations require the specifi-cation of an initial conditionsuch that the output is uniquely defined by the input. Such an initial condition can be viewed as an extra input. That’s why, for an integrator, one sometimes uses the notation

Z

x0

y u

or, for a general system, the block description

˙ x = f(x, u) y = h(x, u) x0 y u

Block-Diagrams

42/45 A block-diagram is obtained by interconnecting individual blocks (by equating signals). Hence such diagrams (should) always correspond in a precise fashion to algebraic equations.

Generic examples Series interconnection: ˙ x1 = f(x1, u1) y1 = h(x1, u1) u1 _x˙ y2 2 = f(x2, u2) y2 = h(x2, u2) Parallel interconnection: ˙ x1 = f(x1, u) y1 = h(x1, u) y ˙ x2 = f(x2, u) y2 = h(x2, u) + u

Example: Mass-Spring-Damper System

¨ q= 1 mu− 1 mc( ˙q)− k mq, y=q

Start from chain of integrators

˙ q(0) ˙ q ¨ q q(0) q Z Z

and obtain block-diagram with equations of motion:

˙ q(0) ˙ q ¨ q q(0) q 1 mc(.) k m Z Z + 1 m + − u y

Example: State-Space System

44/45 The standard linear state-space system can be depicted as

B - - R - C A D -j 6 - j 6 - -? u(t) x0 x(t) y(t)

However, there is no need for such an explicit implementation in e.g. Simulink, since one can just use either a State-Space block (defined byA, B, C, D) or an LTI Systems block (defined by an ss-object).

Covered in Lecture 1

• Models of dynamical systems

differential equations, inputs, outputs, state, reduction to first order, linear systems, compact notation, Simulink simulation

• Linearization

• System interconnections

modularity in modeling, interconnection=equating signals feedback equations, relation to block diagrams

• Aspects of qualitative analysis of dynamic responses

nonzero initial condition, external input signal, equilibria, stability, limit cycles, finite escape time