Modern Control Systems (MCS) Lecture-35-36 Design of Control Systems in Sate Space Reduced Order State Observer Dr. Imtiaz Hussain email: imtiaz.hussain@faculty.muet.edu.pk URL :http://imtiazhussainkalwar.weebly.com/

Introduction The state observers discussed in previous lecture was designed to estimate all the state variables. In practice however, some of the state variables may be accurately measured. Therefore, such accurately measurable state variables need not be estimated. In that case a reduced order state observer may be designed to estimate only those state variables which are not directly measurable.

Introduction Suppose that the state vector x is an n-vector and the output vector y is an m-vector that can be measured. Since m output variables are linear combinations of the state variables, m state variables need not be estimated. We need to estimate only n-m state variables. Then the reduced-order observer becomes an (n-m)th order observer. Such an (n-m)th order observer is the minimum-order observer.

Introduction Following figure shows the block diagram of a system with a minimum-order observer.

Minimum Order State Observer To present the basic idea of the minimum-order observer we will consider the case where the output is a scalar (that is, m=1). Consider the system where the state vector x can be partitioned into two parts xa (a scalar) and xb [an (n-1)-vector]. Here the state variable xa is equal to the output y and thus can be directly measured, and xb is the unmeasurable portion of the state vector. 𝒙 =𝑨𝒙+𝑩𝑢 𝑦=𝑪𝒙

Minimum Order State Observer Then the partitioned state and output equations become

Minimum Order State Observer Then the partitioned state and output equations become The equation of measured portion of the state is given as Or The terms on the left hand side of above equation can be measure, therefore this equation serves as an output equation.

Minimum Order State Observer Then the partitioned state and output equations become The equation of unmeasurable portion of the state is given as Noting that terms Abaxa and Bbu are known quantities. Above equation describes the dynamics of the unmeasured portion of the state.

Minimum Order State Observer The design procedure can be simplified if we utilize the design technique developed for the full-order state observer. Let us compare the state equation for the full-order observer with that for the minimum-order observer. The state equation for the full-order state observer is The state equation for the minimum order state observer is The output equations for the full order and minimum order observers are

Minimum Order State Observer List of Necessary Substitutions for Writing the Observer Equation for the Minimum-Order State Observer Table–1

Minimum Order State Observer The observer equation for the full-order observer is given by : Then, making the substitutions of Table–1 into above equation, we obtain Error dynamics are given as

Minimum Order State Observer The characteristic equation for minimum order state observer is

Minimum Order State Observer Design methods 1. Using Transformation Matrix

Minimum Order State Observer Design methods 1. Using Ackerman’s Formula

Example-1 Consider a system Assume that the output y can be measured accurately so that state variable x1 (which is equal to y) need not be estimated. Let us design a minimum-order observer. Assume that we choose the desired observer poles to be at

Example-1

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