A state-space model with a state, input, output and transmission matrices:

Its response to a unit-step input:

A state-space model specified using only state, input and output matrices:

The transmission matrix is assumed to be a zero matrix:

A state-space model specified using only state and input matrices:

The states are assumed to be the outputs:

Both models have the same output response:

A discrete-time model with a sampling period of 0.1:

Test its controllability and observability:

A model with 3 states, 2 inputs and 1 output:

Count the number of inputs and outputs:

A model with 3 states, 1 input and 2 outputs:

The model dimensions:

A multiple-input multiple-output (MIMO) model:

The model dimensions:

A symbolic state-space model:

Compute the analytical unit-step response:

Specify the state variables:

They are visible in the AffineStateSpaceModel representation:

If no state variables are specified, they are chosen automatically:

Specify the state, input and output variables:

The NonlinearStateSpaceModel representation:

Use a state-space model to design a pole-placement controller:

The controller design that places the poles at :

The open- and closed-loop poles:

Plot the response of the closed-loop system to a set of initial conditions:

Descriptor models are used to model algebraic equations:

It modeled the equation , where is the input signal and is the output:

By default, the descriptor matrix e is assumed to be the identity matrix:

It is equivalent to the following model with an identity descriptor matrix:

Both models have the same state response:

Use Automatic to create a descriptor system with all the states as outputs:

Its state and output responses are the same:

It may be possible to convert a descriptor model to a standard model:

A model with a singular descriptor matrix:

It is still possible to convert it to a standard system:

It may not be possible to convert a descriptor model to a standard one in all cases:

Use SystemsModelDelay to model a system with delays:

It has a delayed output response:

A system with two input delays:

Its output response:

A discrete-time system with a delay in the state matrix:

Its transfer function model:

A state-space model of a transfer-function model:

The state-space representation is not unique:

Both yield same original transfer-function model:

A state-space model of a discrete-time transfer-function model:

A state-space model of an improper transfer-function model yields a descriptor system:

A state-space model of an affine state-space model discards all nonlinearities:

The original model cannot be fully recovered from the linearized model:

If the affine state-space model is linear, no information is lost:

And the original model can be fully recovered:

A state-space model of a nonlinear state-space model discards all nonlinearities:

The original model cannot be fully recovered from the linearized model:

Their output responses are not the same:

If the nonlinear state-space model is linear, no information is lost:

And the original model can be fully recovered:

A state-space model of a system of ODEs:

If the ODEs have nonlinear terms they are approximated:

The nonlinear representation does not approximate the nonlinear terms:

A state-space model of a set of difference equations:

A state-space model of a set of differential-algebraic equations:

Prevent elimination of the algebraic equation:

A state-space model of a set of difference-algebraic equations:

Prevent elimination of the algebraic equation:

A state-space model of a delay-differential equation:

Convert a standard state-space model to a descriptor model:

It may sometimes be possible to convert a descriptor state-space model to a standard one:

Obtain the state-space representation of a difference equation as a descriptor model:

By default, a continuous-time model is constructed:

Explicitly construct a continuous-time model:

A discrete-time model with sampling period τ:

Assign a numerical value to τ:

Its output response:

By default, the controllable companion realization is computed:

Explicitly compute the controllable companion realization:

The observable companion realization:

It is the dual of the controllable companion realization:

The controllable and controllable companion realizations for a MIMO transfer-function model:

The observable and observable companion realizations for a MIMO transfer-function model:

They have a dual relationship with the controllable and controllable companion realizations:

Label the inputs, outputs and states:

Label only the inputs and outputs:

Label only the inputs:

Label only the outputs:

Compute a state-space model of a mass-spring damper system using Newton's second law:

Assemble the equation of the system using :

It is linear and can be put into a linear state-space form without any approximations:

The response of the model to a unit-step input:

Compute a state-space model of an inverted pendulum using the Lagrangian:

The position of :

Its velocity:

The kinetic energy of the cart and pendulum:

The potential energy of the pendulum:

The Lagrangian:

The generalized forces:

The equations of motion:

A state-space model:

The nonpositive eigenvalues make it an unstable system:

Compute the state-space model of a vibration absorber using the Lagrangian and the Rayleigh dissipation function:

The kinetic energy of the system:

The potential energy:

The Lagrangian:

The Rayleigh dissipation function:

The system's equations of motion:

Its equilibrium position:

A state-space model of the system:

A numerical model:

Its response to an oscillatory disturbance:

Compute the state-space model of a multidomain system consisting of a motor with a load on a flexible shaft:

The kinetic energy:

The potential energy:

The Lagrangian:

The generalized torques and :

The equations of motion:

A state-space model:

The state response of a numerical model to an input torque from the motor:

Compute an approximate discrete-time state-space model of a ball and beam system starting from continuous-time equations:

The kinetic energy of the rolling ball, assuming no slipping:

Its potential energy, assuming small tilt-angles of the beam:

Its Lagrangian:

Its equation of motion:

The Kirchhoff voltage equation of the tilt-motor, assuming no armature inductance:

The tilt-motor's equation of motion:

The ball and beam's equations:

Construct a numerical state-space model of the ball and beam:

Discretize the model with a sampling period of 50 :

The displacement of the ball due to a nudge while keeping the beam level:

The displacement computed using the discrete-time model:

In both models, the ball will stop rolling due to friction:

An input voltage that tilts the beam back and forth once:

The displacement of the ball due to the tilts:

The displacement computed using the discrete-time model:

In both models, the ball ends up balanced on a level beam:

Compute state-space models of a pendulum about various equilibrium positions and compare them:

The sinusoidal term makes the model nonlinear:

The equilibrium positions of the pendulum are vertically downward and upward:

A state-space model of the pendulum linearized around a generic equilibrium value :

The state-space models around the two equilibrium positions for a set of parameter values:

The response of the model linearized around 0 is stable, while the one around 180° is unstable:

This is because the two equilibrium points have stable and unstable eigenvalues:

Compute the state-space model of a Wilberforce pendulum that has coupled dynamics and compare the efficacy of different inputs on the pendulum:

A model of the system:

A set of numerical values for the parameters:

Construct a state-space model of the system:

Its output response reveals the coupled dynamics between and :

The system is independently controllable with either the force or torque :

A system specification where is the sole feedback input:

And one where the torque is the sole feedback input:

A pole placement controller to damp the oscillations for each system:

Obtain the closed-loop systems:

The longitudinal oscillations are damped more effectively by the torque :

The rotational oscillations are damped more effectively by the force :

Obtain the feedback gains of each model:

It takes less effort to dampen the oscillations using :

This can also be seen by quantifying the control effort:

Compute the state-space model of a lathe's cutting process. The delays in the model are necessary to capture the chattering behavior of the system:

A model of the lathe's cutting process:

A state-space model:

The chattering can be seen in the delay model:

Obtain a delay-free approximation:

But the delay-free model does not capture the same chattering:

State-space models allow for state-feedback control techniques like pole placement. Place the poles of the inverted pendulum on the left-hand side of the -plane to balance the pendulum: »

A controller that balances the pendulum using a set of closed-loop poles:

The closed-loop system balances the pendulum when the cart is disturbed:

Verify the closed-loop poles:

The control effort:

State-space models are the basis for computing optimal state feedback gains that minimize a cost function. Dampen the oscillations of a flexible shaft using optimal control: »

Set the input current as the sole feedback input:

A set of state and control weight matrices and :

Compute the optimal controller that minimizes the cost function :

The oscillations are damped by the controller:

The optimal feedback gains:

The control effort:

State-space models are also used to solve tracking problems. Design a controller that tracks the position of a ball on a beam: »

Set the system specification to track the ball's position:

A set of state and control weight matrices and :

Compute the tracking controller:

Obtain the closed-loop system:

The closed-loop system places the ball in the middle of the beam:

The controller model:

The control effort:

State-space model are useful in modeling aerospace systems. Construct a state-space model of a satellite's attitude dynamics starting from Euler's equations of motion:

Euler's equations with principal moments of inertia , , :

An operating point:

The operating point is an equilibrium point:

Construct a state-space model:

The satellite's attitude is unregulated if disturbed:

Verify the controllability of the model:

State-space models are used in model analysis. Construct a state-space model of a Harrier VTOL jet and assess its controllability:

The kinetic energy:

The potential energy:

The Lagrangian:

The generalized forces:

The equations of motion:

Its equilibrium position:

A state-space model:

A numerical model:

Both inputs are needed for the aircraft to be fully controllable:

State-space models are useful for the analysis of MIMO systems where multiple inputs affect a given output. Use the state-space model representation to compare the effectiveness of the aileron and the rudder on the yaw dynamics of a Boeing 747 using state feedback control:

A state-space model of the aircraft's lateral motion from a set of state, input and output matrices:

Obtain a state-space model with the rudder and another with aileron as the sole input:

The closed-loop systems of both systems with an LQR controller:

Using the rudder results in a faster yaw response:

As well as a smaller roll angle:

State-space models are useful for the design of discrete-time state feedback controllers. Obtain a state-space model of the 747's longitudinal dynamics and improve its handling qualities using discrete-time state-feedback control:

A nonlinear model of the aircraft's longitudinal dynamics:

A linearized state-space model:

Its response to an initial perturbation in the pitch angle is sluggish and oscillatory:

This is because the eigenvalues of the linear system are close to the imaginary axis:

A set control weights and sampling period:

Compute the optimal controller that minimizes an approximated discrete-time cost function:

The discrete-time closed-loop system:

The handling qualities are improved:

The control effort:

Discrete-time controllers can also be designed by approximating the continuous-time models. Design a discrete-time controller to stabilize a Harrier VTOL jet by approximating its continuous-time state-space model: »

A state-space model of the Harrier's horizontal dynamics:

Discretize the model with a sampling period of :

Without a controller, the jet's horizontal position is unregulated if its pitch is disturbed:

Design a discrete-time state feedback controller to stabilize the jet:

The discrete-time controller model:

The closed-loop system:

The jet is stabilized with respect to an initial disturbance by the controller:

State-space models can also be used for the design of output feedback controllers such as the estimator regulator. Design an estimator regulator that tracks the angular rate required for a satellite to maintain its nadir-pointing orientation using a discrete-time model: »

A model of the satellite:

Discretize the model with a sampling period of 0.3:

The radius and gravitational constant mass product of the Earth:

The semimajor axis of the satellite's circular orbit for an altitude of 470 km:

The orbital period is around 94 minutes:

The angular rate is the number of degrees over the orbital period:

Set the system specification to track the satellite's angular rate in the direction:

Compute a set of estimator gains:

And a state-feedback controller:

Assemble the estimator regulator:

The closed-loop system:

The satellite now tracks the required angular rate to keep its nadir-pointing orientation:

The controller model:

The control effort: